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THE NEGATIVE K-THEORY OF NORMAL SURFACES CHARLES WEIBEL Abstract We relate the negative K-theory of a normal surface to...

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THE NEGATIVE K-THEORY OF NORMAL SURFACES CHARLES WEIBEL

Abstract We relate the negative K-theory of a normal surface to a resolution of singularities. The only nonzero K-groups are K−2 , which counts loops in the exceptional fiber, and K−1 , which is related to the divisor class groups of the complete local rings at the singularities. We also verify two conjectures of Srinivas about K0 -regularity and K−1 of a surface. This paper gives a geometric interpretation for the negative K-theory of a normal surface X. If R is the semilocal ring of X at its finitely many singularities, then the negative K-theory of X and R are the same by [W3, Theorem 1.2]. Thus we also describe the negative K-theory of any excellent 2-dimensional normal semilocal ring R. As we explain on the next page, our results also give an almost complete classification of projective R[T ]-modules. Our interpretation relates the groups K−j (X) to a resolution of singularities X˜ → X of the surface. For starters (see Theorem 4.4), we show that K−j (X) = 0 for all j ≥ 3, even if X is singular. (This confirms a guess in [W1, p. 180].) If X is normal and j = 2, we prove that K−2 (X) ∼ = Zλ , where λ denotes the number of “loops” in the exceptional fiber E; the number of “loops” in a curve is made precise in Definition 2.1. We show in Theorem 5.3 that the group K−1 (R) is the quotient of the Picard group of the infinitesimal thickening nE (which is independent of n for large n) by the image ˜ If Cl(R) denotes the divisor class group of R, K−1 (R) is also isomorphic to of Pic(X). h Cl(R )/ Cl(R), where R h is the henselization of R; this isomorphism was conjectured by Srinivas [Sr2, p. 597]. We intend to study the intriguing relationship between K−1 (X) and the Brauer group in a separate paper [BW]. We also prove several results about K-regularity. Our stable result (Theorem 4.4) is that every 2-dimensional excellent noetherian R is “K−2 -regular.” This implies that K−j (R) = K−j (R[t1 , . . . , tn ]) for every j ≥ 2, and all n, and settles the case dim(R) = 2 of a second guess we made in [W1, p. 186], at least for excellent rings. DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 1, Received 4 October 1999. Revision received 15 May 2000. 2000 Mathematics Subject Classification. Primary 19D35; Secondary 14J17, 19E08.

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To explain this implication, recall that a ring R is called Kq -regular if Kq (R) = Kq (R[t1 , . . . , tn ]) for every n. The passage from K−2 -regularity to K−j -regularity is given by Vorst’s theorem [Vor1, Corollary 2.1], which states that if R is Kq -regular then R is also Kq−1 -regular. We also give a criterion for a surface X to be K−1 -regular, and even K0 -regular, involving the vanishing of the cohomology H 1 (E, L ) of the conormal bundle L (see Proposition 5.8). This criterion not only sharpens the K0 -regularity results of K. Coombes and Srinivas [CS], but it also verifies [Sr2, Conjecture B]. The key geometric idea behind our results is that one can resolve the singularities of a surface by first blowing up along a locally complete intersection and then normalizing. This is more subtle than the usual resolution of singularities, in which one alternatively blows up closed points and then normalizes. Our na¨ıve technique does not extend to 3-dimensional normal domains that are not Cohen-Macaulay, since there is no regular sequence of length 3 to blow up. As promised, we now explain why these calculations give the main part of a classification of projective modules over the Laurent polynomial rings R[T ] = R[t1 , t1 , t1−1 , . . . , tn , tn−1 ], or even polynomial rings R[S, T ] = R[T ][s1 , . . . , sm ] over R[T ]. To avoid discussions of extended projective modules, we assume for simplicity that R is a (commutative noetherian) semilocal domain, so that all projective R-modules are free. We are interested in singular rings because if R is regular, then R. Swan proved in [Sw] that every projective R[S, T ]-module is free. Let P be a (finitely generated) projective module of rank r over R[S, T ], where R is a 2-dimensional semilocal ring. The obstruction to P being free is the class of [P ] − r in the Grothendieck group K0 (R[S, T ]). If [P ] = 1, it is well known that P is free (see [Bass, p. 466]). If [P ] = r and r = 2, then P is free by [Sw, Theorem 1.1]. In fact, if r > 2, then the isomorphism classes of rank r projective modules are in one-to-one correspondence with the elements of K0 (R[S, T ])/Z; this cancellation result follows from the Bhatwadekar-Lindel-Rao theorem in [BLR] that P ∼ = P0 ⊕ R[S, T ]r−2 for some rank 2 projective module P0 . If R is essentially of finite type over an algebraically closed field, we expect that this cancellation result extends to rank 2 projective R[T ]-modules, that is, that the isomorphism classes of rank 2 projectives are also in one-to-one correspondence with the elements of K0 (R[S, T ])/Z. For projective R-modules, this is a theorem of M. Murthy and Swan [MS]. S. Bhatwadekar [Bhat] has recently proven that this is also true for R[s1 ], that is, when m = 1 and n = 0. Now Bass’s fundamental theorem in [Bass, p. 669] says that there is a natural decomposition of K0 (R[S, T ]) involving the groups K−j (R) and the nil-groups N i K−j (R) for j = 0, 1, . . . , n and i ≥ 0. Specifically,

THE NEGATIVE K-THEORY OF NORMAL SURFACES

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n K0 (R[S, T ]) = K0 (R) ⊕ K−1 (R)n ⊕ K−2 (R)(2) ⊕ · · · ⊕ K−n (R)

⊕ NK0 (R)2n+m ⊕ · · · ⊕ N i K−j (R)aij ⊕ · · · ⊕ N m K−n (R), where the number aij of copies of the group N i K−j (R) equals the coefficient of N i Lj in the formal expansion of (1 + N)m (1 + 2N + L)n . Our main result (Theorem 4.4) says that most of these terms are unnecessary when dim(R) = 2. This paper is organized as follows. In Section 1 we cover the necessary material on resolving singularities by blowing up a locally complete intersection ideal and by normalizing. In Sections 2–3 we cover the modifications necessary to compute the K-theory of these blow-ups. In Sections 4–5 we prove the main theorems for K−j , and in Section 6 we give several examples to illustrate our techniques. Section 7 is a technical continuation of Section 2, concerning SK1 -regularity of nonreduced curves. Finally, Section 8 is devoted to K0 -regularity and a verification of [Sr3, Conjecture B]. Notation. If X is a scheme, we write X[t] and X[t, t −1 ] for the product of X with Spec(Z[t]) and Spec(Z[t, t −1 ]), respectively. By a surface we mean a 2-dimensional quasi-projective scheme over an excellent (noetherian) base ring; this includes the usual surfaces of algebraic geometry. 1. Reduction ideals In this section we discuss the role that reduction ideals and their Rees algebras play in blow-ups. The following material was developed in discussions with W. Vasconcelos, and we refer the reader to [Vas] for material about reduction ideals. Throughout, R is a commutative noetherian ring. Definition 1.1 An ideal I is a reduction of an ideal J if there is an integer n such that I J n = J n+1 . We give two alternate interpretations of reduction ideals. First, recall that an element x ∈ R is called integral over I if there are ai ∈ I i such that x m + a1 x m−1 + · · · + ai x m−i + · · · + am = 0.

(1.1.1)

lemma 1.2 Assume that R is noetherian, and assume that I ⊂ J . Then I is a reduction of J if and only if J is integral over I in the sense that every x ∈ J is integral over I . Proof Suppose that I is a reduction of J . If x ∈ J , then xJ n ⊂ I J n . Using the trick of Nakayama’s lemma [Mat, Theorem 2.1], it follows that x satisfies equation (1.1.1)

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modulo annR (J n ). Since the left-hand side of (1.1.1) is in J , its n+1st power vanishes, so that x is integral over I . Conversely, suppose that J is integral over I . Then we can find an integer m such that each generator x of J satisfies x m ∈ I J m−1 . Letting n be (m − 1) times the number of generators of J , this implies that I J n = J n+1 . 1.3. Zariski’s criterion Here is another interpretation of reduction ideals. Let R be a domain with quotient field F , and let denote the set of all valuations on F which are nonnegative on R. Then the integral closure of I coincides with the intersection of the submodules Rv I of F (v ∈ ) by [ZS, p. 350]. This establishes Zariski’s criterion: I is a reduction of J just in case Rv I = Rv J for all valuations v ∈

.

lemma 1.4 (Vasconcelos) Let I ⊂ J be ideals in a domain R. Suppose that for each x ∈ I there are natural numbers n and r such that x r J n+1 ⊆ I r+1 J n . Then I is a reduction of J . Proof In order to apply Zariski’s criterion, choose a valuation v ∈ on F . Since I ⊂ J , we have v(I ) ≥ v(J ). Pick x ∈ I such that v(x) = v(I ). Applying v to the inclusion x r J n+1 ⊆ I r J n+1 = I r+1 J n yields the inequality r · v(x) + (n + 1)v(J ) ≥ (r + 1) · v(x) + n · v(J ), so v(J ) ≥ v(I ). Hence v(I ) = v(J ). This proves that v(I ) = v(J ) for every valuation v. By Zariski’s criterion, I is a reduction of J . To prove the next result, we need some basic facts about the blowing up of Spec(R) along I , that is, the scheme Proj(R[I t]). If x ∈ I and T denotes xt, then R[I t][T −1 ] ∼ = R [T , T −1 ], where R = R[I /x]. Thus the affine open subset D+ (xt) of Proj (R[I t]) is isomorphic to Spec(R ). theorem 1.5 Let I ⊂ J be ideals in a noetherian domain R. Then I is a reduction of J if and only if Proj(R[J t]) → Proj(R[I t]) is a finite morphism. Proof If I is a reduction of J , then B is finite over A because it is generated over A by the finitely generated R module J t ⊕ · · · ⊕ J n t n . For each x in I , it follows that the


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B (xt) of Proj(B) is finite over the affine open subset D A (xt) of affine open subset D+ + Proj(A). Since (J t)n+1 ⊂ (I t)B, these form a cover of Proj(B). Hence Proj(B) is finite over Proj(A). Conversely, suppose that Proj(B) is finite over Proj(A). Then for each x in I , B (xt) → D A (xt) is finite; that is, R[J /x] is finite over R[I /x]. If the restriction to D+ + y ∈ J , then y/x must satisfy an integral equation over R[I /x] of the form (y/x)n+1 + an (y/x)n + · · · + a0 , where all the ai belong to I r /x r for some r. It follows that x r y n+1 ∈ I r+1 J n . Since J is finitely generated, it follows that for some n and r we have x r J n+1 ⊆ I r+1 J n . Using Lemma 1.4, we see that I is a reduction of J .

The following result, which we cite from [Mat, Theorems 14.14 and 17.4], shows that reductions generated by systems of parameters are abundant. proposition 1.6 Suppose that (R, m) is a d-dimensional local domain with infinite residue field. If J = (x1 , . . . , xr )R is an m-primary ideal, then for any sufficiently general d × r matrix (rij ) over R, the elements yi = rij xj satisfy the following: (1) the ideal I = (y1 , . . . , yd ) is a reduction of J ; (2) the elements y1 , . . . , yd form a system of parameters of R; (3) if R is Cohen-Macaulay, the yi form a regular sequence on R. Remark 1.6.1 When R/m is finite, the conclusions still hold for some power J e of J , except that the now meaningless term “sufficiently general” must be replaced by some probabilistic statement like “most.” This was proven by D. Northcott and D. Rees [NR, Theorem 3.4]; I am grateful to J. Lipman and Vasconcelos for helping me locate this reference. The Northcott-Rees proof is essentially like the proof in [Mat, Theorem 14.14] for R/m infinite. The key step (see [NR, p. 355] or [Mat, Theorem 14.14, step 2]) is to show that, for infinitely many e, some (or most) d-tuples of elements of the finite vector space J e /mJ e generate a primary ideal of S = ⊕J e /mJ e . But this is guaranteed by graded Noether normalization (see [BH, Theorem 1.5.17]). The conversion from existence to “most” follows as in [Mat, Theorem 14.14, step 3]. Example 1.7 Suppose that (R, m) is a 2-dimensional normal local ring with R/m infinite. Then every m-primary ideal J = (x1 , . . . , xr )R has a reduction ideal of the form I = (y1 , y2 )R, where y1 , y2 forms a regular sequence. The blow-up along I is X = Proj(S), where S = R[Y1 , Y2 ]/(y1 Y2 − y2 Y1 ) is the Rees algebra of I . By Theorem 1.5, the blow-up X = Proj(R[J t]) along J is finite over X .

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1.8. Affine Cone Let C be a curve in P2k defined by a homogeneous polynomial F (x, y, z) of degree d. Let R be the localization of k[x, y, z]/(F ) at M = (x, y, z). Then the following are equivalent: F (0, 0, 1) = 0; the ideal I = (x, y) is a reduction of M; x, y is a regular sequence in R. Here is an application of these ideas. Recall that a zero-dimensional subscheme Y of X is a locally complete intersection if Y has an affine neighborhood U = Spec(R) in which the ideal I of R defining Y is generated by a regular sequence y1 , y2 . theorem 1.9 Let R be a 2-dimensional normal domain that is excellent. Then there is a regular surface X˜ birational over Spec(R), and any such X˜ is obtained as the normalization of a blow-up along a locally complete intersection ideal. Proof Without loss of generality we can localize R at a singular maximal ideal m. It is classical (see [Lip2]) that X = Spec(R) can be desingularized by a sequence of blow-ups at closed points, followed by normalization. By [Hart, Theorem II.7.17] this desingularization X˜ → X is the blow-up along an ideal J ; we can assume that J is m-primary by [Lip2, p. 155]. If the residue field R/m is infinite, we can choose a reduction ideal I for J generated by a regular sequence. Let X be the blow-up along I ; X = Proj(R[I t]). Then X˜ is integral over X , so it is the normalization of X . If the residue field R/m is finite, we can choose a reduction ideal I for some J e generated by a regular sequence. Since Proj(R[J t]) ∼ = Proj(R[J e t]), X˜ is again integral over the blow-up X along I and hence is the normalization of X . We conclude this section by noting that this technique probably does not work for all 3-dimensional normal domains. For example, there are normal domains that are not Cohen-Macaulay (CM). And even if X is Cohen-Macaulay, once we blow up and normalize we might have a normal scheme X that is not Cohen-Macaulay. If X is not CM, no blow-up along a complete intersection produces an arithmetically Cohen-Macaulay scheme, by the following result of Vasconcelos. lemma 1.10 Suppose that R is a local ring, and suppose that I is an ideal generated by a regular sequence xi , . . . , xn . Then R is CM ←→ R[I t] is CM .


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Proof (Vasconcelos) Let J be the ideal of R[X] = R[X1 , . . . , Xn ] generated by the determinants Xi xj − Xj xi (i, j = 1, . . . , n). Then R[I t] = R[X]/J , the ideal J has height n − 1, and pdR[X] R[I t] = n − 1. By the Auslander-Buchsbaum equality (see [W6, Theorem 4.4.15]), we have d − G(R) = d + n − G(R[X]) = (d + 1) − G(R[I t]), where d = dim(R). Now observe that R is CM exactly when d = G(R), while R[I t] is CM exactly when d + 1 = G(R[I t]). Of course, we do not not know if it is possible for X = Proj(R[I t]) to be CM. Nor do we know if we can arrange for the normalization X of X to be CM. 2. Classical K-theory of projective curves For the next two sections, we need to collect some facts about the classical K-theory of projective curves and nonnormal surfaces, whose singular locus is a projective curve. These facts differ slightly from the affine case, where one may consult [Bass]. The reader may skip these technical sections without interrupting the flow of ideas. We first recall what is known for reduced curves. The following elementary definition, due to L. Roberts [Rob, p. 49], is extremely useful. Definition 2.1 The graph 0 of a noetherian curve Y is the bipartite graph defined as follows. Let S be the singular locus of Y , and let π : Y˜ → Y be the normalization. Then 0 has one vertex for each point of S and one vertex for every component of the normalization Y˜ of Y . There is an edge of 0 for each point of π −1 (S), connecting the corresponding component of Y˜ to the singular point of Y . The number λ(Y ) of loops in a curve Y is defined to be the number of loops in its graph 0. Because λ is the dimension of H 1 (0; Z) ∼ = Zλ , it can be calculated using the Euler characteristic of 0: if 0 has c connected components, V vertices, and E edges, then λ = c − V + E. Note that the connected components of 0 correspond to the connected components of Y . Alternatively, we can calculate λ using the e´ tale cohomology of the curve Y . The following result is proven in [W3, Lemma 2.1] and [W5, Theorem 7.9], and it is implicit in [Rob, p. 47]. lemma 2.2 Let 0 be the bipartite graph of a noetherian curve Y with finite normalization. If 0 has λ loops, then

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Het1 (Y, Z) ∼ = H 1 (0; Z) ∼ = Zλ . The number of loops enters K-theory as the rank of LPic(Y ). Recall that the group LPic(Y ) is defined to be the cokernel of Pic(Y [t]) ⊕ Pic(Y [t −1 ]) → Pic(Y [t, t −1 ]). [W5, Theorem 7.5] states that LPic(Y ) ∼ = Het1 (Y, Z). lemma 2.3 Let Y be a 1-dimensional reduced noetherian scheme. Then (1) Kq (Y ) = 0 for all q ≤ −2, (2) Y is K−1 -regular, and K−1 (Y ) ∼ = LPic(Y ) ∼ = Het1 (Y, Z). If Y has finite normalization, then K−1 (Y ) ∼ = Zλ , where λ is the number of loops in Y . Proof By [W3, Theorem 1.2], we may assume that Y is affine. By [BM, Theorem 8.1] (or [W5, Example 1.7]), K−1 (Y ) ∼ = LPic(Y ) and Kq (Y ) = 0 for q ≤ −2. Examples 2.4 If Y is the node, the graph 0 has two vertices and two parallel edges, forming one loop. Thus K−1 (Y ) ∼ = Z. If Y is a cusp, 0 has two vertices connected by one edge; here there are no loops, and K−1 (Y ) = 0. An interesting class of examples is given by the exceptional fiber Y of a resolution of singularities. For example, resolving the rational double point x 2 + y 3 + z5 = 0 yields a union Y of eight projective lines, intersecting according to the Dynkin diagram E8 (see [Hart, p. 420]). In this case, the graph 0 is the subdivision of the Dynkin diagram. Since 0 has no loops, K−1 (Y ) = 0. Now we turn to calculations of K∗ (Y ). If Y is affine, the following result is proven in [Bass, p. 685]. The last part is standard as well (see [Hart, Exercise III.4.6]). For convenience we write U (Y ) = H 0 (Y, OY× ) for the global units of Y . lemma 2.5 Let Y be a 1-dimensional noetherian scheme that is not reduced, and let Y be a subscheme defined by an ideal N of OY with N 2 = 0. Then (1) Kq (Y ) = 0 for q ≤ −2; (2) K−1 (Y ) ∼ = K−1 (Y ) ∼ = K−1 (Yred ), which is a free abelian group; (3) Y is K−1 -regular, and Kq (Y [t1 , . . . , tp ]) = Kq (Y ) for all p > 0 and q ≤ −1; (4) K0 (Y ) ∼ = H 0 (Y, Z) ⊕ Pic(Y ), and there is an exact sequence


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0 −→ H 0 (Y, N ) −→ U (Y ) −→ U (Y ) −→ H 1 (Y , N ) −→ Pic(Y ) −→ Pic(Y ) −→ 0; (5) if H 1 (Y , N ) = 0, then Pic(Y [t1 , . . . , tp ]) ∼ = Pic(Y [t1 , . . . , tp ]) for all p ≥ 0; ∼ in particular, Pic(Y ) = Pic(Y ). Proof For (1)–(4) it suffices to prove that the relative K-groups Kn (Y, N ) vanish for n < 0 and for n = 0 are isomorphic to the cohomology group H 1 (Y, N ). To check this we use the Brown-Gersten spectral sequence of [TT]: pq E2 = H p Y, K−q N =⇒ K−p−q (Y, N ). Here Kn N denotes the Zariski sheaf associated to the presheaf Kn (OY , N ). Now it is well known (see [Bass]) that Kn N = 0 for n ≤ 0 and K1 N ∼ = 1+ N ∼ = N . This 1 ∼ ∼ yields K0 (Y, N ) = H (Y, N ), sequence (4), and Kn (Y ) = Kn (Y ) for n ≤ −1. To establish K−1 -regularity, we consider the functors Kn (Y [t]), and so on. The same argument applies here: we replace Kn N by Kn N [t], the sheaf on Y associated to the presheaf Kn (OY [t], N [t]). Because N 2 = 0, we have K1 N [t] ∼ = N [t]. Hence K0 (Y [t], N [t]) ∼ = H 1 (Y, K1 N [t]) ∼ = H 1 (Y, N ) ⊗ Z[t]. Porism 2.5.1 More generally, if Y is defined by a nilpotent ideal I , then the proof shows that Kn (Y, I ) = 0 for n < 0 and K0 (Y, I ) ∼ = H 1 (Y, 1 + I ). Alternate proof It is well known that K0 (Y ) ∼ = H 0 (Y, Z) ⊕ Pic(Y ). Since H 0 (Y, Z) = H 0 (Y , Z), the K0 part follows from the cohomology sequence associated to the short exact sequence of sheaves on Y : 1 −→ N −→ OY× −→ OY× −→ 1. Replacing Y by Y [T ] yields the other parts since again we have K0 (Y [T ]) ∼ = H 0 (Y [T ], Z) ⊕ Pic(Y [T ]).

corollary 2.6 Let Y be any curve, and let I be a nilpotent ideal sheaf defining a subscheme Y0 . If H 1 (Y0 , I i /I i+1 ) vanishes for all i ≥ 1, then K0 (Y, I ) vanishes, Pic(Y ) ∼ = Pic(Y0 ), and Pic(Y [t1 , . . . , tp ]) ∼ = Pic(Y0 [t1 , . . . , tp ]) for all p. Recall that a reduced curve Y is seminormal if and only if Y is Pic-regular, that is, if and only ifPic(Y ) ∼ = Pic(Y [t1 , . . . , tp ]) for all p. Of course this is the same as saying that Y is K0 -regular, since K0 (Y [T ]) = H 0 (Y, Z) ⊕ Pic(Y [T ]) for curves.

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proposition 2.7 Let Y be a curve defined over a field k, and let I denote the nilradical ideal of OY . (a) If Y is Pic-regular, then Yred is a seminormal curve and Pic(Y ) ∼ = Pic(Yred ). (b) The following are equivalent: (1) Y is Pic-regular; (2) N p Pic(Y ) = 0 for some p > 0; (3) Yred is seminormal, and H 1 (Y, I ) = 0. If Y is complete and k is perfect, this is equivalent to (4) Yred is seminormal, and Pic(Y ) ∼ = Pic(Yred ). Proof The sequence of Lemma 2.5(4) is a direct summand of the corresponding sequence for Y [T ] = Y [t1 , . . . , tp ]. The same is true for the variation in Porism 2.5.1 with Y = Yred . Since U (Y ) = U (Y [T ]), the complementary exact sequence is Pic(Y [T ]) Pic(Y [T ]) 0 −→ H 1 Y, 1 + t I [t] −→ −→ −→ 0. Pic(Y ) Pic(Y ) The final term vanishes if and only if Y is seminormal. Since H 1 (Y, −) is right exact, the first term has H 1 (Y, I /I 2 ) ⊗ (Z[T ]/Z) as a quotient. For (b), suppose first that Y is defined over a field k of characteristic zero. Then the exponential defines an isomorphism between any nilpotent ideal I and its multiplicative sheaf 1+ I . Similarly, we have I [t] ∼ = 1+ I [t]. If I is the nilradical of OY , we can repeat the above argument (identifying Z[T ] with t1 · · · tp Z[T ]) to get exact sequences for all p > 0 (and p = 0 if Y is complete and k perfect): (2.7.1) 0 −→ H 1 (Y, I ) ⊗ Z[T ] −→ N p Pic(Y ) −→ N p Pic Yred −→ 0. The middle terms vanish exactly when the side terms vanish, and the left sides vanish exactly when H 1 (Y, I ) does. If k has positive characteristic, M. Artin has shown in [Art1, Lemma 1.4] that H 1 (Y, I ) and H 1 (Y, 1 + I ) have filtrations with isomorphic subquotients; thus H 1 (Y, I ) = 0 is equivalent to K0 (Y, I ) = 0 by Porism 2.5.1. Artin’s proof shows that the same is true for H 1 (Y, I [T ]) and H 1 (Y, 1 + I [T ]), so the same argument goes through. proposition 2.8 Suppose that C is a (not necessarily reduced) curve on a surface X defined by an invertible ideal I of OX , and suppose that Y is a curve on X defined by an ideal J of OX , where I n ⊆ J ⊂ I for some n.


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Suppose that the conormal bundle L = I /I 2 has H 1 (C, L i ) = 0 for i = 1, . . . , n − 1. Then Pic(Y ) ∼ = Pic(C[t1 , . . . , tp ]) for = Pic(C) and Pic(Y [t1 , . . . , tp ]) ∼ all p. Proof First consider the curve Y = Yn defined by the ideal I n . Setting I¯ = I /I n , the multiplicative sheaf 1 + I¯ on Y has a filtration whose associated graded sheaves are the L i = I i /I i+1 for i = 1, . . . , n − 1. If H 1 (C, L i ) = 0 for i = 1, . . . , n − 1, then Corollary 2.6 says that the result holds for Yn . In the general case, set I¯ = I /J . Since H 1 (Y, –) is right exact, the surjection 1 + I /I n → 1 + I¯ implies that we also have K0 (Y, I¯ ) = H 1 (Y, 1 + I¯ ) = 0 and hence Pic(Y ) = Pic(C), as claimed. A curve Y on X is called a nilpotent thickening of C if the ideal J defining Y satisfies I n ⊆ J ⊂ I for some n, that is, if the hypothesis of Proposition 2.8 holds. Variation 2.8.1 Let Yn be the curve on X defined by the ideal I n , and set L = I /I 2 . Suppose that H 1 (C, L i ) = 0 for i ≥ n0 . Then Pic(Yn ) ∼ = Pic(Yn0 ) for all n ≥ n0 . This follows either by induction from Lemma 2.5(5) or from Corollary 2.6 with Y0 = Yn0 , using the proof of Proposition 2.8. The point is that the groups K0 (Yn , I ) = H 1 (Y, 1 + I /I n ) are independent of n when n ≥ n0 . For example, an ample line bundle L always has H 1 (C, L i ) = 0 for large i (see [Hart, Lemma III.5.2]). Therefore Variation 2.8.1 applies when the conormal bundle I /I 2 of C is ample. Example 2.9 (Affine cones) Suppose that C ⊂ Pn is a smooth projective curve of genus g ≤ 2, or, more generally, that C is embedded in Pn by a complete linear system of degree greater than or equal to 2g − 1. Let X be the affine cone of C, and let X be the blow-up at the cone point. Then the exceptional divisor E on X is isomorphic to C. If Y is any nilpotent thickening of E on X , then Y is always Pic-regular with Pic(Y ) = Pic(C). To see this, note that E is defined by an invertible ideal I of OX , and note that the degree of L = I /I 2 is −E · E > 0 (see [Hart, Example V.1.4.1 and Exercise V.5.7]). By Proposition 2.8, it suffices to show that H 1 (C, L i ) = 0 for all i ≥ 1. For g ≤ 1 this is easy because L has positive degree. It holds for g = 2 by Riemann-Roch since the degree of L must be at least 5 (see [Hart, Exercise IV.3.1]). If C is embedded in |D| for a divisor D of degree d ≥ 2g − 1, then |D| ∼ = Pd−g

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by Riemann-Roch. By Serre duality, the dimension of H 1 (C, L i ) equals l(K − iD), and this is zero because K − iD has degree less than zero. Example 2.10 Suppose that X is a projective surface, and suppose that C ⊂ X is isomorphic to P1A for some Artinian local ring A. Also suppose that the ideal sheaf I defining C is invertible, and suppose that L = I /I 2 is OP1 (j ) for some j ≥ 0. Let C → Y be a nilpotent thickening on X . Then Pic(Y ) = Pic(C) = Z and Y is Pic-regular. This follows from Proposition 2.8 since H 1 (P1A , O (i)) = 0 for all i ≥ −1. Example 2.10.1 To show that the hypothesis on L is critical, consider a line C in P2 defined, say, by the equation Z = 0 in S = k[X, Y, Z]. The conormal bundle is L ∼ = O (−1), and its nilpotent thickenings Yn are defined by Z n = 0. We claim that Pic(Y2 ) ∼ = Pic(C) = Z but Pic(Y3 ) ∼ = Z ⊕ k and hence Pic(Yn ) = Pic(C) = Z for n > 2. This claim follows from the exact sequence in Lemma 2.5(4), which shows that Pic(Yn+1 ) → Pic(Yn ) is a surjection with kernel H 1 (P1 , O (−n)) ∼ = 0 1 H (P , O (n − 2)). I am grateful to the referee for pointing out this example. Example 2.11 Suppose that C is a rational cubic cusp lying on a smooth projective surface X over k, and suppose that C is the inverse image of a point under some proper map X → X. If C → Y is a nilpotent thickening on X , then Pic(Y ) ∼ = Pic(C) ∼ = k ⊕ Z. Moreover, because C is not seminormal, the curve Y cannot be Pic-regular. Indeed, if L denotes the line bundle I /I 2 on C and p : P1k → C is the normalization, then the pullback p ∗ L has degree −C · C > 0 (see [Hart, Example V.1.4.1]), so p ∗ L ∼ = OP1 (8) for some 8 ≥ 1. If ι is the inclusion of the cusp point, then there is an exact sequence of sheaves on C: 0 −→ L i −→ p∗ OP1 (i8) ⊕ ι∗ (k) −→ ι∗ (k[ε]) −→ 0. The map H 0 (P1k , O (i8)) → k[ε] is onto. From this and Example 2.10 we see that H 1 (C, L i ) = 0 for all i ≥ 1. The assertions now follow from Proposition 2.8. Example 2.12 Suppose that C is a divisor with normal crossings on a projective surface, with ample conormal bundle L . Suppose either that C has no loops in the sense of Definition 2.1 or that each of its irreducible components C1 , . . . , Ce meets at most two other components. Then H 1 (C, L i ) ∼ = ⊕j H 1 (Cj , L i |Cj ) for all i ≥ 1. (Use the method of


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Example 2.11 and the calculation that ⊕H 0 (Cj , L i |Cj ) → ⊕k is onto.) As an illustration, suppose that each Cj is a projective line. Then H 1 (C, L i ) = 0 for all i ≥ 1 as each L |Cj is ample. If Y is any nilpotent thickening of C, then Pic(Y ) ∼ = Pic(C) ∼ = Ze and Y is Pic-regular by Proposition 2.8. If each Cj is a projective line and there are no loops, then we also have H 1 (C, OC ) = 0 (same calculation). Lipman has observed (private communication) that the assumption that L is ample implies that X has rational singularities. Indeed, it follows from induction and H 1 (C, L i ) = 0 that each H 1 (X, OX /I i ) vanishes. Taking the inverse limit, the theorem on formal functions [Hart, Theorem III.11.1] implies that H 1 (X, OX )x = 0 at each singular point x, which is the definition of a rational singularity (cf. [Lip1, Theorem 12.2]). Example 2.13 (Rational singularities) Suppose that k is perfect and that X has rational singularities. Choose X → X such that the reduced exceptional divisor C has normal crossings and smooth branches. Then every nilpotent thickening Y of C on X is Pic-regular and has Pic(Y ) ∼ = Pic(C). The following proof is due to Lipman (cf. [Lip1, Theorem 12.2]). Artin has shown in [Art2, p. 130] that H 1 (Y, OY ) = 0 for every such Y (since the singularities of X are rational and Y has a thickening Y which is a divisorial cycle). Since k is perfect, H 0 (Y, OY ) → H 0 (C, OC ) is a split surjection. The cohomology sequence for OY → OC shows that H 1 (Y, I /I n ) = 0. As C is seminormal, we see from Proposition 2.7 that Y is Pic-regular and Pic(Y ) ∼ = Pic(C). 3. Mayer-Vietoris sequences In the next section we need to know the lower K-theory of a nonnormal surface X . As in the affine case, this can be done using a Mayer-Vietoris sequence for finite maps. We say that a closed subscheme Y ⊂ X is conducting (for a finite morphism π X → X ) if the ideal I of OX defining Y is also an ideal of π∗ OX . theorem 3.1 (Mayer-Vietoris) Let π : X → X be a finite morphism of noetherian schemes such that a conducting subscheme Y has dimension at most 1. Setting Y = Y ×X X , there is an exact “Mayer-Vietoris” sequence K1 (X ) ⊕ K1 (Y ) −→ K1 (Y ) −→ K0 (X ) −→ K0 (X ) ⊕ K0 (Y ) −→ K0 (Y ) −→ K−1 (X ) −→ K−1 (X ) ⊕ K−1 (Y ) −→ K−1 (Y ) −→ K−2 (X ) −→ K−2 (X ) −→ 0. Moreover, Kq (X ) ∼ = Kq (X ) for all q ≤ −3, and N p Kq (X ) ∼ = N p Kq (X ) for all q ≤ −2 and all p > 0.

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In the affine case, when one considers finite ring extensions, this sequence is due to H. Bass and M. Murthy (see [Bass, p. 677]). The scheme version presented here is slightly more delicate and was proven in [PW2, Theorem A.3] and [BPW, Theorem 3.3]. Note that we are using the vanishing of K−2 (Y ) and K−2 (Y ) to terminate the sequence. Example 3.2 The blow-up X of a complete intersection on the affine cone X = Spec(R) of a smooth plane conic C is an instructive case in point. Specifically, let R = k[x, y, z]/ (xy = z2 ), S = R[xt, yt], and X = Proj(S). Then the normalization of X is X = Proj(S ), where S = R[xt, yt, zt]. Now the conductor from S to S is the homogeneous prime ideal I = (x, y, z)S , and S/I = k[X, Y ], S /I = k[X, Y, Z]/(XY = Z 2 ). Hence the associated sheaf I = I˜ defines the conductor subscheme Y = Proj(S/I ) = P1k , and Y = Y ×X X is the plane conic C. In fact, X is a line bundle over C, and its zero-section is the inclusion of Y in X . Since X , Y , and Y are smooth and Y ⊂ X induces an isomorphism on K-theory, it follows that X is K0 -regular, Kq (X ) = 0 for all q < 0, and K0 (X ) ∼ = K0 P1k ∼ = Z ⊕ Z. Example 3.3 Here is a singular (nonnormal) surface with K−2 (X ) = Z/2. Let Y be the configuration of four lines in the plane X = Spec(k[x, y]) defined by f = xy(1−x)(1−y). Let Y be the plane curve u(v + 1 − u2 ) = 0, and let X be the surface obtained from X by gluing along the evident degree two maps s : Y → Y . That is, X = Spec(R), where R is the subring of k[x, y] generated by u = x(1 − x), v = y(1 − y) − 1 + u2 , and I = f k[x, y]. The calculation that the map from K−1 (Y ) = Z to K−1 (Y ) = Z is multiplication by 2 is left as an exercise for the reader. The claim about K−2 (X ) follows immediately upon plugging this calculation into the Mayer-Vietoris sequence. 3.4. H 0 –LPic sequence 0 (X ) of the rank map K0 (X ) → The image of K1 (Y ) → K0 (X ) lies in the kernel K 0 H (X , Z). In fact, the Mayer-Vietoris sequence in Theorem 3.1 surjects onto the “H 0 –LPic sequence” of [W5, Proposition 7.8]: 0 −→ H 0 (X , Z) −→ H 0 (X , Z) ⊕ H 0 (Y , Z) −→ H 0 (Y , Z) −→ LPic(X ) −→ LPic(X ) ⊕ LPic(Y ) −→ LPic(Y ).

(3.4)

Since K−1 (Y ) ∼ = LPic(Y ) for curves by Lemma 2.3, we see that the H 0 –LPic sequence (3.4) may be completed to end in LPic(X ) ⊕ LPic(Y ) −→ LPic(Y ) −→ K−2 (X ) −→ K−2 (X ) −→ 0. (3.4.1)


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−1 (X) to be the kernel of K−1 (X) → LPic(X), a diagram chase shows Defining K that the kernel sequence is exact: 0 (X ) −→ K 0 (X ) ⊕ K 0 (Y ) K1 (X ) ⊕ K1 (Y ) −→ K1 (Y ) −→ K −1 (X ) −→ K −1 (X ) −→ 0. 0 (Y ) −→ K −→ K

(3.4.2)

We saw in Theorem 1.9 that there is a blow-up X of any normal surface whose normalization is nonsingular. This provides motivation for the following result. proposition 3.5 Let X be a surface whose normalization X is regular. Then 2 (X , Z); (1) there is an isomorphism K−2 (X ) ∼ = Hnis (2) X is K−2 -regular, and Kq (X ) = 0 for all q ≤ −3; (3) there is an exact sequence −1 (X ) −→ 0; Pic(X ) −→ Pic(X ) × Pic(Y ) −→ Pic(Y ) −→ K (4) there are isomorphisms × × 2 2 −1 (X ) ∼ K X , OX ∼ X , OX = Hzar = Hnis 1 (X , Z) ⊕ H 2 (X , O × ). and a noncanonical isomorphism K−1 (X ) ∼ = Hnis nis X

Proof 2 (X , Z) = 0, the first assertion is immediate from a Since K−2 (X ) = 0 and Hnis comparison of (3.4.1) with the cohomology sequence 1 1 1 2 2 Hnis (X , Z) ⊕ Hnis (Y , Z) −→ Hnis (Y , Z) −→ Hnis (X , Z) −→ Hnis (X , Z).

Assertion (2) follows from Theorem 3.1. For assertion (3), we use the “Units-Pic” sequence: 1 −→ U (X ) −→ U (X ) × U (Y ) −→ U (Y ) −→ Pic(X ) −→ Pic(X ) × Pic(Y ) −→ Pic(Y ). As pointed out in [W5, Proposition 7.8], this is part of a long cohomology sequence. Since X is regular, it is well known that we have H 2 (X , OX× ) = 0 (for both Zariski and Nisnevich cohomology). So this sequence ends in Pic(X ) × Pic(Y ) −→ Pic(Y ) −→ H 2 X , OX× −→ 0. There is a natural surjection from the K1 –K0 part of (3.4.2) onto the Units-Pic sequence, essentially due to Bass and Murthy (see [Bass, p. 482]). As K−1 (X ) = 0, −1 (X ) = 0 to H 2 (X , O × ). this induces an isomorphism of cokernels, from K

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4. K-theory and resolutions The following theorem, due to R. Thomason [T], is the key to our calculations in K-theory. We first state it for X of arbitrary dimension and then restrict to surfaces. theorem 4.1 (Thomason [T]) Let X be a quasi-projective scheme, and let i : Y → X be a locally complete intersection of pure codimension d. Let f : X → X be the blow-up of X along Y . Then, for all q, Kq (X ) ∼ = Kq (X) ⊕ Zd−1 ⊗ Kq (Y ) . A similar result holds for the functors N p Kq (X). Indeed, if Y denotes the pullback Y ×X X , then Y is locally isomorphic to PdY , so we have K∗ (Y ) ∼ = Zd ⊗ K∗ (Y ). Moreover, the closed immersion i : Y → X defines a map i∗ : K∗ (Y ) → K∗ (X ). Thomason [T] proves that there is a split exact sequence (−i∗ ,λ·fY∗ )

(f ∗ ,i∗ )

0 −→ K∗ (Y ) −−−−−−−→ K∗ (X) × K∗ (Y ) −−−−−→ K∗ (X ) −→ 0. To obtain the result for N p Kq (X), use the case p = 0 with X replaced by X[t1 , . . . , tp ], the product of X with affine space Ap . Application 4.2 (Murthy [Mur]) Let R = k[x, y, z]/(xy = z2 ) be the affine cone of a smooth plane conic. Then R is K0 -regular, Kq (R) = 0 for q < 0, and K0 (R) = Z. These facts are immediate from Theorem 4.1, given our calculation in Example 3.2 for the blow-up X = Proj(R[xt, yt]) along the complete intersection ideal (x, y)R. An alternative proof of these facts was given in 1969 by Murthy [Mur, Proposition 5.2 and Example 6.2]. corollary 4.3 Let R be a 2-dimensional normal local ring, and let x, y be a regular sequence. Set X = Proj(R[U, V ]/(xV − yU )) and A = R/(x, y). Then Kq (X ) ∼ = Kq (R) ⊕ Kq (A),

for all q.

In particular, Kq (X ) ∼ = Kq (R) for all q < 0, while K0 (X ) ∼ = Z⊕Z and Pic(X ) ∼ =Z 1 on the class of the divisor Y ×X X = PA . For q ≤ 0 the ring R is Kq -regular if and only if X is, and N p Kq (R) = p N Kq (X ) for all p > 0. If R is not regular, then X is not K1 -regular. Indeed, X is the blow-up of Spec(R) along the subscheme Spec(A), which has codimension 2. And A is K0 -regular by [Bass, p. 685].


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Remark 4.3.1 When R is not regular, the maximal ideal m of R cannot be generated by x and y, so the nilradical of A is nonzero. By [Bass, p. 671], this means that X is never K1 -regular, even though R might be. When R is not regular, the blow-up X is never normal; indeed, it is easy to see that on the affine open D+ (xt) = Spec(R[y/x]) of X the height-one prime ideal m[y/x] is singular. This implies that if X → X is its normalization, then the conductor subscheme Y is Y ×X X = P1A up to a nilpotent ideal. theorem 4.4 Let X be a 2-dimensional excellent noetherian scheme. (1) If q ≤ −3, then Kq (X) = 0. (2) If q ≤ −2, then X is Kq -regular; that is, Kq (X) ∼ = Kq (X[t1 , . . . , tp ]) for all p. (3) If X is normal, then K−2 (X) ∼ = Zλ , where λ is the number of loops in the exceptional divisors of a resolution of singularities of X. Proof Using the Mayer-Vietoris sequences in Theorem 3.1 and Lemma 2.5 for curves, we see that we may assume that X is normal (cf. [W1, Proposition 2.8]). Since normal surfaces have isolated singularities, [W3, Theorem 0.1] says that we may replace X by Spec of any of its local rings. That is, we may assume X = Spec(R) for R a normal local ring. Choose a resolution of singularities X → X, with X the blow-up along an ideal J of R. By Theorem 1.9, X is the normalization of a blow-up X along a complete intersection. The Mayer-Vietoris sequences in Proposition 3.5 and Corollary 4.3 give Kq (R) = Kq (X ) = 0 for q ≤ −3, and N p Kq (R) = N p Kq (X ) = 0 for p > 0 and q ≤ −2. For q = −2, we have K−2 (R) = K−2 (X ). Since X has no negative K-theory, the Mayer-Vietoris sequence ends in K−1 (Y ) → K−1 (Y ) → K−2 (R) → 0. Now ∼ 1 , so K (Y ) = 0 by Lemma 2.3. Since Y is the exceptional fiber E Yred = Pk −1 red over the singular point of X = Spec(R), we have K−1 (Y ) = K−1 (E) ∼ = Zλ by Proposition 3.5. Exercise 4.5 Suppose that X is a surface that is not normal. Use the Mayer-Vietoris sequence (3.4.1) for the normalization to show that the group K−2 (X) is a finitely generated group. As we saw in Example 3.3, this group need not be torsion free. For extra credit, find 2 (X , Z). a surface X proper over X so that K−2 (X) is isomorphic to Hnis

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Example 4.6 (Affine cones) Suppose that R is the homogeneous coordinate ring of a reduced curve C in Pn . Then the blow-up X of Spec(R) at the maximal ideal m is an A1 -bundle over C, whose exceptional fiber is C = Proj(R). From Theorem 4.4 we have Kq (R) = 0 for all q ≤ −3. We claim that we also have K−2 (R) = 0. This follows easily from Theorem 4.4 only when C is projectively normal, that is, when R is normal and C is a smooth curve. Suppose now that R is not normal; it might not even be Cohen-Macaulay. In this case, we pass to the ring R = m−i , which is not only Cohen-Macaulay but also finite over R (see [GW, Theorem 1.5]). Since the conductor ideal is m-primary, we see from the Mayer-Vietoris sequence in Section 3.4 that K−1 (R) ∼ = K−1 (R ) and K−2 (R) ∼ = K−2 (R ). Since R is Cohen-Macaulay, the scheme X is finite over a blow-up X along a local complete intersection by Proposition 1.6. Using Lemma 2.5, we see that K−2 (X ) ∼ = K−2 (C) = 0 and that K−1 (X ) → K−1 (Y ) ∼ = K−1 (C) is an iso ∼ morphism. Since K−2 (R ) = K−2 (X ) by Theorem 4.1, the claim now follows from sequence (3.4.1). An interesting nonnormal family of examples arises when C is a union of projective lines. If C is connected, then R is the seminormalization of R (see [GW, Corollary 5.9]). Otherwise, R is the product of the seminormalizations of the affine cones of the connected components of C (see [GW, Lemma 6.4]). We describe K−1 (R) in Example 5.8.1, at least in the case when C is a smooth curve. 5. K−1 and Class groups We can also describe K−1 (X) and say when X is K−1 -regular. Let X be a resolution of singularities of a normal surface X, with exceptional fibers Ei over the singular points yi of X. Choose a blow-up X along a local complete intersection Y so that X is finite over X . Choose a conductor subscheme Y which is a nilpotent thickening is a disjoint union of copies of Y ×X X ∼ = P1Y , and set Y = Y ×X X . Then Yred 1 of Pk , and E = Yred is the disjoint union of the Ei . proposition 5.1 Let X be a normal surface, with resolution of singularities X . Suppose that X has e singularities, and suppose that Y and Y are as above. Then K−1 (X) is presented by the exact sequences Pic(X) −→ Pic(X ) −→ Pic(Y ) −→ K−1 (X) −→ 0. Moreover, there is a natural isomorphism N Pic(Y ) ∼ = NK−1 (X).


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Proof −1 (X ) By [W5], we have LPic(X ) = LPic(X) = 0, so K−1 (X) ∼ = K−1 (X ) = K by Theorem 4.1. The result now follows from Proposition 3.5, given the vanishing N Pic(X ) = N Pic(X) = N Pic(Y ) = 0, Example 2.10 for Y , and the calculation in Lemma 5.2. lemma 5.2 π → X be a blow-up along a local complete Let X be a normal surface, and let X − intersection Y of dimension zero. Then the fiber Y ×X X is isomorphic to P1Y , with conormal bundle O (1), and the inclusion j : P1Y =→ X induces a split surjection j ∗ : Pic(X ) → Pic(P1Y ) with kernel Pic(X). More precisely, the following composition is an isomorphism: j∗ j∗ π∗ 0 (X ) −− 0 P1Y = Pic P1Y . K0 (Y ) −−→ K0 P1Y −−→ K →K Proof The question being local on X, we may assume that X = Spec(R) for a local ring R, and we may assume that Y = Spec(R/I ) for I = (x, y)R. The structure sheaf OP1 of P1Y is defined by the invertible ideal I on X = Proj(R[I t]) associated to the 0 (X ) sends the generator to [O 1 ] = 1 − [I ]. ideal I R[I t], so the map K0 (Y ) → K P Applying j ∗ yields 1 − [I /I 2 ]. But I /I 2 is the structure sheaf O (1) on P1Y , and this sheaf generates Pic(P1Y ) ∼ = Z. The following theorem verifies [Sr2, Conjecture A]. Note that if R is semilocal, then the henselization and completion of R are, respectively, equal to the product (over the maximal ideals {mi } of R) of the henselizations and completions of the Rmi . Since Cl(R) ⊆ ⊕ Cl(Rmi ) ⊆ Cl(R h ), our result is compatible with the description of K−1 (R) in [W3, Theorem 3.7]. theorem 5.3 Let R be a normal semilocal ring of dimension 2, essentially of finite type over a field for the henselization and (or over an excellent Dedekind domain). Write R h and R completion of R. Then we have the following: = 0; (1) K−1 (R h ) = K−1 (R) are isomorphic: Cl(R h ) ∼ (2) the class groups of R h and R = Cl(R); h ∼ ∼ (3) K−1 (R) = Cl(R )/ Cl(R) = Cl(R)/ Cl(R). Proof Then X = X ×R We first consider S = Spec(R). S is a resolution of singularities

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) → lim Pic(Yn ) of S. Grothendieck’s existence theorem implies that the map Pic(X ← − is an isomorphism (see [Art3, (3.6)] or [Hart, Exercise II.9.6]). By Proposition 5.1, = 0. we have K−1 (R) and K−1 (R h ) = 0. In someArtin approximation implies that Cl(R h ) ∼ = Cl(R) what more detail, let Xh denote X ×R Spec(R h ). By [Art3, Theorem 3.5], Pic(Xh ) is a dense subgroup of lim Pic(Yn ), so we also have Pic(Xh ) ∼ = Pic(Yn0 ). Again by ← − Proposition 5.1, we have K−1 (R h ) = 0. Cl(R) also follows from Proposition 5.1, The isomorphism K−1 (R) −→ Cl(R)/ given Srinivas’s observation in [Sr2, p. 597] that for n 0 we have an exact sequence Pic(X ) −→ Pic(Yn ) −→

Cl(R) −→ 0. Cl(R)

Remark 5.3.1 If we knew that K−1 (R h ) = 0, we could prove (3) as follows. The map R → R h = Rih is an analytic isomorphism along mi , so (as in [W3, Theorem 3.7], but using [TT, Theorem 7.1]) there is an exact sequence 0 Spec(Rih ) − {mi } −→ K−1 (R) −→ ⊕K−1 Rih . 0 Spec(R) − {mi } −→ ⊕K K The first two terms are class groups, and the final term is zero. corollary 5.4 Let X be a normal surface, and let Rih (i = 1, . . . , e) be the Hensel local rings of X at its singular points. Then there is an exact sequence Cl(X) −→ ⊕ Cl Rih −→ K−1 (X) −→ 0. Proof Let R be the semilocal ring of X at the singular locus; clearly Cl(X) → Cl(R) is onto. By [W3, Theorem 1.2], K−1 (X) ∼ = K−1 (R), so the result follows from Theorem 5.3. Remark 5.4.1 Since K−1 (X) ∼ = H 2 (X , OX× ) by Proposition 3.5(4), the Leray spectral sequence for X → X and OX× degenerates to yield the exact sequence 0 −→ H 2 X, OX× −→ K−1 (X) −→ ⊕i K−1 OX,yi −→ 0. This is the local-to-global sequence of [W3, Theorem 0.2]; the group H 2 (X, OX× ) is zero unless X has more than one singular point.


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Example 5.5 (The Es singularity) Let R = k[x, y, z]/(x 2 + y 3 + z5 ). It is well known that R and R h are unique factorization domains with an isolated singularity at the origin (see [Hart, p. 420], for example). By Corollary 5.4, we see that K−1 (R) = 0. In this case, we also have K−2 (R) = 0 by Theorem 4.4 and Example 2.4. The singularity is resolved by 8 successive blow-ups of closed points, and the exceptional fiber is a union of 8 projective lines, which intersect each other according to the Dynkin diagram E8 . From this it is easy to calculate that Pic(Y ) = Z8 , as in Example 2.13 or as in [Art1, Theorem 1.7]. This surface was discovered by Klein; the binary icosahedral group G = SL2 (F5 ) acts on the plane, and R is the invariant subring k[u, v]G . Following Lipman [Lip1, p. 225], the Néron-Severi group NS(E) = ⊕NS(Ei ) is the free abelian group on the irreducible components Ci of E. Let H denote the cokernel of the endomorphism of NS(E) given by the intersection matrix (Ci · Cj ). A lemma of P. Du Val (see [dVal], [Mum, p. 6], or [Lip1, Lemma 14.1]) states that the intersection matrix is negative definite, so H is a finite group. We note in passing that H arose in D. Mumford’s 1961 study of the topology of normal singularities on a complex surface X: if M is the intersection of X(C) with a small sphere about the singularity, then H is the torsion subgroup of H1 (M, Z) (see [Mum, p. 11]). Now consider the map θ : Pic(X ) → Pic(E) → NS(E), and set Pic0 (X ) = ker θ and G = coker θ . Since the map Pic(X ) → NS(E) sends [Ci ] to the ith row of this matrix, G is a quotient of H . Hence G is also a finite group. In fact, Lipman’s main exact sequence [Lip1, Proposition 14.2] is 0 −→ Pic0 (X ) −→ Cl(R) −→ H −→ G −→ 0. 5.6. Rational singularities If R has a rational singularity, then the group K−1 (R) is the finite group G studied by Lipman. Indeed, Cl(R h ) ∼ = H by [Lip1, Proposition 17.1], so Lipman’s main sequence shows that G is the cokernel of Cl(R) → Cl(R h ), which by Theorem 5.3 is K−1 (R). Conversely, if k is algebraically closed and Cl(R h ) is finite, then R must have a rational singularity by [Lip1, Theorem 17.4]. However, P. Salmon showed in [Salm] that R = k0 (u)[x, y, z]/(x 2 + y 3 + uz6 ) has Cl(R h ) = 0, and hence K−1 (R) = 0, even though R does not have a rational singularity. proposition 5.7 Let X be a normal surface over an algebraically closed field of characteristic zero.

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Then K−1 (X) is the direct sum of a divisible group and the finite group G. In particular, K−1 ⊗ Q/Z vanishes. Proof By Lemma 2.5, Pic(Y ) → NS(Y ) = NS(E) is a surjection whose kernel is a divisible group. Also, Pic(X ) → NS(X ) is a surjection whose kernel is divisible. By the 5-lemma, the cokernel of Pic(X ) → Pic(Y ) contains a divisible group, and the quotient is the the cokernel G. We now turn to the question of K−1 -regularity. proposition 5.8 Let X → X be a resolution of singularities of a normal surface X, and let L be the conormal bundle of the exceptional fibers Ej . That is, L = I /I 2 , where I is the invertible ideal defining the (reduced) exceptional divisors Ej on X. If X is K−1 -regular, then (1) every exceptional fiber Ej must be a seminormal curve, and (2) each H 1 (Ej , L ) must vanish. Conversely, if each Ej is seminormal and H 1 (Ej , L i ) = 0 for all i ≥ 1, then X is K−1 -regular. Proof Because K-regularity is a local question, we may assume that X has an isolated sin = E. Because NK (X) ∼ N Pic(Y ) by Propogularity, and we may assume Yred = −1 sition 5.1, we are reduced to the assertions about N Pic(Y ) made in Propositions 2.7 and 2.8. 5.8.1. Affine cones Let R be the homogeneous coordinate ring of a smooth curve C in Pn . As noted in Example 4.6, the blow-up X of Spec(R) at the vertex point y is an A1 -bundle over C, and the inverse image of y is isomorphic to C. If C is not projectively normal, then the normalization of R is the ring R = ⊕0(C, OC (n)) (see [Hart, Exercise II.5.14]), and we have K−1 (R) ∼ = K−1 (R ) by Theorem 3.1. By Proposition 5.8, the K−1 -regularity of R and R is connected with the vanishing of H 1 (C, OC (1)). Suppose first that C has genus g ≤ 2, or suppose that C is embedded in Pn by a complete linear system D of degree greater than or equal to 2g − 1. By Example 2.9, the cohomology groups vanish, and so R is K−1 -regular. Moreover, from Proposition 5.1 we have K−1 (R) = 0. If char(k) = 0 and deg(D) ≥ 2g + 1, this follows from the Srinivas-Varley theorem that K0 (R) = Z (see [Sr1], [Var]), which implies that R


23

is K0 -regular by [MP, Corollary 1.4]. At the other extreme, suppose that H 1 (C, OC (1)) = 0. Then R is not K−1 -regular by Proposition 5.8. A fortiori, R cannot be K0 -regular. This recovers the theorem of Coombes and Srinivas, who proved in [Sr1] and [CS] that if H 1 (C, OC (1)) is nonzero then NK0 (R) = 0. If R is K0 -regular, then K0 (R) = Z because R is a graded ring with R0 = k. The converse need not hold; Srinivas also proved in [Sr1] that if R is normal and k is algebraically closed of finite characteristic, then K0 (R) = Z always holds. theorem 5.9 Let X be a normal surface over a perfect field. Let X → X be a resolution of singularities such that the exceptional divisor E has smooth components and normal crossings. Then the following are equivalent: (1) X is K−1 -regular; (2) NK−1 (X) = 0; (3) N p K−1 (X) = 0 for some p > 0; (4) H 1 (E, I /I n ) = 0 for every n, where I is the ideal of OX defining E; = E. (5) Pic(Y ) ∼ = Pic(E) for every curve Y on X with Yred Proof This result is immediate from Propositions 5.1 and 2.7 because any reduced curve with smooth components and normal crossings is seminormal. Remark 5.9.1 When k = C, Srinivas proved in [Sr2, Theorem 2] that if a normal surface X is K0 -regular (e.g., if NK0 (X) = 0), then Pic(Y ) → Pic(E) is an isomorphism for = E. Since K -regularity implies K -regularity, every curve Y on X with Yred 0 −1 Srinivas’s result is a consequence of Theorem 5.9. 6. Examples We begin with an example of some historical importance. Example 6.1 (Bloch and Murthy) The ring R = k[x, y, z]/(z2 − x 3 − y 7 ) is a 2-dimensional normal domain. Bloch and Murthy discovered in November 1979 (while we were writing up [W1]) that NK0 (R) = 0 when k = C; their argument is sketched in [Sr1, p. 259]. For any field k, a desingularization π : X → Spec(R) can be chosen so that the curve E = π −1 (0) is a rational cubic cusp. By Example 2.4 and Theorem 4.4, we have K−2 (R) = 0.

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Since R is not K−1 -regular by Theorem 5.9, it is not K0 -regular either. This recovers the Bloch-Murthy result for all fields k. To compute K−1 (R), we proceed as follows. By Example 2.11 we have Pic(Y ) ∼ = Pic(E) = k ⊕ Z. Moreover, the image of Pic(X ) → Pic(Y ) is the Z summand. Hence the sequence in Proposition 5.1 yields K−1 (R) ∼ = k. Example 6.2 (Mumford [Mum, p. 16]) Let C0 be a smooth cubic in P2 , and let p1 , . . . , p15 be points on C0 which are in general position except that on C0 the divisor pi ≡ 5H , where H is a hyperplane section. Blow up these points on P2 to get a smooth surface X , with Ei the exceptional divisor over pi . On X the proper transform C of C0 has C · C = −6, and the linear system of quintics through the pi contracts C on X to yield a normal projective surface X having one singular point y. By Theorem 4.4, K−2 (X) = 0. In this case, the map from Pic(X ) ∼ = Z16 to Pic(C) sends the Ei −Ej to nonzero points xij in the abelian variety Pic0 (C) ∼ = Pic(C)/[E1 ]. Because the conormal bundle L of the elliptic curve C on X has degree 6, H 1 (C, L i ) = 0 for all i ≥ 0 by Riemann-Roch. By Proposition 2.8, Pic(Y ) = Pic(C) for every nilpotent thickening Y of C. Thus Proposition 5.1 yields K−1 (X) ∼ = Pic 0 (C)/subgroup generated by the xij . This example shows that K−1 (X) does not have a reasonable algebraic structure. This example was originally given by Mumford in order to show that the homology of a normal singularity was not reflected by the ideal class group Cl(R) of the local ring R. Mumford conjectured in [Mum] that Cl(R) was equal to Cl(R); A. Grothendieck observed in [GB, p. 75] that this was false. We can now see an Cl(R) ∼ other reason why the class groups are not equal, since Cl(R)/ = K−1 (X) by Theorem 5.3. Example 6.3 (Srinivas [Sr2]) Let C be an irreducible sextic in P2 with 10 nodes, let X → P2 be the blow-up at these 10 nodes, and let E be the strict transform of C. Then E 2 = −4, and E contracts on X to yield a normal projective surface X having one singular point y of multiplicity 4. Since E ∼ = P1 , we have K−2 (X) = 0. In this case, K−1 (X) ∼ = Z/2. One can either see this by calculating class groups ˆ ∼ as in [Sr2, p. 597]—the local ring R = OX,y has Cl(R) = Z/2 and Cl(R) = Z/4—or 11 ∼ see it geometrically by observing that the image of Pic(X ) = Z in Pic(E) ∼ =Z has index two. The 10 exceptional curves Ei of X → P2 have E · Ei = 2, and the inverse image L of the general line in P2 has E · L = 6 (see [Sr2, p. 629]).


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6.4. Rational double points Suppose for simplicity that k has characteristic zero. We say that R has a rational is isomorphic to k[[x, y, z]]/(f ) for one of the following f : double point if R ∼ Cl(R) = Z/(n + 1), (Z/2)2 for n even, ∼ Cl(R) = Z/4 for n odd,

(An )

f = zn+1 + xy

(Dn ), n ≥ 4

f = z2 + xy 2 + x n−1

(E6 )

f = z2 + y 3 + x 4

(E7 )

f = z2 + x 3 y + y 3

∼ Cl(R) = Z/3, ∼ Cl(R) = Z/2,

(E8 )

f = z2 + x 3 + y 5

=0 Cl(R)

(see [Durf, Characterization A5]). In each case, the exceptional curve C consists of n copies of P1 , meeting each other according to the associated Dynkin diagram [Durf, Characterization A3]. As in Example 2.4, C has no loops because its graph 0 is the subdivision of the Dynkin diagram. Thus K−2 (R) = K−1 (C) = 0. We also know that the group K−1 (R) is finite by Theorem 5.3 because it is a quotient of Cl(R), which is listed above. Examples when R is a UFD are given in [Sr3]; in these cases K−1 (R) = Cl(R). By Example 2.13, Pic(Y ) = Pic(C) for every nilpotent thickening Y of C. Hence rational double point singularities are all K−1 -regular by Theorem 5.9. If k = C, Srinivas showed in [Sr2, Corollary 4.4] that R is K0 -regular by applying [MP] to the calculation of K0 (X) in [Sr3] and [MK].

6.5. Quotient singularities If G is a finite subgroup of SL2 (k) acting on the plane (hence on k[u, v]), we can form the invariant subring k[u, v]G . A rational surface singularity Spec(R) is called a quo is isomorphic to the completion of some k[u, v]G . tient singularity if its completion R By [Durf, Characterization A5], a rational double point is a quotient singularity with embedding dimension 3. Now suppose that k = C. Srinivas showed in [Sr2, Corollary 4.4] that every quotient singularity R is K0 -regular. To do this, he observed that R has a finite cover S, e´ tale over Spec(R) − {m}, which is a rational double point. Since N p Kq (R) is a C-vector space for p > 0, and the kernel of N p Kq (R) → N p Kq (S) has exponent n for q ≤ 0, N p Kq (R) is a subgroup of N p Kq (S). The result now follows from Example 6.4. Since the kernel of Kq (R) → Kq (S) also has exponent n, it follows from Theorem 4.4 that K−2 (R) = 0 and that K−1 (R) is finite. Example 6.3, which is a quotient singularity, shows that K−1 (R) need not be zero.

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6.6. Reid’s method This example clarifies the calculations by L. Reid [R, p. 198]. Let F (x, y, z) be a homogeneous polynomial of degree d, defining a reduced curve C in P2k . Then the equation f (x, y) = F (x, y, 1) defines an affine curve C0 . Consider the subring R = k[f, xf, yf ] of k[x, y]. Then the map from k[u, v, w] to k[x, y] sending u, v, and w to xf , yf , and f determines an isomorphism R∼ =

k[u, v, w] . − F (u, v, w)

w d+1

If C has d distinct points at infinity, then R is a 2-dimensional normal domain whose only singularity is at the maximal ideal m. This is easy to see from the Jacobian criterion, using Euler’s formula u(∂F /∂u) + v(∂F /∂v) + w(∂F /∂w) = d · F to see that the singularities lie on the locus w = F (u, v, 0) = 0. Let X denote the blow-up of X = Spec(R) along m. The affine open D+ (wt) of X is just Spec(k[x, y]). From this it is easy to see that X is nonsingular and that the exceptional fiber of X → X is the curve C = Proj(k[u, v, w]/F ). Hence K−2 (R) ∼ = K−2 (X ) ∼ = K−1 (C). If we choose F so that C is a node, or any other plane curve with K−1 (C) = Z, then K−2 (R) = Z. Suppose that F (1, 0, 0) = 0. Then v, w is a regular sequence, and the ideal I = (v, w) is a reduction of m, since ud ∈ I md−1 and hence md = I md−1 . Set A = R/I = k[u]/(ud ), and let X be the blow-up along I . Not only is X finite over X , but the ideal md−1 R[mt] lies in the conductor from R[mt] to R[I t]. Therefore we may analyze Pic(Y ) as in Section 2 in order to describe K−1 (R). 7. SK1 -regularity for nonreduced curves This section is in some sense a technical continuation of Section 2, where we considered K0 -regularity for nonreduced curves. It is used in Section 8 to discuss K0 regularity of normal surfaces. Recall that there is a natural decomposition K1 (Y ) = U (Y ) ⊕ SK1 (Y ), where U (Y ) denotes the global units of Y . Using Lemma 2.5 and Porism 2.5.1, the argument ¯ replaced by Ki (Y, N ) yields an exact sequence for of [PW1, p. 369] with Ki (D) every closed subscheme Y defined by an ideal N of OY : K2 (Y ) −→ K2 (Y ) −→ SK1 (Y, N ) −→ SK1 (Y ) −→ SK1 (Y ) −→ 0.

(7.0)

Replacing Y with the product Y [T ] of Y with Spec(k[t1 , . . . , tp ]), this sequence is · · · K2 Y [T ] −→ SK1 Y [T ], N [T ] −→ SK1 (Y [T ]) −→ SK1 (Y [T ]) −→ 0. (7.0[T ])


27

We begin with a result that could have been included in Lemma 2.5. Recall (say, from [W2]) that if N is a nilpotent ideal of a ring A, then K2 (A, N) is generated by Dennis-Stein symbols "x, a#, where x ∈ N and a ∈ A. Write ⊗Y for ⊗OY . lemma 7.1 For any scheme Y , let N be an ideal of OY with N Then there is an exact sequence of sheaves ψ

2

= 0, defining a subscheme Y .

ϕ

N ⊗Y N −−→ K2 N −→ N ⊗Y CY −→ 0.

Here ψ(x ⊗ y) = "x, y# and ϕ("x, a#) = x ⊗ da for sections x, y of N and a of OY . If Y is a curve, then SK1 (Y, N ) ∼ = H 1 (Y, K2 N ) and there is an exact sequence ψ H 1 Y, N ⊗Y N −−→ H 1 Y, K2 N −→ H 1 Y, N ⊗Y CY −→ 0. ˜ N = (N ⊗Y N )/{x ⊗y +y ⊗x} because In fact, ψ factors through the quotient N ∧ "x, y# + "y, x# = 0 in K2 (A, N) (see [W2, Theorem 1.3]). Proof The first assertion is just the sheafification of [W2, Theorem 1.3]. Now suppose that Y is a curve. The Brown-Gersten spectral sequence used in the proof of Lemma 2.5 yields SK1 (Y, N ) ∼ = H 1 (Y, K2 N ) because of the sequence in Lemma 2.5(4) and the extension 0 −→ H 1 Y, K2 N −→ K1 (Y, N ) −→ H 0 (Y, N ) −→ 0. Applying the right exact H 1 (Y, −) yields the final sequence. Porism 7.1.1 Here is a variation, for π : Y [T ] → Y . Let Ki N [T ] denote the sheaf on Y associated to the presheaf U % → Ki (U [T ], N [T ]). Then SK1 (Y [T ], N [T ]) equals H 1 (Y, K2 N [T ]), by the same argument applied to the Brown-Gersten spectral sequence H p (Y, K−q N [T ]) ⇒ K−p−q (Y [T ], N [T ]) of [TT]. Since H i (Y, N [T ]) is H i (Y, N ) ⊗ k[T ] and since ψ(xt i ⊗ yt j ) = ψ(xt i+j , y) when xy = 0, the proof of Lemma 7.1 yields an exact sequence ψ H 1 Y, N ⊗Y N )⊗k k[T ] −−→ H 1 Y, K2 N [T ] −→ H 1 Y, N ⊗Y π∗ CY [T ] −→ 0. Moreover, N ⊗Y π∗ CY [T ] ∼ = N ⊗Y CY ⊗k k[T ] ⊕ N ⊗k Ck[T ] . corollary 7.2 Assume that Y is a curve over a field k, and assume that N is a locally principal ideal of OY with N 2 = 0, defining a subscheme Y . If char(k) = 2, assume in addition

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that either N ⊆ (ann N )2 or H 1 (Y, N ⊗Y N ) = 0. Then SK1 (Y, N ) ∼ = H 1 Y , N ⊗ Y CY , SK1 Y [T ], N [T ] ∼ = SK1 (Y, N ) ⊗ k[T ] ⊕ H 1 (Y, N ) ⊗ Ck[T ] . Thus SK1 (Y [T ], N [T ]) ∼ = SK1 (Y, N ) if and only if both SK1 (Y, N ) and H 1 (Y, N ) vanish. Proof By [W2, Lemma 1.2 and Theorem 1.3], the assumptions imply that the sheaf map ψ = 0 and that K2 N ∼ = N ⊗Y CY . Similarly, K2 N [T ] ∼ = = N [T ] ⊗ CY [T ] ∼ N ⊗ CY [T ] ⊕ N ⊗ Ck[T ] . The rest follows from Porism 7.1.1. corollary 7.3 Let Y , Y , and N be as in Corollary 7.2. If H 1 (Y, N ) = 0, then SK1 (Y, N ) ∼ = H 1 (Y , N ⊗Y CY /k ) and SK1 (Y [T ], N [T ]) ∼ = SK1 (Y, N ) ⊗ k[T ]. Proof Tensor the “first fundamental exact sequence” [W6, Sequence 9.2.6] with N to get N ⊗k Ck −→ N ⊗Y CY −→ N ⊗Y CY /k −→ 0.

Now apply the right exact functor H 1 (Y , −), and use Corollary 7.2. Porism 7.3.1 More generally, the proof shows that the quotient of SK1 (Y, N ) by the image of ψ and H 1 (Y, N ) ⊗ Ck is H 1 (Y , N ⊗ CY /k ). Remark 7.3.2 If H 1 (Y, N ) is nonzero and if Y is defined over k0 , the proof yields a surjection from NSK1 (Y ) to H 1 (Y, N ) ⊗ Ck/k0 ⊗ tk[t]. This map was constructed by Srinivas [Sr2, Section 3]. Example 7.4 (Fat P1 ) (a) Suppose that C = P1A for some Artinian A, and suppose that C → Y is a nilpotent thickening defined by an ideal I such that L = I /I 2 is an ample line bundle on C. We claim that SK1 (Y ) ∼ = SK1 (P1A ) ∼ = A× . ∼ To see this, recall that CC = OC (−2)⊕(CA ⊗A OC ). Inductively, the claim holds for any subscheme Y of Y defined by a nonzero ideal N ⊆ I i with IN = 0. Then N is a quotient of L i , so H 1 (Y, N ) = H 1 (Y, N ⊗Y N ) = 0. Moreover, we


29

∼ N ⊗Y CC . It follows from Corollary 7.3 that SK1 (Y, N ) = ∼ have N ⊗Y CY = 1 ∼ H (Y, N ⊗ CC/k ) = 0, yielding the inductive step that SK1 (Y ) = SK1 (Y ). (b) Now suppose that X is the blow-up of a surface X along a local complete intersection Y = Spec(A). We claim that for every nilpotent thickening Y of the fiber C = Y ×X X ∼ = P1A we have SK1 (Y ) ∼ = A× and SK1 (Y [T ]) ∼ = A[T ]× . To see this, note that the fiber is defined by an invertible ideal I of OX with conormal bundle I /I 2 ∼ = O (1). If Yn ⊂ X is defined by I n , then Yn ⊂ is defined by N = I n /I n+1 ∼ Yn+1 = OP1 (n). Thus N ⊗ CYn ∼ = OP1 (n − 2). Y Y ) ∼ SK (Y ) ∼ SK (Y ) for Hence SK1 (Yn+1 , N ) = 0, and hence SK1 (Yn+1 = 1 n = 1 1 all n ≥ 1 by Corollary 7.2. By Example 2.10 and Corollary 7.3, we even have SK1 (Yn+1 [T ], N [T ]) = 0, and hence SK1 (Yn [T ]) ∼ = SK1 (Y1 [T ]) ∼ = A[T ]× . proposition 7.5 Let Y be a curve, proper over a perfect field k. Let Y be a subscheme defined by a nilpotent ideal N of OY such that IN = 0, where I is the nilradical. Assume that Pic(Y ) ∼ = Pic(Y ). Then the map SK1 (Y, N ) → SK1 (Y ) factors through the 1 quotient H (Y, N ⊗ CY /k ) of SK1 (Y, N ). Similarly, the map SK1 (Y [T ], N [T ]) → SK1 (Y [T ]) factors through the quotient H 1 (Y, N ⊗ CY /k ) ⊗ k[T ] of SK1 (Y [T ], N [T ]). Proof Let W denote the subgroup of SK1 (Y, N ) generated by the image of ψ and H 1 (Y, N ) ⊗ Ck . We will show that W is in the image of K2 (Y ) → SK1 (Y, N ); by (7.0) this implies that W vanishes in SK1 (Y ). The first assertion will then follow from Remark 7.3.1; the second assertion will be proven analogously, using (7.0[T ]). Since Y is proper, the three rings k = H 0 (Y, OY /I ), A = H 0 (Y, OY ), and B = H 0 (Y, OY /N ) are finite-dimensional over k. As k is perfect, k is smooth and A → k is a split surjection. The assumption on Pic(Y ) implies that B/A surjects onto H 1 (Y, N ). In order to lift elements of H 1 (Y, N ), cover Y by two affine opens Ui = Spec(Ai ) with affine intersection U12 = Spec(A12 ), and set N12 = H 0 (U12 , N ). Then each λ ∈ H 1 (Y, N ) is represented by an n ∈ N12 ; the assumption that λ comes from B/A implies that there are ai ∈ Ai such that a2 = a1 + n in A12 . These ai then define an element b of B. Subtracting an element of k if necessary, we can assume that a1 and b are nilpotent. Hence a1 n = 0. For each s, x ∈ A[T ], the Dennis-Stein symbols "ai s, x# of K2 (Ai [T ]) both map to "bs, x# in K2 (B[T ]), while in K2 (A12 [T ]) we have "a1 s, x#"ns, x# = "a2 s, x# because a1 n = 0. From this we see that the composition

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K2 (B[T ]) −→ K2 (Y [T ]) −→ SK1 Y [T ], N [T ] ∼ = H 1 Y [T ], K2 N [T ] sends "bs, x# to the class σ ∈ H 1 (Y [T ], K2 N [T ]) represented by "ns, x#. When x ∈ H 0 (Y, N ), then σ corresponds to ψ(λs ⊗ x) in Lemma 7.1 and Porism 7.1.1. When x, s ∈ k[T ], σ corresponds to λ ⊗ s dx in H 1 (Y, N ) ⊗ Ck[T ] . Since the s dx form a k-basis of Ck[T ] , the result follows from Porism 7.3.1 or from Lemma 7.1. proposition 7.6 Let C be a seminormal curve over a perfect field k, and let N be an ample line bundle. Then H 1 (C, N ⊗ CC/k ) = 0. Proof Write ι for the inclusion of the singular locus S = Spec(ki ) in C, p : Cj → C for the normalization, and p −1 S = Spec(kij ) for the fiber. Because C is seminormal, the ki and kij are fields. Because k is perfect, each Cki /k and Ckij /k vanishes. Then for certain ki -vector spaces Ti we have exact sequence 0 −→ ⊕ι∗ (Ti ) −→ CC/k −→ ⊕p∗ CCj /k −→ 0.

(7.6.1)

Tensor this sequence with N , and write Nj for the pullback of N to Cj . We have H 1 (C, N ⊗ CC/k ) ∼ = ⊕H 1 (Cj , Nj ⊗ CCj /k ). Since N is ample and since the Cj are smooth, Serre duality yields H 1 (Cj , Nj ⊗ CCj /k ) = 0. We are done. Recall that by T. Vorst’s theorem [Vor2], every seminormal curve C over a perfect field is also SK1 -regular. corollary 7.7 Let Y be a Pic-regular curve over a perfect field k such that C = Yred is defined by an ideal N of OY with N 2 = 0 and such that N is (a quotient of ) an ample line bundle over C. Then SK1 (Y ) ∼ = SK1 (C), and Y is SK1 -regular. Proof By Proposition 2.7, the Pic-regularity of Y implies that C is seminormal, so SK1 (C) = SK1 (C[T ]) by Vorst. By Proposition 7.6, H 1 (C, N ⊗ CC/k ) = 0. But then Theorem 7.5 implies that SK1 (Y ) ∼ = SK1 (C), and SK1 (Y [T ]) ∼ = SK1 (C[T ]). Remark 7.7.1 If char(k) = 2, a stronger assertion holds:SK1 (Y, N ) = 0 and SK1 (Y [T ], N [T ]) = 0.


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This follows from Corollary 7.3, which applies because (by Proposition 2.7) the Picregularity of Y implies that H 1 (C, N ) = 0. theorem 7.8 Let Y be a Pic-regular curve, proper over a perfect field k, such that C = Yred is defined by a locally principal ideal I of OY . Assume that L = I /I 2 is ample on C. Then SK1 (Y ) ∼ = SK1 (C), and Y is SK1 -regular. Proof The case I 2 = 0 is just Corollary 7.7. In general, suppose I n+1 = 0, and let Yn denote the subscheme of Y defined by N = I n ; Yn is Pic-regular by Lemma 2.5. Now N is a quotient of L n , which is ample on C, so Proposition 7.6 yields H 1 (C, N ⊗ CC/k ) = 0. By Proposition 7.5, SK1 (Y [T ]) ∼ = SK1 (Yn [T ]). The result now follows by induction on n. 8. K0 -regularity equals K−1 -regularity for normal surfaces In this section we show that a normal surface X over a perfect field is K0 -regular if and only if it is K−1 -regular. Combined with Theorem 5.9, this verifies [Sr2, Conjecture B] that X is K0 -regular if and only if Pic(nE) ∼ = Pic(E) for all n. 0 (X), and recall that SK0 (X) denotes the Recall that K0 (X) = H 0 (X, Z) ⊕ K 0 (X) → Pic(X). It is well known that normal kernel of the determinant map K p schemes are Pic-regular, so N K0 (X) = N p SK0 (X) for p > 0. If Y is a curve, the determinant map is an isomorphism, so SK0 (Y ) = 0. Combining (3.4.2) with the “units-Pic” sequence yields the following exact sequence, where X → X and Y → Y are as in Theorem 3.1: SK1 (X ) ⊕ SK1 (Y ) −→ SK1 (Y ) −→ SK0 (X ) −→ SK0 (X ) −→ 0.

(8.1)

(When X is affine, this is the sequence in [Bass, p. 490].) There are of course similar exact sequences, which we refer to as N p (8.1), in which “SK” is formally replaced by “N p SK.” Note that if X is regular and p > 0, then the two terms NSKi (X ) vanish. theorem 8.2 Suppose that π : X → X is a resolution of singularities of a normal surface X, with exceptional fibers Ej over the singular points of X. Then for sufficiently large nilpotent thickenings Yj of the exceptional fibers Ej we have N p K0 (X) ∼ = ⊕j N p SK1 Yj .

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Proof By Theorem 1.9, X is finite over some blow-up X of X along a local complete intersection Y = Spec(A). Choosing a conducting subscheme j : Y ⊂ X , we have SK1 (Y ) ∼ = SK1 (P1Y ) ∼ = A× by Example 7.4(b). Now the map j ∗ : SK1 (X ) → SK1 (P1Y ) is onto because it is split by j∗ π ∗ as in the proof of Lemma 5.2. Hence the map N p SK1 (Y ) → N p SK1 (Y ) in sequence N p (8.1) factors through N p SK1 (X ), which is zero for p > 0. The result now follows from sequence N p (8.1). Example 8.3 Let k be a nonperfect field of characteristic 2, and set R = k[x, y, w]/(w 3 −y 2 +αx 2 ), where α ∈ k is an element with no square root. Using [Hart, Exercise III.10.1], one can show that R[1/x] and R[1/y] are regular, so by Serre’s criterion R is a normal domain with one singular point. Applying Reid’s method (see Section 6.6) to the projective curve C defined by f (x, y, z) = y 2 − αx 2 , we see that C is the exceptional fiber of a resolution of singularities X → X = Spec(R). In particular, its conormal bundle is ample. Now C is seminormal but not SK1 -regular, by [Vor2, Theorem A]. Since the √ normalization of C is a projective line P18 over 8 = k( α), the argument of Example 2.12 shows that every nilpotent thickening Y of C on X is Pic-regular. By Proposition 5.1 and Theorem 8.2, R is K−1 -regular but not K0 -regular. We can now establish [Sr3, Conjecture B]. theorem 8.4 Suppose that X is a K−1 -regular surface over a perfect field. Then X is K0 -regular. Proof By Proposition 5.1, our K−1 -regularity assumption implies that every conductor subscheme Y is Pic-regular. Now combine Theorems 7.8 and 8.2. Acknowledgments. I am deeply grateful to Wolmer Vasconcelos for several conversations about reduction ideals, to Joe Lipman for generous correspondence about [Lip1] and [Lip2] and for pointing out [NR], to Mark Walker for several useful conversations centered around Lemma 5.2, to Luca Barbieri-Viale in connection with Theorem 5.3 and Example 6.2, and to Srinivas for useful correspondence about his Conjecture B. I am also indebted to Andrew Nestler and the referee for a careful reading and for suggesting several improvements, including Example 2.10.1.


33

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Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, USA; [email protected]

BRAID GROUP ACTIONS ON DERIVED CATEGORIES OF COHERENT SHEAVES PAUL SEIDEL and RICHARD THOMAS

Abstract This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim X ≥ 2, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

1. Introduction 1.1. Derived categories of coherent sheaves Let X be a smooth complex projective variety, and let D b (X) be the bounded derived category of coherent sheaves. It is an interesting question how much information about X is contained in D b (X). Certain invariants of X can be shown to depend only on D b (X). This is obviously true for K(X), the Grothendieck group of both the abelian category Coh(X) of coherent sheaves and of D b (X). A deep result of D. Orlov [40] implies that the topological ∗ (X) is also an invariant of D b (X); hence, so are the sums of its even K-theory Ktop and odd Betti numbers. Because of the uniqueness of Serre functors (see [2]), the dimension of X, and whether it is Calabi-Yau (ωX ∼ = OX ) or not, can be read off from D b (X). Using Orlov’s theorem quoted above, one can prove that the Hochschild cohomology of X, H H ∗ (X) = Ext∗X×X (O , O ), depends only on D b (X). As pointed out by Kontsevich [30, p. 131], it is implicit in the work of M. Gerstenhaber and DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 1, Received 19 January 2000. Revision received 24 June 2000. 2000 Mathematics Subject Classification. Primary 14J32, 18, 18E30; Secondary 20F36, 53D40. Authors’ work supported by Max Planck Institute of Mathematics, Bonn, and Hertford College, Oxford.

37

38

S. Schack [15] that

SEIDEL AND THOMAS

H H r (X) ∼ =

p+q=r

H p X, q T X .

Thus, for Calabi-Yau varieties, dim H H r (X) = p+q=r hp,n−q (X); in mirror symmetry these are the Betti numbers of the mirror manifold. Finally, a theorem of A. Bondal and Orlov [5] says that if the canonical sheaf ωX or its inverse is ample, X can be entirely reconstructed from D b (X). Contrary to what this list of results might suggest, there are in fact nonisomorphic varieties with equivalent derived categories. The first examples are due to S. Mukai: abelian varieties in [37] and K3 surfaces in [38]. Examples with nontrivial ωX have been found by Bondal and Orlov [4]. This paper is concerned with a closely related object, the self-equivalence group Auteq(D b (X)). Recall that an exact functor between two triangulated categories C, D is a pair (F, νF ) consisting of a functor F : C → D and a natural isomorphism νF : F ◦ [1]C ∼ = [1]D ◦ F (here [1]C , [1]D are the translation functors) with the property that exact triangles in C are mapped to exact triangles in D. The appropriate equivalence relation between such functors is graded natural isomorphism, which means natural isomorphism compatible with the maps νF (see [5, Section 1]). Ignoring set-theoretic difficulties, which are irrelevant for C = D b (X), the equivalence classes of exact functors from C to itself form a monoid. Auteq(C) is defined as the group of invertible elements in this monoid. Known results about Auteq(D b (X)) parallel those for D b (X) itself. It always contains a subgroup A(X) ∼ = (Aut(X)Pic(X))×Z generated by the automorphisms of X, the functors of tensoring with an invertible −1 is ample, sheaf, and the translation. Bondal and Orlov [5] have shown that if ωX or ωX b then Auteq(D (X)) = A(X). Mukai’s arguments in [37] imply that Auteq(D b (X)) is bigger than A(X) for all abelian varieties. (Recent work of Orlov [41] describes Auteq(D b (X)) completely in this case.) Our own interest in self-equivalence groups comes from Kontsevich’s homological mirror conjecture in [30]. One consequence of this conjecture is that, for Calabi-Yau varieties to which mirror symmetry applies, the group Auteq(D b (X)) should be related to the symplectic automorphisms of the mirror manifold. This conjectural relationship is rather abstract and difficult to spell out in concrete examples. Nevertheless, as a first and rather naive check, one can look at some special symplectic automorphisms of the mirror and try to guess the corresponding self-equivalences of D b (X). Having made this guess in a sufficiently plausible way (which means that the two objects show similar behaviour), the next step might be to take some unsolved questions about symplectic automorphisms and translate them into one about Auteq(D b (X)). Using the smoother machinery of sheaf theory, one stands a good chance of solving this analogue; this in turn provides a conjectural answer, or “mirror symmetry prediction,” for the original problem. The present paper is an experiment

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BRAID GROUP ACTIONS

in this mode of thinking. We now state the main results independently of their motivation; the discussion of mirror symmetry is taken up again in Section 1.2. Let X, Y be two (as before, smooth complex projective) varieties. The FourierMukai transform (FMT) by an object P ∈ D b (X × Y ) is the exact functor L P : D b (X) −→ D b (Y ), P (G ) = Rπ2∗ π1∗ G ⊗ P , where π1 : X × Y → X, π2 : X × Y → Y are the projections. This is a very practical way of defining functors. Orlov [40] has proved that any equivalence D b (X) → D b (Y ) can be written as an FMT. Earlier work of A. Maciocia [32] shows that if P is an equivalence, then P must satisfy a partial Calabi-Yau condition, P ⊗ π1∗ ωX ⊗ π2∗ ωY−1 ∼ = P . T. Bridgeland [7] provides a partial converse to this. Now take an object E ∈ D b (X) which is a complex of locally free sheaves. We define the twist functor TE : D b (X) → D b (X) as the FMT with P = Cone η : E ∨ E −→ O , (1.1) where E ∨ is the dual complex, is the exterior tensor product, ⊂ X × X is the diagonal, and η is the canonical pairing. Since quasi-isomorphic E give rise to isomorphic functors TE , one can use locally free resolutions to extend the definition to arbitrary objects of D b (X). Definition 1.1 (a) E ∈ D b (X) is called spherical if HomrD b (X) (E , E ) is equal to C for r = 0, dim X and zero in all other degrees, and if in addition E ⊗ ωX ∼ = E.

(b) An (Am )-configuration, m ≥ 1, in D b (X) is a collection of m spherical objects E1 , . . . , Em such that 1 |i − j | = 1, ∗ dimC HomD b (X) Ei , Ej = 0 |i − j | ≥ 2. Here, as elsewhere in the paper, Homr (E , F ) stands for Hom(E , F [r]), and Hom∗ (E , F ) is the total space r∈Z Homr (E , F ). theorem 1.2 The twist TE along any spherical object E is an exact self-equivalence of D b (X). Moreover, if E1 , . . . , Em is an (Am )-configuration, the twists TEi satisfy the braid relations up to graded natural isomorphism: TEi TEi+1 TEi ∼ = TEi+1 TEi TEi+1 ∼ TEi TEj = TEj TEi

for i = 1, . . . , m − 1, for |i − j | ≥ 2.

40

SEIDEL AND THOMAS

We should point out that the first part, the invertibility of TE , was also known to Kontsevich, Bridgeland, and Maciocia. Let ρ be the homomorphism from the braid group Bm+1 to Auteq(D b (X)) defined by sending the standard generators g1 , . . . , gm ∈ Bm+1 to TE1 , . . . , TEm . We call this a weak braid group action on D b (X). (There is a better notion of a group action on a category which requires the presence of certain additional natural transformations (see [11]); we have not checked whether these exist in our case.) There is an induced representation ρ∗ of Bm+1 on K(X). Concretely, the twist along an arbitrary E ∈ D b (X) acts on K(X) by

(1.2) (TE )∗ (y) = y − [E ], y [E ], where [F ], [G ] = i (−1)i dim Homi (F , G ) is the Mukai pairing (see [38]) or “Euler form.” If dim X is even, then ρ∗ factors through the symmetric group Sm+1 . The odd-dimensional case is slightly more complicated, but still ρ∗ is far from being faithful, at least if m is large. For ρ itself we have the following contrasting result. theorem 1.3 Assume that dim X ≥ 2. Then the homomorphism ρ generated by the twists in any (Am )-configuration is injective. The assumption dim X = 1 cannot be removed; indeed, there is a B4 -action on the derived category of an elliptic curve which is not faithful (see Section 3.4). 1.2. Homological mirror symmetry and self-equivalences We begin by recalling Kontsevich’s homological mirror conjecture in [30]. On the one hand, one takes Calabi-Yau varieties X and their derived categories D b (X). On the other hand, using entirely different techniques, it is thought that one can attach to any compact symplectic manifold (M, β), with zero first Chern class a triangulated category, the derived Fukaya category D b Fuk(M, β). (Despite the notation, this is not constructed as the derived category of an abelian category.) Kontsevich’s conjecture is that whenever X and (M, β) form a mirror pair, there is a (noncanonical) exact equivalence (1.3) D b (X) ∼ = D b Fuk(M, β). A more prudent formulation would be to say that (1.3) should hold for the generally accepted constructions of mirror manifolds. Before discussing this conjecture further, we need to explain what D b Fuk(M, β) looks like. This is necessarily a tentative description since a rigorous definition does not exist yet. Moreover, for simplicity we have omitted some of the more technical aspects.

BRAID GROUP ACTIONS

41

Let (M, β) be as before, of real dimension 2n. To simplify things we assume that π1 (M) is trivial; this excludes the case of the 2-torus, so that n ≥ 2. Recall that a submanifold Ln ⊂ M is called Lagrangian if β|L ∈ %2 (L) is zero. Fol˜ which are lowing Kontsevich [30, p. 134], one considers objects, denoted by L, Lagrangian submanifolds with some extra structure. We call such objects graded Lagrangian submanifolds and the extra structure the grading. This grading amounts approximately to an integer choice. In fact, there is a free Z-action, denoted by ˜ ] for j ∈ Z, on the set of graded Lagrangian submanifolds; and if L is L˜ → L[j a connected Lagrangian submanifold, all its possible gradings (assuming that there are any) form a single orbit of this action. For details we refer to [49]. For any pair (L˜ 1 , L˜ 2 ) of graded Lagrangian submanifolds, one expects to have a Floer cohomology group H F ∗ (L˜ 1 , L˜ 2 ), which is a finite-dimensional graded R-vector space satisfying H F ∗ (L˜ 1 , L˜ 2 [j ]) = H F ∗ (L˜ 1 [−j ], L˜ 2 ) = H F ∗+j (L˜ 1 , L˜ 2 ). Defining this is a difficult problem; a fairly general solution has been announced recently by K. Fukaya, Kontsevich, Y.-G. Oh, H. Ohta, and K. Ono. The most essential property of D b Fuk(M, β) is that any graded Lagrangian submanifold L˜ defines an object in this category. The translation functor (which is part of the structure of D b Fuk(M, β) as a triangulated category) acts on such objects ˜ by L˜ → L[1]. The morphisms between two objects of this kind are given by the degree zero Floer cohomology with complex coefficients: HomD b Fuk(M,β) L˜ 1 , L˜ 2 = H F 0 L˜ 1 , L˜ 2 ⊗R C. (Floer groups in other degrees can be recovered by changing L˜ 2 to L˜ 2 [j ].) Composition of such morphisms is given by certain products on Floer cohomology, which were first introduced by Donaldson. There is also a slight generalisation of this: any ˜ E) consisting of a graded Lagrangian submanifold, together with a flat unipair (L, tary vector bundle E on the underlying Lagrangian submanifold, defines an object of D b Fuk(M, β). The morphisms between such objects are a twisted version of Floer cohomology. It is important to keep in mind that D b Fuk(M, β) contains many objects other than those that we have described. This is necessarily so because it is triangulated; there must be enough objects to complete each morphism to an exact triangle, and these objects do not usually have a direct geometric meaning. However, ˜ E) generate the category D b Fuk(M, β) it is expected that the objects of the form (L, in some sense. Remark 1.4 In the traditional picture of mirror symmetry, M carries a C-valued closed 2-form βC with real part β. What we have said concerns the Fukaya category for im(βC ) = 0. Apparently, the natural generalisation to im(βC ) = 0 would be to take objects

42

SEIDEL AND THOMAS

˜ E, A) consisting of a graded Lagrangian submanifold L, ˜ a complex vector bun(L, dle E on the underlying Lagrangian submanifold L, and a unitary connection A on E with curvature FA = −βC | L ⊗ idE . The point is that to any map w : (D 2 , ∂D 2 ) → (M, L) one can associate a complex number trace monodromy of A around w | ∂D 2 ∗ exp − w βC , rank(E) D2 which is invariant under deformations of w. These numbers, as well as certain variations of them, would be used as weights in the counting procedure that underlies the definition of Floer cohomology. For simplicity, we stick to the case im(βC ) = 0 in our discussion. In parallel with graded Lagrangian submanifolds, there is also a notion of graded symplectic automorphisms; in fact, these are just a special kind of graded Lagrangian submanifolds on (M, −β) × (M, β). The graded symplectic automorphisms form a topological group Sympgr (M, β) that is a central extension of the usual symplectic automorphism group Symp(M, β) by Z. Sympgr (M, β) acts naturally on the set of graded Lagrangian submanifolds. Moreover, the central subgroup Z is generated by a graded symplectic automorphism denoted by [1], which maps each graded Lagrangian ˜ submanifold L˜ to L[1]; we refer again to [49] for details. Because D b Fuk(M, β) is defined in what are essentially symplectic terms, every graded symplectic automorphism of M induces an exact self-equivalence of it. Moreover, an isotopy of graded symplectic automorphisms gives rise to an equivalence between the induced functors. Thus one has a canonical map π0 Sympgr (M, β) −→ Auteq D b Fuk(M, β) . Now we return to Kontsevich’s conjecture. Assume that (M, β) has a mirror partner X such that (1.3) holds. Then there is an isomorphism between Auteq(D b Fuk(M, β)) and Auteq(D b (X)). Combining this with the canonical map above yields a homomorphism (1.4) µ : π0 Sympgr (M, β) −→ Auteq D b (X) . Somewhat oversimplifying, and ignoring the conjectural nature of the whole discussion, one can say that symplectic automorphisms of M induce self-equivalences of the derived category of coherent sheaves on its mirror partner. Note that the map µ depends on the choice of equivalence (1.3) and hence is not canonical. Remark 1.5 One can see rather easily that the central element [1] ∈ Sympgr (M, β) induces the

BRAID GROUP ACTIONS

43

translation functor on D b Fuk(M, β) and hence on D b (X). Passing to the quotient yields a map µ¯ : π0 Symp(M, β) −→ Auteq D b (X) /(translations). This simplified version may be more convenient for those readers who are unfamiliar with the “graded symplectic” machinery. 1.3. Dehn twists and mirror symmetry A Lagrangian sphere in (M, β) is a Lagrangian submanifold S ⊂ M which is diffeomorphic to S n . One can associate to any Lagrangian sphere a symplectic automorphism τS called the generalised Dehn twist along S, which is defined by a local construction in a neighbourhood of S. (See [48] or [49] for details; strictly speaking, τS depends on various choices, but since the induced functor on D b Fuk(M, β) is expected to be independent of these choices, we ignore them in our discussion.) These maps are symplectic versions of the classical Picard-Lefschetz transformations. In particular, their action on H∗ (M) is given by x − ([S] · x)[S] if dim(x) = n, (τS )∗ (x) = (1.5) x otherwise, where · is the intersection pairing twisted by a dimension-dependent sign. As explained in [49, Section 5b], τS has a preferred lift τ˜S ∈ Sympgr (M, β) to the graded symplectic automorphism group. Suppose that (M, β) has a mirror partner X such that Kontsevich’s conjecture (1.3) holds. Choose some lift S˜ of S to a graded Lagrangian ˜ Then submanifold, and let E ∈ D b (X) be the object that corresponds to S. ˜ S˜ = H F ∗ S, ˜ S˜ ⊗R C. (1.6) Hom∗D b (X) (E , E ) ∼ = Hom∗D b Fuk(M,β) S, ˜ S) ˜ is isomorphic to the ordinary cohomology The Floer cohomology group H F ∗ (S, ∗ H (S; R); this is not true for general Lagrangian submanifolds, but it holds for spheres. Therefore E must be a spherical object. (This motivated our use of the word spherical.) A natural conjecture about the homomorphism µ introduced in Section 1.2 is that (1.7) µ [τ˜S ] = [TE ], where TE is the twist functor as defined in Section 1.1. Roughly speaking, the idea is that twist functors and generalised Dehn twists correspond to each other under mirror symmetry. At present this is merely a guess, which can be motivated, for example, by comparing (1.2) with (1.5). But supposing that one wanted to actually prove this claim, how should one go about it? The first step would be to observe that for any

44

SEIDEL AND THOMAS

F ∈ D b (X) there is an exact triangle

Hom∗ (E , F ) ⊗C E

F

TE (F )

[1]

Here Hom∗ (E , F ) is the graded group of homs in D b (X), Hom∗ (E , F )⊗C , E is the corresponding direct sum of shifted copies of E , and the first arrow is the evaluation map. This exact triangle determines TE (F ) up to isomorphism; moreover, it does so in purely abstract terms, which involves only the triangulated structure of the category D b (X). Hence, if there was an analogous abstract description of the action of τ˜S on D b Fuk(M, β), one could indeed prove (1.7). (This is slightly imprecise since it ignores a technical problem about nonfunctoriality of cones in triangulated categories.) The first step towards such a description will be provided in [50]. Note that here, for the first time in our discussion of mirror symmetry, we have made essential use of the triangulated structure of the categories. Now define an (Am )-configuration of Lagrangian spheres in (M, β) to be a collection of m ≥ 1 pairwise transverse Lagrangian spheres S1 , . . . , Sm ⊂ M such that 1 |i − j | = 1, |Si ∩ Sj | = (1.8) 0 |i − j | ≥ 2. Such configurations occur in Kähler manifolds that can be degenerated into a manifold with a singular point of type (Am ) (see [48] or [28]). The generalised Dehn twists τ˜S1 , . . . , τ˜Sm along such spheres satisfy the braid relations up to isotopy inside Sympgr (M, β). For n = 2, and ignoring the issue of gradings, this was proved in [48, appendix]; the argument given there can be adapted to yield the slightly sharper and more general statement that we are using here. Thus, by mapping the standard generators of the braid group to the classes [τ˜Si ], one obtains a homomorphism from Bm+1 to π0 (Sympgr (M, β)). It is a difficult open question in symplectic geometry whether this homomorphism, which we denote by ρ , is injective (see [28] for a partial result). We now see what mirror symmetry has to say about this. Assume, as before, that Kontsevich’s conjecture holds, and let E1 , . . . , Em ∈ D b (X) be the objects corresponding to some choice of gradings S˜1 , . . . , S˜m for the Sj . We already know that each Ei is a spherical object. An argument similar to (1.6) but based on (1.8) shows that E1 , . . . , Em is an (Am )-configuration in D b (X) in the sense of Definition 1.1. Hence the twist functors TEi satisfy the braid relations (see Theorem 1.2) and generate a homomorphism ρ from Bm+1 to Auteq(D b (X)). Assuming that our claim (1.7) is true, one would have a commutative diagram

45

BRAID GROUP ACTIONS

Bm+1

π0 Sympgr (M, β)

ρ

ρ

µ

Auteq D b (X)

Since dimC X = n ≥ 2, we have Theorem 1.3, which says that ρ is injective. In the diagram above this would clearly imply that ρ is injective. Thus we are led to a conjectural answer “based on mirror symmetry” to a question of symplectic geometry. conjecture 1.6 Let (M, β) be a compact symplectic manifold with π1 (M) trivial and c1 (M, β) = 0, and let (S1 , . . . , Sm ) be an (Am )-configuration of Lagrangian spheres in M for some m ≥ 1. Then the map ρ : Bm+1 → π0 (Sympgr (M, β)) generated by the generalised Dehn twists τ˜S1 , . . . , τ˜Sm is injective. 1.4. A survey of the paper Section 2 introduces spherical objects and twist functors for derived categories of fairly general abelian categories. The main result is the construction of braid group actions, Theorem 2.17. Section 3.1 explains how the abstract framework specializes in the case of coherent sheaves; this recovers the definitions presented in Section 1.1 and, in particular, in Theorem 1.2. More generally, in Section 3.2 we consider singular and quasi-projective varieties, as well as equivariant sheaves on varieties with a finite group action; the latter give rise to what are probably the simplest examples of our theory. In Section 3.3 we present a more systematic way of producing spherical objects, which exploits their relations with the (much studied) exceptional objects on Fano varieties. Elliptic curves provide the only example where both sides of the homological mirror conjecture are completely understood; in Section 3.4 the group of symplectic automorphisms and the group of autoequivalences of the derived category are compared in an explicit way. Section 3.5 gives more explicit examples on K3 surfaces. Finally, Section 3.6 puts our results in the framework of mirror symmetry for singularities; this was the underlying motivation for much of this work. Section 4 contains the proof of the faithfulness result, Theorem 2.18. For the benefit of the reader, we provide here an outline of the argument, in the more concrete situation stated as Theorem 1.3; the general case does not differ greatly from this. Let E1 , . . . , Em be a collection of spherical objects in D b (X), and set E = E1 ⊕ · · · ⊕ Em . For a fixed m and dimension n of the variety, the endomorphism algebra End∗ (E ) = Hom∗ Ei , Ej i,j

46

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is essentially the same for all (E1 , . . . , Em ). More precisely, after possibly shifting each Ei by some amount, one can achieve that End∗ (E ) is equal to a specific graded algebra Am,n depending only on m, n. Moreover, one can define a functor . naive : D b (X) → Am,n -mod into the category of graded modules over Am,n by mapping F to Hom∗ (E , F ). By a result of [28], the derived category D b (Am,n -mod) carries a weak action of Bm+1 , and one might hope that . naive would be compatible with these two actions. A little thought shows that this cannot possibly be true; Am,n -mod can be embedded into D b (Am,n -mod) as the subcategory of complexes of length 1, but the braid group action on D b (Am,n -mod) does not preserve this subcategory. Nevertheless, the basic idea can be saved, at the cost of introducing some more homological algebra. Take resolutions Ei of Ei by bounded below complexes of injective quasi-coherent sheaves. Then one can define a differential graded algebra end(E ) whose cohomology is End∗ (E ). The quasi-isomorphism type of end(E ) is independent of the choice of resolutions, so it is an invariant of the (Am )-configuration E1 , . . . , Em . As before, there is an exact functor hom(E , −) : D b (X) → D(end(E )) to the derived category of differential graded modules over end(E ). Now assume that end(E ) is formal, that is to say, quasi-isomorphic to the differential graded algebra Am,n = (Am,n , 0) with zero differential. Quasi-isomorphic differential graded algebras have equivalent derived categories, so what one obtains is an exact functor . : D b (X) −→ D Am,n , which replaces the earlier . naive . A slight modification of the arguments of [28] shows that there is a weak braid group action on D(Am,n ); moreover, in contrast to the situation above, the functor . now relates the two braid group actions. Still borrowing from [28], one can interpret the braid group action on D(Am,n ) in terms of low-dimensional topology and, more precisely, geometric intersection numbers of curves on a punctured disc. This leads to a strong faithfulness result for it, which through the functor . implies the faithfulness of the original braid group action on D b (X). This argument by reduction to the known case of D(Am,n ) hinges on the formality of end(E ). We prove that this assumption is always satisfied when n ≥ 2. This has nothing to do with the geometric origin of end(E ); in fact, what we show is that Am,n is intrinsically formal for n ≥ 2, which means that all differential graded algebras with this cohomology are formal. There is a general theory of intrinsically formal algebras, which goes back to the work of S. Halperin and J. Stasheff [19] in the commutative case; the Hochschild cohomology computation necessary to apply this theory to Am,n is the final step in the proof of Theorem 2.18.

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2. Braid group actions 2.1. Generalities Fix a field k; all categories are assumed to be k-linear. If S is an abelian category, Ch(S) is the category of cochain complexes in S and cochain maps, K(S) is the corresponding homotopy category (morphisms are homotopy classes of cochain maps), and D(S) is the derived category. The variants involving bounded (below, above, or on both sides) complexes are denoted by Ch+ (S), Ch− (S), Chb (S), and so on. Let (Cj , δj )j ∈Z be a cochain complex of objects and morphisms in Ch(S), that is to say, Cj ∈ Ch(S) and δj ∈ HomCh(S) (Cj , Cj +1 ) satisfying δj +1 δj = 0. Such a complex is exactly the same as a bicomplex in S. In this case we write {· · · → C−1 → C0 → C1 → · · · } for the associated total complex, obtained by collapsing the bigrading; this is a single object in Ch(S). The same notation is applied to bicomplexes of objects of Ch(S) (which are triple complexes in S). For C, D ∈ Ch(S), let hom(C, D) be the standard cochain complex of k-vector spaces whose cohomology is H i hom(C, D) = HomiK(S) (C, D); that is, homi (C, D) i (φ) = dD φ − (−1)i φdC . Now suppose = j ∈Z HomS (C j , D j +i ) with dhom(C,D) that S contains infinite direct sums and products. Given an object C ∈ Ch(S) and a cochain complex b of k-vector spaces, one can form the tensor product b ⊗ C and the complex of linear maps lin(b, C), both of which are again objects of Ch(S). They are defined by choosing a basis of b and taking a corresponding direct sum (for b ⊗ C) or product (for lin(b, C)) of shifted copies of C, with a differential that combines db and dC . The outcome is independent of the chosen basis up to canonical isomorphism. The definition of b ⊗ C is clear, but for lin(b, C) there are two possible choices of signs. Ours is fixed to fit in with an evaluation map b ⊗ lin(b, C) → C. To clarify the issue, we now spell out the definition. Take a homogeneous basis (xi )i∈I of the q total space b, and write db (xi ) = j zj i xj . Then linq (b, C) = i∈I Ci , where Ci is a copy of C shifted by deg(xi ). The differential d q : linq (b, C) → linq+1 (b, C) q q q+1 has components dj i : Ci → Cj , which are given by q dj i

 deg(xi ) d   C (−1) deg(x ) i = (−1) zij · idC   0

i = j, deg(xi ) = deg(xj ) + 1, otherwise.

One can verify that the map b ⊗ lin(b, C) → C, xj ⊗ (ci )i∈I → cj , is indeed a morphism in Ch(S). Moreover, there are canonical monomorphic cochain maps

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b ⊗ hom(D, C) −→ hom(D, b ⊗ C), hom(D, C) ⊗ b −→ hom lin(b, D), C , hom B, lin(b, C) ⊗ D −→ lin b, hom(B, C) ⊗ D ,

(2.1)

where b is as before and B, C, D ∈ Ch(S). These maps are isomorphisms if b is finite-dimensional, and they are quasi-isomorphisms if b has finite-dimensional cohomology. From now on S is an abelian category and S ⊂ S a full subcategory such that the following conditions hold: (C1) S is a Serre subcategory of S (this means that any subobject and quotient object of an object in S lies again in S and that S is closed under extension); (C2) S contains infinite direct sums and products; (C3) S has enough injectives, and any direct sum of injectives is again injective (this is not a trivial consequence of the definition of an injective object); (C4) for any epimorphism f : A A with A ∈ S and A ∈ S , there is a B ∈ S and a g : B → A such that fg is an epimorphism (because S is a Serre subcategory, g may be taken to be mono): g

B

A f

A

lemma 2.1 Let X be a noetherian scheme over k, and let S = Qco(X), S = Coh(X) be the categories of quasi-coherent, respectively, coherent sheaves. Then properties (C1)– (C4) are satisfied. Proof (C1) and (C2) are obvious. S has enough injectives by [20, Chapter II, Theorem 7.18]. Moreover, X is locally noetherian, which implies that direct sums of injectives are again injective (see [20, p. 121] and the references quoted therein). This proves (C3). Finally, we need to verify that a diagram as in (C4) with A quasi-coherent and A coherent can be completed with a coherent sheaf B . Such a B certainly exists locally, and, replacing it by its image in A (which is also coherent), we may extend it to be a coherent subsheaf on all of X (see [17, Chapter I, Theorem 6.9.7]). Since X is quasi-compact, repeating this a finite number of times and taking the union yields a B whose map to A is globally surjective. As indicated by this example, our main interest is in D b (S ). However, we find it

49

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convenient to replace all complexes by injective resolutions. These resolutions may exist only in S, and they are not necessarily bounded. The precise category we want to work with is defined as follows. Definition 2.2 K ⊂ K + (S) is the full subcategory whose objects are those bounded below cochain complexes C of S-injectives which satisfy H i (C) ∈ S for all i, and H i (C) = 0 for i 0. We now prove, in several steps, that K is equivalent to D b (S ). First, let D ⊂ D + (S) be the full subcategory of objects whose cohomology has the same properties as in Definition 2.2. The assumption that S has enough injectives implies that the obvious functor K → D is an equivalence. Now let ChbS (S) be the category of bounded b (S) be the cochain complexes in S whose cohomology objects lie in S , and let DS b corresponding full subcategory of D (S). It is a standard result (proved by truncating b (S) → D is an equivalence. The cochain complexes) that the obvious functor DS b (S) and D b (S ). final step (and the only nontrivial one) is to relate DS lemma 2.3 For any C ∈ ChbS (S), there is an E ∈ Chb (S ) and a monomorphic cochain map ι : E → C which is a quasi-isomorphism. Proof Recall that, as an abelian category, S has fibre products. The fibre product of two maps f1 : A1 → A, f2 : A2 → A is the kernel of f1 ⊕ 0 − 0 ⊕ f2 : A1 ⊕ A2 → A. If f1 is mono (thought of as an inclusion), we write f2−1 (A1 ) for the fibre product, and if both f1 and f2 are mono, we write A1 ∩ A2 . In the latter case, one can also define the sum A1 + A2 as the image (kernel of the map to the cokernel) of f1 ⊕ f2 . Let N be the largest integer such that C N = 0. Set E n = 0 for all n > N. For n ≤ N, define E n ⊂ C n (for brevity, we write the monomorphisms as inclusions) inductively as follows. By invoking (C4), one finds subobjects F n , Gn ⊂ C n which lie in S and complete the diagrams n −1 n+1 ker dCn dC E dCn

Fn

E n+1 ∩ im dCn

and

Gn

H n (C)

Set E n = F n + Gn (this is again in S ), and define dEn = dCn | E n . Since E n is a subobject of C n for any n, E is a bounded complex. Consider the obvious map

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SEIDEL AND THOMAS

j n : ker dEn = E n ∩ ker dCn → H n (C). The definition of Gn implies that j n is an epimorphism, and the definition of F n−1 yields ker j n = E n ∩ im dCn−1 = im dEn−1 . It follows that the inclusion induces an isomorphism H ∗ (E) ∼ = H ∗ (C). From Lemma 2.3 it now follows by standard homological algebra (see [14, Proposition b (S) is an equivalence of categories. III.2.10]) that the obvious functor D b (S ) → DS Combining this with the remarks made above, one gets the following proposition. proposition 2.4 There is an exact equivalence (canonical up to natural isomorphism) D b (S ) ∼ = K. 2.2. Twist functors and spherical objects Definition 2.5 Let E ∈ K be an object satisfying the following finiteness conditions: (K1) E is a bounded complex; (K2) for any F ∈ K, both Hom∗K (E, F ) and Hom∗K (F, E) have finite (total) dimension over k. Then we define the twist functor TE : K → K by ev (2.2) TE (F ) = hom(E, F ) ⊗ E −−→ F . This expression requires some explanation; ev is the obvious evaluation map. The grading is such that if one ignores the differential, TE (F ) = F ⊕(hom(E, F )⊗E)[1]. In other words, TE (F ) is the cone of ev. Since E is bounded and F is bounded below, hom(E, F ) is again bounded below. Hence hom(E, F ) ⊗ E is a bounded below complex of injectives in S. (This uses property (C3) of S.) Its cohomology H ∗ (hom(E, F ) ⊗ E) is isomorphic to Hom∗K (E, F ) ⊗ H ∗ (E) (because hom(E, F ) is quasi-isomorphic to Hom∗K (E, F ), which is finite-dimensional) and so is bounded, and the finiteness conditions imply that each cohomology group lies in S . Therefore hom(E, F ) ⊗ E lies in K, and the same holds for TE (F ). The functoriality of TE is obvious, and one sees easily that it is an exact functor. Actually, for any F, G ∈ K there is a canonical map of complexes (TE )∗ : hom(F, G) → hom(TE (F ), TE (G)). In fancy language, this means that TE is functorial on the differential graded category that underlies K. proposition 2.6 If two objects E1 , E2 ∈ K satisfying (K1), (K2) are isomorphic, the corresponding functors TE1 , TE2 are isomorphic.

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Proof Take cones of the rows of the following commutative diagram: hom(E1 , F ) ⊗ E1

F

hom(E2 , F ) ⊗ E1

F

hom(E2 , F ) ⊗ E2

F

Here the vertical arrows are induced by a quasi-isomorpism of complexes E1 → E2 .

Note also that TE[j ] is isomorphic to TE for any j ∈ Z. Definition 2.7 For an object E as in Definition 2.5, we define the dual twist functor TE : K → K by TE (F ) = {ev : F → lin(hom(F, E), E)}. Here the grading is such that F lies in degree zero; ev is again some kind of evaluation map. To write it down explicitly, choose a homogeneous basis (ψi ) of hom(F, E). q Then linq (hom(F, E), E) = i Ei , where Ei is a copy of E[deg(ψi )], and the ith component of ev is simply ψi itself. TE is again an exact functor from K to itself. lemma 2.8 TE is left adjoint to TE . Proof Using the maps from (2.1) and condition (K2), one constructs a chain of natural (in F, G ∈ K) quasi-isomorphisms hom F, TE (G) = hom F, hom(E, G) ⊗ E −→ hom(F, G) ←− hom(E, G) ⊗ hom(F, E) −→ hom(F, G) −→ hom lin hom(F, E), E , G −→ hom(F, G) = hom TE (F ), G . Here the chain map hom(E, G) ⊗ hom(F, E) → hom(F, G) is just composition. The reader may easily check that the required diagrams commute. Taking H 0 on both sides yields a natural isomorphism HomK (F, TE (G)) ∼ = HomK (TE (F ), G).

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Definition 2.9 An object E ∈ K is called n-spherical for some n > 0 if it satisfies (K1), (K2) of Definition 2.5 and, in addition, (K3) HomiK (E, E) is equal to k for i = 0, n and zero in all other degrees; j n−j (K4) the composition HomK (F, E) × HomK (E, F ) → HomnK (E, E) ∼ = k is a nondegenerate pairing for all F ∈ K, j ∈ Z. One can also define zero-spherical objects; these are objects E for which Hom∗K (E, E) j is 2-dimensional and concentrated in degree zero, and such that the pairings HomK −j (E, F ) × HomK (F, E) → Hom0K (E, E)/k · idE are nondegenerate. (This means in particular that Hom0K (E, E) is isomorphic to k[t]/t 2 as a k-algebra.) We do not pursue this further; the interested reader can easily verify that the proof of the next proposition extends to this case. proposition 2.10 If E is n-spherical for some n > 0, both TE TE and TE TE are naturally isomorphic to the identity functor IdK . In particular, TE is an exact self-equivalence of K. Proof 1 TE TE (F ) is a total complex    hom(E, F ) ⊗ E        

δ

  hom E, lin hom(F, E), E ⊗ E     γ

α

F

β

lin hom(F, E), E

    

(2.3)

Here α = ev, β = ev , γ is a map induced by ev, and δ is a map induced by ev . We need to know a little more about δ. By the very definition of ev by duality, δ’s induced map on cohomology Hom∗K (E, F ) ⊗ H ∗ (E) −→ Hom∗K (F, E)∨ ⊗ Hom∗K (E, E) ⊗ H ∗ (E)

(2.4)

is dual to the composition Hom∗K (F, E) ⊗ Hom∗K (E, F ) → Hom∗K (E, E), tensored with the identity map on H ∗ (E). This second pairing is, by the conditions (K3) and (K4) on E, perfect when we divide Hom∗K (E, E) by its degree zero piece (k · idE ). Thus the following modification of the map (2.4), Hom∗K (E, F ) ⊗ H ∗ (E) −→ Hom∗K (F, E)∨ ⊗

Hom∗K (E, E) ⊗ H ∗ (E), k · idE

is an isomorphism. 1

We thank one of the referees for simplifying our original proof of this result.

(2.5)

BRAID GROUP ACTIONS

53

We now enlarge slightly the object in the top right-hand corner of (2.3) to produce a new, quasi-isomorphic, complex QE (F ). The last equation in (2.1) gives a map hom(E, lin(hom(F, E), E)) ⊗ E @→ lin(hom(F, E), hom(E, E) ⊗ E). Since hom(F, E) has finite-dimensional cohomology, this is a quasi-isomorphism. The map γ¯ : lin(hom(F, E), hom(E, E) ⊗ E) → lin(hom(F, E), E) induced by ev : hom(E, E) ⊗ E → E naturally extends γ . In fact, γ¯ splits canonically. Define the map φ : lin(hom(F, E), E) → lin(hom(F, E), hom(E, E) ⊗ E) induced by k → hom(E, E), 1 → idE . From the definition of γ¯ , it follows that γ¯ ◦ φ = id. This splitting gives a way of embedding an acyclic complex {id : lin(hom(F, E), E) → lin(hom(F, E), E)} into our enlarged complex QE (F ); the cokernel is hom(E, E) δ⊕α ⊗E ⊕F . hom(E, F ) ⊗ E −−−→ lin hom(F, E), k · idE There is an obvious map of F to this, and everything we have done is functorial in F . Thus to prove that TE TE ∼ = IdK we are left with showing that the cokernel hom(E, E) δ ⊗E (2.6) hom(E, F ) ⊗ E −→ lin hom(F, E), k · idE is acyclic; that is, the arrow induces an isomorphism on cohomology. But passing to cohomology yields (2.5), which we already noted was an isomorphism. The proof that TE TE ∼ = IdK is similar; one passes from TE TE (F ) to a quasiisomorphic but slightly smaller object, which then has a natural map to F . The details are almost the same as before, and we leave them to the reader. 2.3. The braid relations lemma 2.11 Let E1 , E2 ∈ K be two objects such that E1 satisfies conditions (K1), (K2) of Definition 2.5 and E2 is n-spherical for some n > 0. Then TE2 (E1 ) also satisfies (K1), (K2), and TE2 TE1 is naturally isomorphic to TTE2 (E1 ) TE2 . Proof Since E1 and E2 are bounded complexes, so are hom(E1 , E2 ) and TE2 (E1 ). Lemma 2.8 says that Hom∗K (F, TE2 (E1 )) ∼ = Hom∗K (TE 2 (F ), E1 ). By assumption on E1 , this implies that Hom∗K (F, TE2 (E1 )) is always finite-dimensional. Similarly, the finitedimensionality of Hom∗K (TE2 (E1 ), F ) follows from Proposition 2.10 since Hom∗K (TE2 (E1 ), F ) ∼ = Hom∗K (E1 , TE 2 (F )). We have now proved that TE2 (E1 ) satisfies (K1), (K2). TE2 TE1 (F ) is a total complex

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  hom E2 , hom(E1 , F ) ⊗ E1 ⊗ E2       

hom(E1 , F ) ⊗ E1

 hom(E2 , F ) ⊗ E2     F

   

where all arrows are evaluation maps or are induced by them. We argue as in the proof of Proposition 2.10. Using (2.1), one sees that the object in the top left-hand corner can be replaced by the smaller quasi-isomorphic hom(E1 , F ) ⊗ hom(E2 , E1 ) ⊗ E2 . More precisely, this modification defines another functor RE1 ,E2 on K which is naturally isomorphic to TE2 TE1 . One can rewrite the definition of this functor as (2.7) RE1 ,E2 (F ) = hom(E1 , F ) ⊗ TE2 (E1 ) −→ TE2 (F ) . The arrow in (2.7) is obtained by composing (TE2 )∗ ⊗id hom(E1 , F ) ⊗ TE2 (E1 ) −−−−−−−→ hom TE2 (E1 ), TE2 (F ) ⊗ TE2 (E1 )

with the evaluation map ev : hom(TE2 (E1 ), TE2 (F )) ⊗ TE2 (E1 ) → TE2 (F ). This means that one has a natural map from RE1 ,E2 (F ) to TTE2 (E1 ) TE2 (F ), given by (TE2 )∗ ⊗ id on the first component and by the identity on the second one. Since (TE2 )∗ is a quasi-isomorphism by Proposition 2.10, this natural transformation is an isomorphism. proposition 2.12 Let E1 , E2 be as before, and assume in addition that HomiK (E2 , E1 ) = 0 for all i. Then TE1 TE2 ∼ = TE2 TE1 . Proof The assumption implies that TE2 (E1 ) is isomorphic to E1 . Hence the result follows directly from Lemma 2.11 and Proposition 2.6. (One can also prove this by a direct computation, without using Lemma 2.11.) proposition 2.13 Let E1 , E2 ∈ K be two n-spherical objects for some n > 0. Assume that the total dimension of Hom∗K (E2 , E1 ) is 1. Then TE1 TE2 TE1 ∼ = TE2 TE1 TE2 . Proof Since the twists are not affected by shifting, we may assume that HomiK (E2 , E1 ) is 1-dimensional for i = 0 and zero in all other dimensions. A simple computation shows that g h TE 1 (E2 ) ∼ TE2 (E1 ) ∼ = E2 −→ E1 }, = E2 −→ E1 ,

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BRAID GROUP ACTIONS

where g and h are nonzero maps. As HomK (E2 , E1 ) is 1-dimensional, it follows that TE2 (E1 ) and TE 1 (E2 ) are isomorphic up to the shift [1]. By applying Lemma 2.11 and Proposition 2.6, one finds that TE1 TE2 TE1 ∼ = TE1 TTE2 (E1 ) TE2 ∼ = TE1 TTE

1

(E2 ) TE2 .

On the other hand, applying Lemma 2.11 to TE 1 (E2 ) and E1 and using Proposition 2.10 shows that TE1 TTE (E2 ) TE2 ∼ = TE2 TE1 TE2 . 1

We now carry over the results obtained so far to the derived category D b (S ). Throughout the rest of this section, Hom always means HomD b (S ) . Definition 2.14 An object E ∈ D b (S ) is called n-spherical for some n > 0 if it has the following properties: (S1) E has a finite resolution by injective objects in S; (S2) Hom∗ (E, F ), Hom∗ (F, E) are finite-dimensional for any F ∈ D b (S ); (S3) Homi (E, E) is equal to k for i = 0, n and zero in all other dimensions; (S4) the composition map Homi (F, E) × Homn−i (E, F ) → Homn (E, E) ∼ = k is a nondegenerate pairing for all F ∈ K and i ∈ Z. Clearly, if E is such an object, any finite resolution by S-injectives is an n-spherical object of K in the sense of Definition 2.9. Using such a resolution and the equivalence of categories from Proposition 2.4, one can associate to E a twist functor TE which, by Proposition 2.10, is an exact self-equivalence of D b (S ). This is independent of the choice of resolution up to isomorphism, thanks to Proposition 2.6. lemma 2.15 In the presence of (S2) and (S3), condition (S4) is equivalent to the following apparently weaker one. (S4 ) There is an isomorphism Hom(E, F ) ∼ = Homn (F, E)∨ which is natural in b F ∈ D (S ). Proof The proof is by a “general nonsense” argument. Take any natural isomorphism as in (S4 ), and let qF : Hom(E, F ) × Homn (F, E) → k be the family of nondegenerate pairings induced by it. Because of the naturality, these pairings satisfy qF (φ, ψ) = qF (φ ◦ idE , ψ) = qE (idE , φ ◦ ψ). Since the pairings are all nondegenerate, qE (idE , −) : Homn (E, E) → k is nonzero and hence by (S3) an isomorphism. We have therefore shown that

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SEIDEL AND THOMAS composition Hom(E, F ) × Homn (F, E) −−−−−−−→ Homn (E, E) ∼ =k

is a nondegenerate pairing for any F , which is the special case i = 0 of (S4). The other cases follow by replacing F by F [i]. lemma 2.16 Let X be a noetherian scheme over k, and S = Qco(X), S = Coh(X). Then condition (S4) or (S4 ) for an object of D b (S ) implies condition (S1). Proof Let E be an object of D b (S ), and let F ∈ S be a coherent sheaf. Since E is bounded and F has a bounded below resolution by S-injectives, one has Homi (E , F ) = 0 0, and [20, for i # 0. Using (S4) or (S4 ), it follows that Homi (F , E ) = 0 for i Proposition II.7.20] completes the proof. Now define an (Am )-configuration (m > 0) of n-spherical objects in D b (S ) to be a collection (E1 , . . . , Em ) of such objects, satisfying 1 |i − j | = 1, ∗ dimk HomD b (S ) Ei , Ej = (2.8) 0 |i − j | ≥ 2. theorem 2.17 Let (E1 , . . . , Em ) be an (Am )-configuration of n-spherical objects in D b (S ). Then the twists TE1 , . . . , TEm satisfy the relations of the braid group Bm+1 up to graded natural isomorphism. That is to say, they generate a homomorphism ρ : Bm+1 → Auteq(D b (S )). This follows immediately from the corresponding results for K (see Propositions 2.12 and 2.13). One minor point remains to be cleared up. Theorem 2.17 states that the braid relations hold up to graded natural isomorphism, whereas before we have only talked about ordinary natural isomorphism. But one can easily see that all the natural isomorphisms we have constructed are graded ones, essentially because everything commutes with the translation functors. We can now state the main result of this paper. theorem 2.18 Suppose that n ≥ 2. Then the homomorphism ρ defined in Theorem 2.17 is injective, and in fact the following stronger statement holds. If g ∈ Bm+1 is not the identity element, then ρ(g)(Ei ) ∼ = Ei for some i ∈ {1, . . . , m}.

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3. Applications 3.1. Smooth projective varieties We now return to the concrete situation of derived categories of coherent sheaves. The main theme is the use of suitable duality theorems to simplify condition (S4 ) in the definition of spherical objects. Throughout, all varieties are over an algebraically closed field k. For the moment we consider only smooth projective varieties X, of dimension n. Let us recall some facts about duality on such varieties. Serre duality says that for any G ∈ D b (X) the composition Homn−∗ G , ωX ⊗ Hom∗ (O , G ) −→ Homn O , ωX = H n (ωX ) ∼ (3.1) =k is a nondegenerate pairing. (The classical form is for a single sheaf G ; the general case can be derived from this by induction on the length of the complex, using the five lemma.) Now let E be a bounded complex of locally free coherent sheaves on X. For all G1 , G2 ∈ D + (X) there is a natural isomorphism Hom∗ G1 ⊗ E , G2 ∼ (3.2) = Hom∗ G1 , G2 ⊗ E ∨ . This is proved using a resolution G2 of G2 by injective quasi-coherent sheaves; the point is that G2 ⊗ E ∨ is an injective resolution of G2 ⊗ E ∨ (see [20, Proposition 7.17]). Setting G = F ⊗ E ∨ in (3.1) for some F ∈ D b (X) and using (3.2) shows that there is an isomorphism, natural in F , ∨ Hom∗ (E , F ) ∼ (3.3) = Homn−∗ F , E ⊗ ωX . Again, by (3.2) and the standard finiteness theorems, Hom∗ (E , F ) ∼ = H∗ (E ∨ ⊗ F ) is of finite total dimension; hence so is Hom∗ (F , E ) by (3.3). Finally, because of the existence of finite locally free resolutions, everything we have said holds for an arbitrary E ∈ D b (X). lemma 3.1 An object E ∈ D b (X) is spherical, in the sense of Definition 2.14, if and only if it satisfies the following two conditions: Homj (E , E ) is 1-dimensional for j = 0, n and zero for all other j ; and E ⊗ ωX ∼ = E. Proof It follows from (3.3) and the previous discussion that the conditions are sufficient. Conversely, assume that E is a spherical object. Then property (S4) and (3.3) imply that the functors Hom(−, E ⊗ ωX ) and Hom(−, E ) are isomorphic. By a general nonsense argument, E must be isomorphic to E ⊗ ωX .

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This shows that the abstract definition of spherical objects specializes to the one in Section 1.1. We now prove the corresponding statement for twist functors. lemma 3.2 Let E ∈ D b (X) be a bounded complex of locally free sheaves, which is a spherical object. Then the twist functor TE as defined in Section 2.3 is isomorphic to the FMT by P = Cone(η : E ∨ E → O ). Proof Let E ∈ K be a bounded resolution of E by injective quasi-coherent sheaves. Let T : K → D + (X) be the functor that sends F to Cone(ev : hom(E , F ) ⊗ E → F ). We show that the diagram K D b (X)

TE T P

K (3.4) D + (X)

where the unlabeled arrows are the equivalence K ∼ = D b (X) and its inclusion into + D (X) commutes up to isomorphism. Since TE is defined using the twist functor TE on K and K ∼ = TE . Take an = D b (X), the commutativity of (3.4) implies that P ∼ b object F ∈ D (X) and a resolution F ∈ K. Then P (F ) = Rπ2 ∗ π1∗ F ⊗ π1∗ E ∨ ⊗ π2∗ E −→ O ⊗ π1∗ F ∼ = Rπ2∗ π1∗ F ⊗ π1∗ E ∨ ⊗ π2∗ E −→ O ⊗ π1∗ F ∼ = π2∗ π1∗ H om E , F ⊗ π2∗ E −→ O ⊗ π1∗ F ∼ = hom E , F ⊗ E −→ F = T (F ), where the arrow in the last line is evaluation. This provides a natural isomorphism that makes the left lower triangle in (3.4) commute. To deal with the other triangle, set up a diagram as in the proof of Proposition 2.6. Example 3.3 Let X be a variety that is Calabi-Yau in the strict sense; that is to say, ωX ∼ = O and i H (X, O ) = 0 for 0 < i < n. Then any invertible sheaf on X is spherical. For the trivial sheaf, the twist TO is the FMT given by the object on X × X which is the ideal sheaf of the diagonal shifted by [1]. This is what Mukai [38] calls the reflection functor.

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lemma 3.4 Let Y ⊂ X be a connected subscheme that is a local complete intersection, with (locally free) normal sheaf ν = (JY /JY2 )∨ . Assume that ωX |Y is trivial, and assume that H i (Y, j ν) = 0 for all 0 < i + j < n. Then OY ∈ D b (X) is a spherical object. Proof Denote by ι the embedding of Y into X. The local Koszul resolution of ι∗ OY gives the well-known formula for the sheaf Exts, Ext j (ι∗ OY , ι∗ OY ) ∼ = ι∗ (j ν). The asi+j i j (i.e., the hypercohomology sumptions and the spectral sequence H (Ext ) ⇒ Ext spectral sequence of H(RH om) = Ext) give Extr (ι∗ OY , ι∗ OY ) = 0 for 0 < r < n. We have Hom(ι∗ OY , ι∗ OY ) ∼ = k, and hence Extn (ι∗ OY , ι∗ OY ) ∼ = k by duality. Example 3.5 Let X be a surface. Then any smooth rational curve C ⊂ X with C · C = −2 satisfies the conditions of Lemma 3.4. Now take a chain C1 , . . . , Cm of such curves such that Ci ∩ Cj = ∅ for |i − j | ≥ 2, and Ci · Ci+1 = 1 for i = 1, . . . , m − 1. Then (OC1 , . . . , OCm ) is an (Am )-configuration of spherical objects. Remark 3.6 As far as Lemma 3.1 is concerned, one could remove the assumption of smoothness and work with arbitrary projective varieties X. Serre duality must then be replaced by the general duality theorem (see [20, Theorem III.11.1]) applied to the projection π : X → Spec k. This yields a natural isomorphism, for G ∈ D − (X), ∨ Extn−∗ G , ωX ∼ = Ext∗ OX , G , where now ωX = π ! (OSpec k ) ∈ D + (X) is the dualizing complex. With this replacing (3.1), one can essentially repeat the same discussion as in the smooth case, leading to an analogue of Lemma 3.1. The only difference is that the condition that E has a finite locally free resolution must be included as an assumption. We do not pursue this further, for lack of a really relevant application. 3.2. Two generalisations We now look at smooth quasi-projective varieties. Rather than aiming at a comprehensive characterisation of spherical objects, we just carry over Lemma 3.4, which provides one important source of examples. Let X be a smooth quasi-projective variety of dimension n, and let Y ⊂ X be a complete subscheme of codimension c; ι denotes the embedding Y @→ X. Complete X ¯ Then Y ⊂ X¯ is closed, and X is smooth, so ι∗ OY has a finite to a projective variety X. locally free resolution; thus we may use Serre duality (see [20, Theorem III.11.1])

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¯ and the methods of (3.3), to conclude that on X, ∨ Hom ι∗ OY , F ∼ = Homn F , ι∗ OY ⊗ ωX on X. By continuing as in the projective case and using the same spectral sequence as in Lemma 3.4, one obtains the following result.2 lemma 3.7 Assume that H i (Y, j ν) = 0 for all 0 < i + j < n, and assume that ι∗ ωX is trivial. Then ι∗ OY is a spherical object in D b (X). One can now, for example, extend Example 3.5 to quasi-projective surfaces. For subschemes of codimension 1, we later provide a stronger result, Proposition 3.15, which can be used to construct more interesting spherical objects. The other generalisation at which we want to look is technically much simpler. Let X be a smooth n-dimensional projective variety over k with an action of a finite group G. We assume that char(k) = 0; this implies the complete reducibility of G-representations, which we use in an essential way. Let QcoG (X) be the category whose objects are G-equivariant quasi-coherent sheaves and whose morphisms are the G-equivariant sheaf homomorphisms. One can write G (3.5) HomQcoG (X) E1 , E2 = HomQco(X) E1 , E2 with respect to the obvious G-action on HomQco(X) (E1 , E2 ). Because taking the invariant part of a G-vector space is an exact functor, it follows that a G-sheaf is injective in QcoG (X) if and only if it is injective in Qco(X). This can be used to show that QcoG (X) has enough injectives and also that S = QcoG (X) and its Serre subcategory S = CohG (X) of coherent G-sheaves satisfy conditions (C1)–(C4) from Section 2.1. As a further application, one derives a formula similar to (3.5) for the derived category G (3.6) HomD + (QcoG (X)) F1 , F2 = HomD + (Qco(X)) F1 , F2 for all F1 , F2 ∈ D + (QcoG (X)). This allows one to carry over the usual finiteness results for coherent sheaf cohomology, as well as Serre duality, to the equivariant context. The same argument as in the nonequivariant case now leads to the following lemma. lemma 3.8 b (X) = D b (Coh (X)) of coherent equivAn object E in the derived category DG G ariant sheaves is spherical if and only if the following two conditions are satisfied: 2 We

thank one of the referees for simplifying our original version of this proof.

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Hom

j b (X) (E , E ) is 1-dimensional for j DG

= 0, n and zero in other degrees; and E ⊗ωX

is equivariantly isomorphic to E .

Finally, one can combine the two generalisations and obtain an equivariant version of Lemma 3.7. This is useful in examples that arise in connection with the McKay correspondence. We concentrate on the simplest of these examples, which also happens to be particularly relevant for our purpose. Consider the diagonal subgroup G∼ = Z/(m+1) of SL2 (k). Write R for its regular representation, and write V1 , . . . , Vm for its (nontrivial) irreducible representations. Let X be a smooth quasi-projective surface with a complex symplectic form, carrying an effective symplectic action of G. Choose a fixed point x ∈ X; the tangent space Tx X must necessarily be isomorphic to R as a G-vector space. For i = 1, . . . , m, set Ei = Ox ⊗ Vi ∈ CohG (X). The Koszul resolution of Ox , together with (3.6), shows that r G HomrD b (X) Ei , Ej ∼ = R ⊗ Vi∨ ⊗ Vj . G

This implies that each Ei is a spherical object and that these objects form an (Am )b (X). configuration, so that we obtain a braid group action on DG Example 3.9 In particular, we have a braid group action on the equivariant derived category of coherent sheaves over A2 , with respect to the obvious linear action of G. (This is probably the simplest example of a braid group action on a category in the present paper.) Let π : Z → X/G be the minimal resolution. This is again a quasi-projective surface with a symplectic form; it can be constructed as a Hilbert scheme of G-clusters on X. The irreducible components of π −1 (x) are smooth rational curves C1 , . . . , Cm which are arranged as in Example 3.5, so that their structure sheaves generate a braid group action on D b (Z). A theorem of M. Kapranov and E. Vasserot [24] provides an equivalence of categories b (X) ∼ (3.7) DG = D b (Z), which takes Ej to OCj up to tensoring by a line bundle (see [24, p. 7]). This means that the braid group actions on the two categories essentially correspond to each other. Adding the trivial 1-dimensional representation V0 , and the corresponding equivariant b (X) to an action of the affine braid sheaf E0 = Ox = Ox ⊗V0 , extends the action on DG group, except for m = 1. Interestingly, the cyclic symmetry between V0 , V1 , . . . , Vm is not immediately visible on D b (Z); the equivalence (3.7) takes E0 to the structure sheaf of the whole exceptional divisor π −1 (x). Finally, everything we have said carries

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over to the other finite subgroups of SL(2, k) with the obvious modifications. The Dynkin diagram of type (Am ) which occurs implicitly several times in our discussion must be replaced by those of type D/E, and one obtains actions of the corresponding (affine) generalised braid groups. A recent deep theorem of Bridgeland, A. King, and M. Reid [8] extends the equivalence (3.7) to certain higher-dimensional quotient singularities. We consider only one very concrete case. Example 3.10 Let X be the Fermat quintic in P4 with the diagonal action of G = (Z/5)3 familiar from mirror symmetry. The fixed-point set XH of the subgroup H = (Z/5)2 × 1 b (X). consists of a single G-orbit F, whose structure sheaf is a spherical object in DG By considering other subgroups of the same kind, one finds a total of ten spherical objects with no Homs between any two of them. Now let π : Z → X/G be the b (X) ∼ D b (Z) crepant resolution given by the Hilbert scheme of G-clusters. Then DG = by [8], so that one gets corresponding spherical objects on Z. Because of the nature of the equivalence, the object corresponding to OF must be supported on the exceptional set p−1 (F) of the resolution. We have not determined its precise nature, but this is clearly related to Proposition 3.15 and Examples 3.20. 3.3. Spherical and exceptional objects The reader familiar with the theory of exceptional sheaves (see [47]), or with certain aspects of tilting theory in representation theory, will have noticed a similarity between our twist functors and mutations of exceptional objects. (See also [6], and note that their “elliptical exceptional” objects are examples of 1-spherical objects.) The braid group also occurs in the mutation context, but there it acts on collections of exceptional objects in a triangulated category instead of on the category itself. The relation of the two kinds of braid group actions is not at all clear. Here we content ourselves with two observations, the first of which is motivated by examples in [31]. Definition 3.11 Let X, Y be smooth projective varieties, with ωX trivial. A morphism f : X → Y (of codimension c = dim X − dim Y ) is called simple if there is an exact triangle OY −→ Rf∗ OX −→ ωY [−c].

In most applications, Y would be Fano because one could then use the wealth of known results about exceptional sheaves on such varieties. However, the general theory does not require this assumption on Y .

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lemma 3.12 Suppose that c > 0 and

   OY i ∼ R f∗ OX = 0   ω Y

for i = 0, for 0 < i < c, for i = c.

Then f is simple. Proof Rf∗ OX is a complex of sheaves whose cohomology is nonzero only in two degrees; a general argument, valid in any derived category, shows that there is an exact triangle R 0 f∗ OX → Rf∗ OX → (R c f∗ OX )[−c]. proposition 3.13 Suppose that f is simple, and suppose that F ∈ D b (Y ) is an exceptional object, in the sense that Hom(F , F ) ∼ = k and Homi (F , F ) = 0 for all i = 0. Then ∗ b Lf F ∈ D (X) is a spherical object. Proof One can easily show, using, for example, a finite locally free resolution of F and a finite injective quasi-coherent resolution of OX , that Rf∗ Lf ∗ F ∼ = F ⊗L (Rf∗ OX ). Hence, by tensoring the triangle in Definition 3.11 with F , one obtains another exact triangle F → Rf∗ Lf ∗ F → F ⊗ ωY [−c]. This yields a long exact sequence · · · −→ Hom∗ (F , F ) −→ Hom∗ F , Rf∗ Lf ∗ F −→ Hom∗−c F , F ⊗ωY −→ · · ·. The second and third group are Hom∗ (Lf ∗ F , Lf ∗ F ) and Homdim X−∗ (F , F ) by, respectively, adjointness and Serre duality. From the assumption that F is exceptional, one now immediately obtains the desired result. Examples 3.14 (a) (This assumes char(k) = 0.) Consider a Calabi-Yau X with a fibration f : X → Y over a variety Y such that the generic fibres are elliptic curves or K3 surfaces. Clearly f∗ O X ∼ = OY ; relative Serre duality shows that R c f∗ OX ∼ = ωY ; and in the K3-fibred case one has also R 1 f∗ OX = 0. Hence f is simple. (b) (This assumes char(k) = 2.) Let f : X → Y be a 2-fold covering branched over a double anticanonical divisor. One can use the Z/2-action on X to split f∗ OX into two direct summands, which are isomorphic to OY and ωY , respectively; this implies that f is simple. An example, already considered in [31], is a K3 double covering of P2 branched over a sextic. Another example, which is slightly degenerate but still

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works, is the unbranched covering map from a K3 surface to an Enriques surface. (c) Examples with c = −1 come from taking X to be a smooth anticanonical divisor in Y and taking f to be the embedding. Then Rf∗ OX = f∗ OX ∼ = {ωY → OY } with the map given by the section of ωY−1 defining X. Quartic surfaces in P3 are an example considered in [31]. We now describe a second connection between spherical and exceptional objects, this time using pushforwards instead of pullbacks. The result applies to quasi-projective varieties as well, but it is limited to embeddings of divisors. Let X ⊂ PN be a smooth quasi-projective variety, and let ι : Y @→ X be an embedding of a complete connected hypersurface Y . As in the parallel argument in the previous section, we work on the projective completion X¯ of X, in which Y is closed. By the smoothness of X, given that F ∈ D b (Y ), ι∗ F has a finite locally free resolution, and Serre duality on X¯ (see [20, Theorem III.11.1]) yields ∨ Hom(ι∗ F , G ) ∼ = Homdim X G , ι∗ F ⊗ ωX on X. proposition 3.15 Assume that ι∗ ωX is trivial. If F ∈ D b (Y ) is an exceptional object with a finite locally free resolution, then ι∗ F is spherical in D b (X). Proof In view of the previous discussion, what remains to be done is to compute Homi (ι∗ F , ι∗ F ), which, by [20, Theorem III.11.1] applied to ι∗ , is isomorphic to Homi−1 (F , Lι∗ ι∗ F ⊗ ωY ). We need the following result (which, perhaps surprisingly, need not be true without the ι∗ s). lemma 3.16 We have ι∗ Lι∗ (ι∗ F ) ∼ = ι∗ (F ⊗ ωY−1 )[1] ⊕ ι∗ F . Proof Replacing ι∗ F by a quasi-isomorphic complex F of locally free sheaves, the lefthand side of the above equation is ι∗ (F |Y ) = F ⊗ OY , which is quasi-isomorphic to F ⊗ O (−Y ) −→ O ' ι∗ F ⊗ O (−Y ) −→ O , where the arrow is multiplication by the canonical section of O (Y ). Since this vanishes on Y , which contains the support of ι∗ F , we obtain ι∗ (F ⊗ O (−Y )|Y )[1] ⊕ ι∗ F , as required.

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65

By hypothesis we may assume that F is a finite complex of locally free sheaves on Y , so that H om(F , F ) ∼ = F ⊗ F ∨ . Thus, computing Homi (ι∗ F , ι∗ F ) as the (i − 1)th (derived/hyper) sheaf cohomology of the complex of OY -module sheaves Lι∗ ι∗ F ⊗ F ∨ ⊗ ωY , we may push forward to X and there use Lemma 3.16. That is, pick an injective resolution OY → I on Y , so that Homi (ι∗ F , ι∗ F ) is the (i − 1)th cohomology of GY Lι∗ ι∗ F ⊗ F ∨ ⊗ ωY ⊗ I , where G is the global sections functor. Pushing forward to X, this is GX ι∗ Lι∗ ι∗ F ⊗ ι∗ (F )∨ ⊗ ωY ⊗ I , which by Lemma 3.16 is GX ι∗ F ⊗ F ∨ ⊗ I [1] ⊕ GX ι∗ F ⊗ F ∨ ⊗ ωY ⊗ I . This may be brought back onto Y to give the (i − 1)th cohomology of GY (F ⊗ F ∨ ⊗ I )[1] ⊕ GY (F ⊗ F ∨ ⊗ ωY ⊗ I ). This is Homi (F , F ) ⊕ Homn−i (F , F )∨ , where for the second term we have used Serre duality on Y . Since F is exceptional, this completes the proof. 3.4. Elliptic curves The homological mirror conjecture for elliptic curves has been studied extensively by A. Polishchuk and E. Zaslow [45], [44]. (Unfortunately, their formulation of the conjecture differs somewhat from that in Section 1.2, so their results cannot be applied directly here.) Polishchuk [43] and Orlov [41], following earlier work of Mukai [37], have completely determined the automorphism group of the derived category of coherent sheaves. These are difficult results, to which we have little to add. Still, it is perhaps instructive to see how things work out in a well-understood case. We begin with the symplectic side of the story. Let (M, β) be the torus M = R/Z × R/Z with its standard volume form β = ds1 ∧ ds2 . Matters are slightly more complicated than in Section 1.2 because the fundamental group is nontrivial. In particular, the C ∞ -topology on Symp(M, β) is no longer the correct one; this is due to the fact that Floer cohomology is not invariant under arbitrary isotopies, but only under Hamiltonian ones. There is a bi-invariant foliation F of codimension 2 on Symp(M, β), and the Hamiltonian isotopies are precisely those that are tangent to the leaves. To capture this idea one introduces a new topology, the Hamiltonian topology, on Symp(M, β). This is the topology generated by the leaves of F | U , where U ⊂ Symp(M, β) runs over all C ∞ -open subsets. To avoid confusion, we write Symph (M, β) whenever we have the Hamiltonian topology in mind, and we call this the Hamiltonian automorphism group; this differs from the terminology in most of the literature, where the name is reserved for what, in our terms, is the

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connected component of the identity in Symph (M, β). The difference between the two topologies becomes clear if one considers the group Aff(M) = M SL(2, Z) of oriented affine diffeomorphisms of M. As a subgroup of Symp(M, β), this has its Lie group topology, in which the translation subgroup M is connected. In contrast, as a subgroup of Symph (M, β) it has the discrete topology. lemma 3.17 The embedding of Aff(M) into Symph (M, β) as a discrete subgroup is a homotopy equivalence. The proof consists of combining the known topology of Diff + (M), Moser’s theorem that Symp(M, β) ⊂ Diff + (M) is a homotopy equivalence, and the flux homomorphism that describes the global structure of the foliation F. We omit the details. Let π : R → RP1 , s → [cos(πs) : sin(πs)] be the universal covering of RP1 . R) ⊂ SL(2, R) × Diff(R) of pairs (g, g) Consider the subgroup SL(2, ˜ such that g˜ is 1 a lift of the action of g on RP . SL(2, R) is a central extension of SL(2, R) by Z. (Topologically, it consists of two copies of the universal cover.) We define a graded symplectic automorphism of (M, β) to be a pair R) φ, φ˜ ∈ Symph (M, β) × C ∞ M, SL(2, such that g˜ is a lift of Dg : M → SL(2, R); here we have used the standard trivialisation of T M. The graded symplectic automorphisms form a group under the composi˜ ˜ = (φψ, (φ˜ ◦ ψ)ψ). ˜ We denote this group by Symph,gr (M, β), and tion (φ, φ)(ψ, ψ) R)). It is we equip it with the topology induced from Symph (M, β) × C ∞ (M, SL(2, h a central extension of Symp (M, β) by Z. One can easily verify that the definition is equivalent to that in [49], which in turn goes back to ideas of Kontsevich [30]. Even in this simplest example, the construction of the derived Fukaya category D b Fuk(M, β) has not yet been carried out in detail, so we proceed on the basis of guesswork in the style of Section 1.2. The basic objects of D b Fuk(M, β) are pairs (L, E) consisting of a Lagrangian submanifold and a flat unitary bundle on it. Thus, in addition to symplectic automorphisms, the category should admit another group of self-equivalences, which act on all objects (L, E) by tensoring E with some fixed flat unitary line bundle ξ → M. The two kinds of self-equivalence should give a homomorphism def (3.8) γ : G = M ∨ π0 Symph,gr (M, β) −→ Auteq D b Fuk(M, β) , where M ∨ = H 1 (M; R/Z) is the Jacobian, or a dual torus. In order to make the picture more concrete, we now write down the group G explicitly. Take the stan 1 0 dard presentation of SL(2, Z) by generators g1 = 01 11 , g2 = −1 1 and relations

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Z) ⊂ SL(2, R) be the preimage of g1 g2 g1 = g2 g1 g2 , (g1 g2 )6 = 1. Let SL(2, Z) SL(2, Z). One can lift g1 , g2 to elements a1 = (g1 , g˜ 1 ) and a2 = (g2 , g˜ 2 ) in SL(2, which satisfy g˜ 1 (1/2) = 1/4 and g˜ 2 (1/4) = 0. Together with the central element Z), and one can easily work out what the t = (id, s → s − 1), these generate SL(2, relations are:

Z) = a1 , a2 , t | a1 a2 a1 = a2 a1 a2 , (a1 a2 )6 = t 2 , [a1 , t] = [a2 , t] = 1 . SL(2, Z) defines a graded symplectic automorphism of Any element of (g, g) ˜ ∈ SL(2, (M, β); one simply takes φ = g and φ˜ to be the constant map with value g. ˜ Moreover, any translation of M has a canonical lift to a graded symplectic automorphism by R). These two observations taking φ˜ to be the constant map with value 1 ∈ SL(2, Z) of Symph,gr (M, β), which fits together give a subgroup Aff(M) = M SL(2, into a commutative diagram 1

Z

Aff(M)

Aff(M)

1

Symph,gr (M, β)

Symph (M, β)

1

=

1

Z

Hence, in view of Lemma 3.17, π0 (Symph,gr (M, β)) ∼ After spelling out = Aff(M). Z), with respect everything, one finds that G is the semidirect product (R/Z)4 SL(2, Z) on R4 given by to the action of SL(2,     1 1 0 0 1 0 0 0 0 1 0 0 −1 1 0 0   a1 −→  a2 −→  t −→ id . (3.9) 0 0 1 0 ,  0 0 1 1 , 0 0 −1 1

0

0 0 1

We now pass to the mirror dual side. Let X be a smooth elliptic curve over C. We choose a point x0 ∈ X which is the identity for the group law on x. The derived category D b (X) has self-equivalences TO , S

and

Rx , Lx , TOx

(x ∈ X)

defined as follows. TO is the twist by O , which is spherical for obvious reasons; S is the original example of an FMT, S = L with L = O ( − {x0 } × X − X × {x0 }) the Poincaré line bundle. It maps the structure sheaves of points Ox to the line bundles O (x − x0 ), and it was shown to be an equivalence by Mukai [37]. Rx is the selfequivalence induced by the translation y → y + x; Lx is the functor of tensoring with the degree zero line bundle O (x − x0 ); and TOx is the twist along Ox which is spherical by Lemma 3.4. These functors have the following properties:

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[Lx , Ry ] ∼ = id TOx

for all x, y,

(3.10)

is isomorphic to O (x) ⊗ −, S4 ∼ = [−2],

(3.12)

∼ = TO TOx0 TO ∼ = S −1 ,

(3.13)

TOx0 TO TOx0

(3.11)

∼ TOx0 Rx TO−1 = Rx L−1 x , x

(3.14)

∼ TOx0 Lx TO−1 = Lx , x

(3.15)

0

0

TO Rx TO−1 ∼ = Rx , −1 ∼ TO Lx T = Rx Lx . O

(3.16) (3.17)

Most of these isomorphisms are easy to prove; those that present any difficulties are (3.11), (3.12), and (3.13). The first and third of these are proved below, and the second one is a consequence of [37, Theorem 3.13(1)]. Proof of (3.11) (This argument is valid for the structure sheaf of a point on any algebraic curve.) A simple computation shows that the dual in the derived sense is Ox∨ ∼ = Ox [−1]. The formula for inverses of FMTs (for which, see, e.g., [7, Lemma 4.5]) shows that ∼ TO−1 = Q for some object Q fitting into an exact triangle x f

Q −→ O − −→ O(x,x) .

When following through the computation, it is not easy to keep track of the map f , but that is not really necessary. All we need to know is that f = 0, which is true because the converse would violate the fact that Q is an equivalence. Then, since any morphism O → O(x,x) in the derived category is represented by a genuine map of sheaves, f must be some nonzero multiple of the obvious restriction map. It follows that Q is isomorphic to the kernel of f , which is O ⊗ π1∗ O (−x). This is the functor of tensoring with O (−x). Passing to inverses yields means that TO−1 x the desired result. Proof of (3.13) The equality between the first two terms follows from Theorem 2.17 because Ox0 , O form an (A2 )-configuration of spherical objects. By the standard formula for the adjoints of an FMT, the inverse of S is the FMT with L ∨ [1]. By definition, TO is the FMT with O (− )[1]. Using (3.11), it follows that TOx0 TO TOx0 is the FMT with π1∗ O (x0 ) ⊗ O (− )[1] ⊗ π2∗ O (x0 ) ∼ = L ∨ [1]. Equations (3.12) and (3.13) show that (TO TOx0 )6 ∼ = [2]. Therefore one can define a

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homomorphism

Z) −→ Auteq D b (X) SL(2,

by mapping the generators a1 , a2 , t to TO , TOx0 , and the translation [1]; this already occurs in Mukai’s paper [37], slightly disguised by the fact that he uses a different presentation of SL(2, Z). The functors Lx , Rx yield another homomorphism X×X → Auteq(D b (X)); and one can combine the two constructions into a map def Z) −→ Auteq D b (X) . (3.18) γ : G = (X × X) SL(2, Z)-action on X × X Here the semidirect product is taken with respect to the SL(2, indicated by (3.14)–(3.17); explicitly, it is given by the matrices 1 1 1 0 a1 −→ , a2 −→ , t −→ id . (3.19) 0 1 −1 1 lemma 3.18 The group G in (3.8) is isomorphic to the group G in (3.18). Proof Introduce complex coordinates z1 = r1 +ir4 , z2 = r2 −ir3 on R4 /Z4 . Then the action Z) described in (3.9) becomes C-linear and is given by the same matrices as of SL(2, in (3.19). This is sufficient to identify the two semidirect products that define G and G . We should point out that although the argument is straightforward, the change of coordinates is by no means obvious from the geometric point of view; a look back at the definition of G shows that z1 , z2 mix genuine symplectic automorphisms with the extra symmetries of D b Fuk(M, β) which come from tensoring with flat line bundles. The way in which this fits into the general philosophy is that one expects to have a commutative diagram, with the right vertical arrow given by Kontsevich’s conjecture: G

Auteq D b Fuk(M, β)

γ

∼ =

G

(3.20)

∼ = γ

Auteq(D b (X))

To be accurate, one should adjust the modular parameter of X and the volume of (M, β), eventually introducing a complex part βC as in Remark 1.4, so that they are indeed mirror dual. This has not played any role up to now since the groups G and G are independent of the parameters, but it would become important in further study. A theorem of Orlov [41] says that γ is always injective and that it is an

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isomorphism if and only if X has no complex multiplication. Only the easy part of the theorem is important for us here. If X has complex multiplication, then its symmetries induce additional automorphisms of D b (X), which are not contained in the image of γ . Therefore, if the picture (3.20) is correct, the derived Fukaya category for the corresponding values of βC must admit exotic automorphisms that do not come from symplectic geometry or from flat line bundles. It would be interesting to check this claim, especially because similar phenomena may be expected to occur in higher dimensions. We now apply the intuition provided by the general discussion to the specific topic of braid group actions. To a simple closed curve S on (M, β), one can associate a Dehn twist τS ∈ Symph (M, β) which is unique up to Hamiltonian isotopy. This is defined by taking a symplectic embedding ι of (U, θ) = ([−M; M] × R/Z, ds1 ∧ ds2 ) into M for some M > 0, with ι({0} × R/Z) = S, and by using a local model τ : U −→ U, τ (s1 , s2 ) = s1 , s2 − h(s1 ) , where h ∈ C ∞ (R, R) is some function with h(s) = 0 for s ≤ −M/2, h(s) = 1 for s ≥ M/2, and h(s) + h(−s) = 1 for all s. The interesting fact is that the Dehn twists along two parallel geodesic lines are not Hamiltonian isotopic; they differ by a translation that depends on the area lying between the two lines. Now take S1 = R/Z × {0},

S2 = {0} × R/Z,

S3 = R/Z × {1/2}.

This is an (A3 )-configuration of circles. Hence the Dehn twists τS1 , τS2 , τS3 define a homomorphism from the braid group B4 to π0 (Symph (M, β)). However, this is τS1 is Hamiltonian isotopic to a translation that has order 2, so the not injective; τS−1 3

nontrivial braid (g3−1 g1 )2 ∈ B4 gets mapped to the identity element. The natural lift of this homomorphism to Symph,gr (M, β) has the same noninjectivity property. Guided by mirror symmetry, one translates this example into algebraic geometry as follows: E1 = Ox0 , E2 = O , E3 = Ox ∈ D b (X), where x = x0 is a point of order 2 on X, form an (A3 )-configuration of spherical objects. Hence their twist functors generate TE1 is the functor of tensoring with a weak action of B4 on D b (X). By (3.11), TE−1 3 O (x − x0 ). Since the square of this is the identity functor, we have the same relation as in the symplectic case, so that the action is not faithful. 3.5. K3 surfaces Let X be a smooth complex K3 surface. Consider, as in Example 3.5, a chain of embeddings ι1 , . . . , ιm : P1 → X whose images Ci satisfy Ci · Cj = 1 for |i − j | = 1 and Ci ∩ Cj = 0 for |i − j | ≥ 2. One can then use the structure sheaves OCi to define a braid group action on D b (X). However, this is not the only way.

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proposition 3.19 For each i = 1, . . . , m, choose Ei to be either O (−Ci ) or OCi (−1) := (ιi )∗ OP1 (−1). Then the Ei form an (Am )-configuration of spherical objects in D b (X) and hence generate a weak braid group action on that category. The choice can be made for each Ei independently. These multiple possibilities are relevant from the mirror symmetry point of view. This is explained in [53], so we only summarize the discussion here. Suppose that X is elliptically fibred with a section S. Its mirror should be the symplectic 4-manifold (M, β) with M = X and where β is the real part of some holomorphic 2-form on X (hyper-Kähler rotation). The smooth holomorphic curves in M are precisely the Lagrangian submanifolds in (M, β) which are special (with respect to the calibration given by the Kähler form of a Ricci-flat metric on X). In particular, the curves Ci turn into an (Am )-configuration of Lagrangian 2-spheres; hence the generalised Dehn twists along them generate a homomorphism Bm+1 → π0 (Sympgr (M, β)). One can wonder what the corresponding braid group action on D b (X) should be. This question is not really meaningful without a distinguished equivalence between the derived Fukaya category of (M, β) and that of coherent sheaves on X, which is not what is predicted by Kontsevich’s conjecture. But if we adopt the Strominger-Yau-Zaslow [51] picture of mirror symmetry, then conjecturally there should be a distinguished full and faithful embedding of triangulated categories D b Fuk(M, β) @→ D b (X) induced by the particular special Lagrangian torus fibration of M which comes from the elliptic fibration of X. (This fibration may, of course, not be distinguished.) That this is an embedding, and not an equivalence, is a feature of even-dimensional mirror symmetry. This embedding should be an extension of the FMT which takes special Lagrangian submanifolds of M (algebraic curves in X) to coherent sheaves on X using the relative Poincaré sheaf on X ×P1 X that comes from considering the elliptic fibres to be self-dual using the section (see, e.g., [53]). Assuming this, it now makes sense to ask what spherical objects of D b (X) correspond to the special Lagrangian spheres C1 , . . . , Cm . The FMT takes any special Lagrangian submanifold C that is a section of the elliptic fibration to the invertible sheaf O (S − C); if C lies in a fibre of the fibration, it goes to the structure sheaf OC . If we assume that all curves Ci fall into one of these two categories and that S intersects all those that lie in one fibre, then the FMT takes the special Lagrangian submanifolds C1 , . . . , Cm in (M, β) to sheaves E1 , . . . , Em as in Proposition 3.19, tensored with O (S). Then, up to the minor difference of tensoring by O (S), one of the braid group actions mentioned in that proposition would be the correct mirror dual of the symplectic one.

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As mentioned in Section 3.2, such configurations of curves Ci are the exceptional loci in the resolution of any algebraic surface with an (Am )-singularity. Now, (Am )configurations of Lagrangian 2-spheres occur as vanishing cycles in the smoothing of the same singularity. Thus, in a sense, mirror symmetry interchanges smoothings and resolutions. A more striking, though maybe less well-understood, instance of this phenomenon is H. Pinkham’s interpretation of Arnold’s strange duality (see, e.g., [42]), which has been interpreted as a manifestation of mirror symmetry by a number of people (P. Aspinwall and D. Morrison, M. Kobayashi, I. Dolgachev, W. Ebeling, and so on). Each of the 14 singular affine surfaces S(c1 , c2 , c3 ) on Arnold’s list has a natural compactification S(c1 , c2 , c3 ) which has four singular points. One of these points is the original singularity at the origin; the other three are quotient singularities lying on the divisor at infinity, which is a P1 . One can smooth the singular point at the origin; the intersection form of the vanishing cycles obtained in this way is T (c1 , c2 , c3 ) ⊕ H , where T (c1 , c2 , c3 ) is the matrix associated to the Dynkin-type diagram (see Figure 1) and H = 01 01 . On the other hand, one can resolve the three c1

c2

c3

Figure 1

singular points at infinity. Inside the resolution this yields a configuration of smooth rational curves of the form T (b1 , b2 , b3 ) for certain other numbers (b1 , b2 , b3 ). One can also do the two things together; this removes all singularities, yielding a smooth K3 surface X(c1 , c2 , c3 ) with a splitting of its intersection form as T c1 , c2 , c3 ⊕ H ⊕ T b1 , b2 , b3 . Strange duality is the observation that the numbers (b1 , b2 , b3 ) associated to one singularity on the list occur as (c1 , c2 , c3 ) for another singularity, and vice versa. Kobayashi [29] (extended by Ebeling [12] to more general singularities) explains this by showing that the K3’s, X(c1 , c2 , c3 ) and X(b1 , b2 , b3 ), belong to mirror dual families. The associated map on homology interchanges the T (c1 , c2 , c3 ) and T (b1 , b2 , b3 ) summands of the intersection form. (The extra hyperbolic of vanishing

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73

cycles goes to the H 0 ⊕ H 4 of the other K3 surface.) Thus the smoothing of the singular point at the origin in S(c1 , c2 , c3 ) corresponds, in a slightly vague sense, to the resolution of the divisor at infinity of S(b1 , b2 , b3 ). From our point of view, since the rational curves at infinity in X(b1 , b2 , b3 ) can be used to define a braid group action on its derived category, one would like to have a similar configuration of Lagrangian 2-spheres (vanishing cycles) in the finite part of X(c1 , c2 , c3 ). On the level of homology, such a configuration exists of course, but it is apparently unknown whether it can be realized geometrically. (Recall that Lagrangian submanifolds can have many more nonremovable intersection points than their intersection number suggests.) 3.6. Singularities of 3-folds Throughout the following discussion, all varieties are smooth projective 3-folds that are Calabi-Yau in the strict sense. (Some singular 3-folds also occur, but they are specifically designated as such.) Let X be such a variety. Examples 3.20 Any invertible sheaf on X is a spherical object in D b (X). If S is a smooth connected surface in X with H 1 (S, OS ) = H 2 (S, OS ) = 0 (e.g., a rational surface or Enriques surface), the structure sheaf OS is a spherical object, by Lemma 3.4. Similarly, for C a smooth rational curve in X with normal bundle νC ∼ = OP1 (−1) ⊕ OP1 (−1) (usually referred to as a (−1, −1)-curve), OC is spherical. The ideal sheaf JC of such a curve is also a spherical object; this follows from JC [1] ∼ = TO (OC ). Supposing the ground field to be k = C, we now return to the conjectural duality between smoothings and resolutions that played a role in Section 3.5 and that has been considered by many physicists. (Of course, our interest in this is in trying to mirror Dehn twists on smoothings, which arise as monodromy transformations around a degeneration of the smoothing collapsing the appropriate spherical vanishing cycle, by twists on the derived categories of the resolutions.) To explain the approach of physicists (as described in [36], for instance), it is better to adopt the traditional point of view in which mirror symmetry relates the combined complex and (complexified) Kähler moduli spaces of two varieties, rather than Kontsevich’s conjecture, which considers a fixed value of the moduli variables. Then the idea can be phrased as follows. Moving towards the discriminant locus in the complex moduli space of a variety X, which means degenerating it to a singular variety Y , should be mirror dual to going to a “boundary wall” of the complexified Kähler cone of the mirror X (the annihilator of a face of the Mori cone), thus inducing an extremal contraction X → Y . A second application of the same idea, with the roles of the mirrors reversed, shows

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that an arbitrary crepant resolution Z → Y should be mirror dual to a smoothing Z of Y . A case that is reasonably well understood is that of the ordinary double point (ODP: x 2 + y 2 + z2 + t 2 = 0 in local analytic coordinates) singularity. ODPs should be self-dual, in the sense that if Y has d distinct ODPs, then so does Y . (This can be checked for Calabi-Yau hypersurfaces in toric varieties, for instance.) We now review briefly C. Clemens’s work in [9] on the homology of smoothings and resolutions of such singularities. A degeneration of X to a variety Y with d ODPs determines d vanishing cycles in X and hence a map v : Zd → H3 (X). Let v ∨ : H3 (X) → Zd be the Poincaré dual of v. Suppose that Y has a crepant resolution Z that, in local analytic coordinates near each ODP, looks like the standard small resolution. This means that the exceptional set in Z consists of d disjoint (−1, −1)-curves. By [9] and [16], one has ker v ∨ ∼ H3 (Z) = im(v) and exact sequences v∨

H3 (X) −−→ Zd −→ H2 (Z) −→ H2 (X) −→ 0, v

0 −→ H4 (X) −→ H4 (Z) −→ Zd −→ H3 (X). Thus, if there are r relations between the vanishing cycles (the image of v is of rank d − r), the Betti numbers are b2 (Z) = b2 (X) + r,

b3 (Z) = b3 (X) − 2(d − r),

b4 (Z) = b4 (X) + r. (3.21)

Topologically, Z arises from X through codimension 3 surgery along the vanishing cycles, and the statements above can be proved, for example, by considering the standard cobordism between them. More intuitively, one can explain matters as follows. Going from X to Y shrinks the vanishing cycles to points; at the same time, the relations between vanishing cycles are given by 4-dimensional chains that become cycles in the limit Y because their boundaries shrink to points. This means that we lose d − r generators of H3 and get r new generators of H4 . In Z there are d − r relations between the homology classes of the exceptional P1 ’s; these relations are pullbacks of closed 3-dimensional cycles on Y which do not lift to cycles on Z, so going from Y to Z adds r new generators to H2 while removing another d − r generators from H3 . Finally, H4 (Z) = H4 (Y ) for codimension reasons. Mirror symmetry exchanges oddand even-dimensional homology, so if X and Z have mirrors X and Z, then b2 (Z) = b2 (X) − (d − r),

b3 (Z) = b2 (X) + 2r,

b4 (Z) = b4 (X) − (d − r).

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The suggested explanation, in the general framework explained above, is that Z should contain d vanishing cycles with d − r relations between them, obtained from a degeneration to a variety Y with d ODP, and that X should be a crepant resolution of Y . Thus mirror symmetry exchanges ODPs with the opposite number of relations between their vanishing cycles. Moreover, to the d vanishing cycles in the original variety X correspond d rational (−1, −1)-curves in its mirror X. It seems plausible to think that the structure sheaves or ideal sheaves of these curves (possibly twisted by some line bundle) should be mirror dual to the Lagrangian spheres representing the vanishing cycles in X; however, as in the K3 case, such a statement is not really meaningful unless one has chosen some specific equivalence D b (X) ∼ = D b Fuk(X). Remark 3.21 When r = 0, H2 (Y ) ∼ = H2 (X), so the exceptional cycles in Y are homologous to zero. This is not possible if the resolution is algebraic, so we exclude this case and also the case d = r to avoid the same problem on the mirror. Going a bit beyond this, we now propose a possible mirror dual to the (A2d−1 )singularity. Let X be a variety that can be degenerated to some Y with an (A2d−1 )singularity, and let v1 , . . . , v2d−1 ∈ Hn (X) be the corresponding vanishing cycles. The signs are fixed in such a way that vi · vi+1 = 1 for all i. We impose two additional conditions. One is that Y should have a partial smoothing Y (equivalently, X a partial degeneration) having d ODPs, built according to the local model x 2 + y 2 + z2 +

d !

(t − Mi )2 = 0

i=1

with the Mi ’s distinct and small. This means that in the (A2d−1 )-configuration of vanishing cycles in X, one can degenerate the 1st, 3rd, . . . , (2d − 1)th to ODPs. The second additional condition is that Y should admit a resolution Z of the standard kind considered above. Then, according to Remark 3.21, there is at least one relation between v1 , v3 , . . . , v2d−1 . In fact, since the intersection matrix of all vi has only a 1-dimensional nullspace, there must be precisely one relation. Remark 3.22 This relation is in fact v1 + v3 + · · · + v2d−1 = 0. The corresponding situation on Z is that all the exceptional P1 ’s are homologous. This should not be too surprising; they can be moved back together to give the d-times thickened P1 in the resolution of the original (A2d−1 )-singularity that one gets by taking the d-fold branch cover t → t d of the resolution of the ODP x 2 + y 2 + z2 + t 2 = 0. We note in passing that out of the 2d possible ways of resolving the ODPs in Y (differing by flops), at most two

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can lead to an algebraic manifold since an exceptional P1 cannot be homologous to minus another one. In view of our previous discussion, we expect that the mirror X of X admits a contraction X → Y to a variety with d ODPs; any smoothing Z of Y should contain d vanishing cycles with (d − 1) relations between them. By (3.21), these give rise to a (d − 1)-dimensional subspace of H4 (X; C) ∼ = H 1,1 (X). There is a natural basis for the relations between the exceptional P1 ’s in Z , which comes from the even-numbered vanishing cycles v2i . The corresponding basis of the subspace of H 1,1 (X) can be represented by divisors S2 , S4 , . . . , S2d−2 such that S2i intersects only the (i − 1)th and ith exceptional P1 . Based on these considerations and others described below, we make a concrete guess as to what X looks like. Definition 3.23 An (A2d−1 )-configuration of subvarieties inside a smooth 3-fold consists of embedded smooth surfaces S2 , S4 , . . . , S2d−2 and curves C1 , C3 , . . . , C2d−1 such that (1) the canonical sheaf of the 3-fold is trivial along each S2i ; (2) each S2i is isomorphic to P2 with two points blown up; (3) S2i ∩ S2j = ∅ for |i − j | ≥ 2; (4) S2i−2 , S2i are transverse and intersect in C2i−1 , which is a rational curve and exceptional (i.e., has self-intersection −1) both in S2i−2 and S2i . Note that the last condition implies that C2i−1 is a (−1, −1)-curve in the 3-fold. What we postulate is that the mirror X contains such a configuration of subvarieties, with the C2i−1 being the exceptional set of the contraction X → Y . Apart from being compatible with the informal discussion above, there are some more feasibility arguments in favour of this proposal. First, such configurations exist as exceptional loci in crepant resolutions of singularities; Figure 2 represents a toric 3-fold with trivial canonical bundle containing such a configuration. The thick lines represent the C2i−1 ’s joining consecutive surfaces S2i−2 , S2i , which are themselves represented by the nodes. Removing these nodes and lines gives the singularity of which it is a resolution by collapsing the whole chain of surfaces and lines; this singularity we think of as the dual of the (A2d−1 )-singularity. We could have deformed the (A2d−1 )-singularity in X differently, for instance, by degenerating an even-numbered vanishing cycle v2i to an ODP. This should correspond to contracting a P1 in X. Assuming that our guess is right, so that X contains an (A2d−1 )-configuration, this should be the other exceptional curve in the S2i besides C2i−1 and C2i+1 (i.e., the line that we call C2i ∼ = P1 joining C2i−1 and C2i+1 ; in Figure 2 these are represented by the vertical lines). Contracting these curves while not

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Figure 2

contracting C2i−1 , C2i+1 turns S2i into a P1 × P1 . The whole 4-cycle S2i contracts to a lower dimensional cycle only when we contract another of the P1 ’s in it, leaving the final P1 (over which the surface fibres) still uncontracted (on X, this corresponds to degenerating two consecutive vanishing cycles while leaving the others finite). Thus there are contractions of X mirroring various possible partial degenerations of X. A final argument in favour of our proposal, and much of the motivation for it, is that it leads to braid group actions on derived categories of coherent sheaves. These are of interest in themselves, independent of whether or not they can be considered to be mirror dual to the braid groups of Dehn twist symplectomorphisms on smoothings of (A2d−1 )-singularities. proposition 3.24 Let X be a smooth quasi-projective 3-fold, and let S2 , S4 , . . . , S2d−2 , C1 , C3 , . . . , C2d−1 be an (A2d−1 )-configuration of subvarieties in X. Then taking Ei = OCi if i is odd, or OSi if i is even, gives an (A2d−1 )-configuration (E1 , E2 , . . . , E2d−1 ) of spherical objects in D b (X). The assumption that the Si are P2 ’s with two points blown up can be weakened considerably for this result to hold; any other rational surface will do. Proposition 3.24 is a 3-dimensional analogue of Example 3.5 and hence, as a comparison with our discussion of K3 surfaces shows, possibly too naive from the mirror symmetry point of view. There is an alternative way of constructing spherical objects, closer to Proposition 3.19. proposition 3.25 Let X be a smooth projective 3-fold that is Calabi-Yau in the strict sense, containing an (A2d−1 )-configuration as in Proposition 3.24. Take rational curves L2i in S2i such that L2i ∩ C2j +1 = ∅ for all i, j . (The inverse image of the generic line in P2 is such a rational curve in the blow-up of P2 .) Choose

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Ei =

OCi (−1) or JCi OSi (−Li ) or OX (−Si )

if i is odd, if i is even.

Then the Ei , i = 1, 2, . . . , 2d −1, form an (A2d−1 )-configuration of spherical objects in D b (X). Here OCi (−1) is shorthand for ι∗ (OP1 (−1)), where ι : P1 → X is some embedding with image Ci , and OSi (−Li ) should be interpreted in the same way. As in Lemma 3.19, the choice of Ei can be made independently for each i. There are many other interesting configurations of spherical objects which arise in connection with 3-fold singularities. Their mirror symmetry interpretations are mostly unclear. For instance, a slight variation of the situation above yields braid group actions built only from structure sheaves of surfaces. proposition 3.26 Let X be a smooth quasi-projective 3-fold, and let S1 , S2 , . . . , Sm be a chain of smooth embedded rational surfaces in X with the following properties: Si ∩ Si+1 is transverse and consists of one rational curve whose self-intersection in Si and Si+1 is either zero or −2; Si ∩ Sj = ∅ for |i − j | ≥ 2; and ωX |Si is trivial. Then Ei = OSi is an (Am )-configuration of spherical objects in D b (X). The conditions actually imply that every intersection Si ∩ Si+1 is a rational curve with normal bundle ∼ = OP1 ⊕ OP1 (−2) in X. Note also that the presence of rational curves with self-intersection zero forces at least every second of the surfaces Si to be fibred over P1 . Configurations of this kind are the exceptional loci of crepant resolutions of suitable toric singularities. In a different direction, I. Nakamura’s resolutions of abelian quotient singularities using Hilbert schemes of clusters, with their toric representations as tessellations of hexagons (see [39], [10]), lead to situations similar to Proposition 3.24. The nodes of the hexagons in Figure 3 represent surfaces that are the blow-ups of P1 × P1 in two distinct points; the six lines emanating from a node represent the six exceptional P1 ’s in the surface, in which it intersects the other surfaces represented by the other nodes that the lines join. The structure sheaves of these curves and surfaces give rise to twists on the derived category satisfying braid relations according to the Dynkintype diagram obtained by adding a vertex in the middle of each edge. (These added vertices represent the structure sheaves of the curves; see Figure 3.) The McKay correspondence (see Section 3.2) translates this into a group of twists on the equivariant derived category of the 3-fold on which the finite group acted.

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Figure 3

4. Faithfulness 4.1. Differential graded algebras and modules The notions discussed in this section are, for the most part, familiar ones; we collect them here to set up the terminology and also for the reader’s convenience. A detailed exposition of the general theory of differential graded modules can be found in [1, Section 10]. Fix a field k, and fix an integer m ≥ 1. Take the semisimple k-algebra R = k m with generators e1 , . . . , em and relations ei2 = ei for all i, ei ej = 0 for i = j (so 1 = e1 + · · · + em is the unit element). R plays the role of ground ring in the following considerations. In particular, by a graded algebra we always mean a Zgraded unital associative k-algebra A, together with a homomorphism (of algebras, and unital) ιA : R → A0 . This equips A with the structure of a graded R-bimodule, and the multiplication becomes a bimodule map. For the sake of brevity, we denote the bimodule structure by ei a and aei (a ∈ A) instead of ιA (ei )a, respectively, a ιA (ei ). All homomorphisms A → B between graded algebras are required to commute with the maps ιA , ιB . A differential graded algebra (dga) A = (A, dA ) is a graded algebra A together with a derivation dA of degree 1, which satisfies dA2 = 0 and dA ◦ ιA = 0. The cohomology H (A ) of a dga is a graded algebra. A homomorphism of dgas is called a quasi-isomorphism if it induces an isomorphism on cohomology. Two dgas A , B are called quasi-isomorphic if there is a chain of dgas and quasiisomorphisms A ← C1 → · · · ← Ck → B connecting them. (In fact, it is sufficient to allow k = 1 since the category of dgas admits a calculus of fractions; see [26, Lemma 3.2].) A dga A is called formal if it is quasi-isomorphic to its own cohomology algebra H (A ), thought of as a dga with zero differential. By a graded module over a graded algebra A, we always mean a graded right A-module. Through the map ιA , any such module M becomes a right R-module; again, we write xei (x ∈ M) instead of xιA (ei ). A differential graded module (dgm) over a dga A = (A, dA ) is a pair M = (M, dM ) consisting of a graded A-module

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2 = 0 and d (xa) = M and a k-linear map dM : M → M of degree 1 such that dM M (dM x)a + (−1)deg(x) x(dA a) for a ∈ A. The cohomology H (M ) is a graded module over H (A ). For instance, A is a dgm over itself, and as such it splits into a direct sum of dgms Pi = ei A, dA |ei A , 1 ≤ i ≤ m. (4.1)

By definition, a dgm homomorphism M → N is a homomorphism of graded modules which is at the same time a homomorphism of chain complexes. Dgms over A and their homomorphisms form an abelian category Dgm(A ). One can also define chain homotopies between dgm homomorphisms. The category K(A ), with the same objects as Dgm(A ) and with the homotopy classes of dgm homomorphisms as morphisms, is triangulated. The translation functor in it takes M = (M, dM ) to M [1] = (M[1], −dM ), with no change of sign in the module structure. Exact triangles are all those isomorphic to one of the standard triangles involving a dgm homomorphism and its cone. Having mentioned cones, we use the opportunity to introduce a slight generalisation, which is used later on. Assume that one has a chain complex in Dgm(A ), namely, dgms Ci , i ∈ Z, and dgm homomorphisms δi : Ci → Ci+1 such that δi+1 δi = 0. Then one can form a new dgm C by setting C = i∈Z Ci [i] and   ··· δi−1 (−1)i dCi  . dC =  i+1   δi (−1) dC δi+1

i+1

···

We refer to this as collapsing the chain complex (it can also be viewed as a special case of a twisted complex; see, e.g., [3]), and we write C = {· · · → Ci → Ci+1 → · · · }; for complexes of length 2, it specializes to the cone of a dgm homomorphism. Inverting the dgm quasi-isomorphisms in K(A ) yields another triangulated category D(A ), in which any short exact sequence of dgms can be completed to an exact triangle. As usual, D(A ) can also be defined by inverting the quasi-isomorphisms directly in Dgm(A ), but then the triangulated structure is more difficult to see. We call D(A ) the derived category of dgms over A . Warning Even though we use the same notation as in ordinary homological algebra, the expressions K(A ) and D(A ) have a different meaning here. In particular, D(A ) is not the derived category, in the usual sense, of Dgm(A ). For any dga homomorphism f : A → B there is a “restriction of scalars” functor Dgm(B ) → Dgm(A ). This preserves homotopy classes of homomorphisms, takes

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cones to cones, and commutes with the shift functors. Hence it descends to an exact functor K(B ) → K(A ). Moreover, it obviously preserves quasi-isomorphisms, so that it also descends to an exact functor D(B ) → D(A ); we denote any of these functors by f ∗ . The next result, taken from [1, Theorem 10.12.5.1], shows that two quasi-isomorphic dgas have equivalent derived categories. theorem 4.1 If f is a quasi-isomorphism, f ∗ : D(B ) → D(A ) is an exact equivalence. Let A be a dga. The standard twist functors t1 , . . . , tm from Dgm(A ) to itself are defined by ti (M ) = M ei ⊗k Pi −→ M . The tensor product of M ei = (Mei , dM |Mei ) with the dgm Pi of (4.1) is one of complexes of k-vector spaces; it becomes a dgm with the module structure inherited from Pi . The arrow is the multiplication map Mei ⊗k ei A → M, which is a homomorphism of dgms, and we are taking its cone; ti descends to exact functors K(A ) → K(A ) and D(A ) → D(A ), for which we use the same notation. This is straightforward for K(A ). As for D(A ), one needs to show that ti preserves quasi-isomorphisms; this follows from looking at the long exact sequence · · · −→ H (M )ei ⊗k ei H (A ) −→ H (M ) −→ H (ti (M )) −→ · · ·. lemma 4.2 Let f : A → B be a quasi-isomorphism of dgas. Then the following diagram commutes up to isomorphism, for each 1 ≤ i ≤ m: D(B )

ti

f∗

D(A )

D(B ) f∗

ti

D(A )

Proof Let M = (M, dM ) be a dgm over B . Consider the commutative diagram of dgms over A : f ∗M M e i ⊗ k ei A id ⊗(f |ei A)

M ei ⊗k f ∗ (ei B )

id

f ∗M

The upper horizontal arrow is m ⊗ a → mf (a), and the lower one is multiplication.

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The cone of the upper row is ti (f ∗ (M )), while that of the lower one is f ∗ (ti (M )). The two vertical arrows combine to give a quasi-isomorphism between these cones. Now let S ⊂ S be as in Section 2.1, and let K be the category from Definition 2.2. Let E1 , . . . , Em be objects of K, and let E be their direct sum. The chain complex of endomorphisms hom Ei , Ej end(E) := hom(E, E) = 1≤i, j ≤m

has a natural structure of a dga. Multiplication is given by composition of homomorphisms; ιend(E) maps ei ∈ R to idEi ∈ hom(Ei , Ei ), so left multiplication with ei is the projection to hom(E, Ei ), while right multiplication is the projection to hom(Ei , E). In the same way, for any F ∈ K, the complex hom(E, F ) is a dgm over end(E). The functor hom(E, −) : K → K(end(E)) defined in this way is exact because it carries cones to cones. The objects Ei get mapped to the dgms hom(E, Ei ) = ei end(E), which are precisely the Pi from (4.1). We define a functor .E to be the composition quotient functor hom(E,−) K −−−−−−−→ K end(E) −−−−−−−−−→ D end(E) . lemma 4.3 Assume that E1 , . . . , Em satisfy the conditions from Definition 2.5, so that the twist functors TEi are defined. Then the following diagram is commutative up to isomorphism, for each 1 ≤ i ≤ m: TEi

K

K

.E

.E

D end(E)

ti

D end(E)

Proof For F ∈ K, consider the commutative diagram of dgms over end(E), hom(Ei , F ) ⊗k hom(E, Ei )

hom(E, F ) id

hom E, hom(Ei , F ) ⊗k Ei

hom(E, F )

with the following maps: the horizontal arrow in the first row is the composition; that in the second row is induced by the evaluation map hom(Ei , F ) ⊗k Ei → F . The left-hand vertical arrow is the first of the canonical maps from (2.1), which is a quasi-isomorphism since hom(Ei , F ) has finite-dimensional cohomology. The cone

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of the first row is ti (.E (F )), while that of the second row is .E (TEi (F )). The vertical arrows combine to give a natural quasi-isomorphism between these cones. Later on, in our application, the Ei occur as resolutions of objects in D b (S ). The next two lemmas address the question of how the choice of resolutions affects the construction. This is not strictly necessary for our purpose, but it rounds off the picture. lemma 4.4 Let Ei , Ei (1 ≤ i ≤ m) be objects in K such that Ei ∼ = Ei for all i. Then the dgas end(E) and end(E ) are quasi-isomorphic. Proof Choose for each i a map gi : Ei → Ei which is a chain homotopy equivalence. Set Ci = Cone(gi ), and let C be the direct sum of these cones; this is the same as the cone of g = g1 ⊕ · · · ⊕ gm . Let end(C) be the endomorphism dga of C1 , . . . , Cm . An element of end(C) of degree r is a matrix φ11 φ12 φ= φ21 φ22 with φ11 ∈ homr (E, E), φ21 ∈ homr−1 (E, E ), φ12 ∈ homr+1 (E , E), φ22 ∈ homr (E , E ). The differential in end(C) maps φ to −dE φ11 + (−1)r φ11 dE − (−1)r φ12 g −dE φ12 − (−1)r φ12 dE . gφ11 − (−1)r φ22 g + dE φ21 + (−1)r φ21 dE gφ12 + dE φ22 − (−1)r φ22 dE Let C ⊂ end(C) be the subalgebra of matrices that are lower-triangular (φ12 = 0). The formula above shows that this is closed under the differential and hence again a dga. The projection π2 : C → end(E ), π2 (φ) = φ22 , is a homomorphism of dgas. Its kernel is isomorphic (as a complex of k-vector spaces, and up to a shift) to the cone of the composition with g map g ◦ − : hom(E, E) −→ hom(E, E ). Since g is a homotopy equivalence, this cone is acyclic, so that π2 is a quasiisomorphism of dgas. A similar argument shows that the projection π1 : C → end(E), π1 (φ) = (−1)deg(φ) φ11 , is a quasi-isomorphism of dgas. The two maps together prove that end(E) and end(E ) are quasi-isomorphic. As a consequence of this and Theorem 4.1, the categories D(end(E)) and D(end(E )) are equivalent. Actually, we have shown a more precise statement: any choice of

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gi : Ei → Ei yields, up to isomorphism of functors, an exact equivalence (π2∗ )−1 π1∗ : D(end(E)) → D(end(E )). We now see that this equivalence is compatible with the functors .E , .E . lemma 4.5 In the situation of Lemma 4.4, (π2∗ )−1 π1∗ ◦ .E ∼ = .E . Proof The obvious short exact sequence 0 → E → C → E[1] → 0 induces, for any F ∈ K, a short exact sequence of dgms over C : 0 −→ π1∗ hom(E, F )[−1] −→ hom(C, F ) −→ π2∗ hom(E , F ) −→ 0. In the derived category D(C ), this short exact sequence can be completed to an exact triangle by a morphism π2∗ hom(E , F ) −→ π1∗ hom(E, F ).

(4.2)

One can define such a morphism explicitly by replacing the given sequence with a (canonically constructed) quasi-isomorphic one, for which the corresponding morphism can be realized by an actual homomorphism of dgms (cf. [14, Proposition III.3.5]). The advantage of this explicit construction is that (4.2) is now natural in F . Since C is a contractible complex, hom(C, F ) is acyclic, which implies that (4.2) is an isomorphism in D(C ) for any F . This shows that the diagram .E

D end(E)

π1∗

K D(C )

.E π2∗

D end(E )

commutes up to isomorphism, as desired. 4.2. Intrinsic formality Applications of dga methods to homological algebra often hinge on constructing a chain of quasi-isomorphisms connecting two given dgas. For instance, in the situation explained in Section 4.1, one can try to use the dga end(E) to study the twists TEi via the functor .E . What really matters for this purpose is only the quasiisomorphism type of end(E). In general, quasi-isomorphism type is a rather subtle invariant. However, there are some cases where the cohomology already determines the quasi-isomorphism type.

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Definition 4.6 A graded algebra A is called intrinsically formal if any two dgas with cohomology A are quasi-isomorphic or, equivalently, if any dga B with H (B ) ∼ = A is formal. For instance, one can show easily that any graded algebra A concentrated in degree zero is intrinsically formal. (This particular example can be viewed as the starting point for J. Rickard’s theory of derived Morita equivalences (see [46]), as recast in dga language by B. Keller [25].) However, our intended application is to algebras of a rather different kind. An augmented graded algebra is a graded algebra A together with a graded algebra homomorphism MA : A → R which satisfies MA ◦ ιA = idR . Its kernel is a 2-sided ideal, called the augmentation ideal; we write it as A+ ⊂ A. A special case is when A is connected, which means Ai = 0 for i < 0 and ιA : R → A0 is an isomorphism; then there is of course a unique augmentation map, and A+ is the subspace of elements of positive degree. theorem 4.7 Let A be an augmented graded algebra. If H H q (A, A[2 − q]) = 0 for all q > 2, then A is intrinsically formal. We remind the reader that the Hochschild cohomology H H ∗ (A, M) of a graded A-bimodule M is the cohomology of the cochain complex   q " #$ %   C q (A, M) = HomR−R A+ ⊗R · · · ⊗R A+ , M , (∂ q φ) a1 , . . . , aq+1 = (−1)M a1 φ a2 , . . . , aq+1 +

q &

(−1)Mi φ a1 , . . . , ai ai+1 , . . . , aq+1

i=1

− (−1)Mq φ a1 , . . . , aq aq+1 , where HomR−R denotes homomorphisms of graded R-bimodules. (By definition, these are homomorphisms of degree zero.) The signs are M = q deg(a1 ), Mi = deg(a1 ) + · · · + deg(ai ) − i. The bimodules relevant for our application are M = A[s] with the left multiplication twisted by a sign: a ·x ·a = (−1)s deg(a) axa for a, a ∈ A and x ∈ M. Note that the chain complex C ∗ (A, A[s]) depends on s, so the cohomology groups that occur in Theorem 4.7 belong to different complexes. We give a proof of Theorem 4.7 for lack of an accessible reference, and also because our framework (in which dgas may be nonzero in positive and negative

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degrees) differs slightly from the usual one. However, the result is by no means new. Originally, the phenomenon of intrinsic formality was discovered by Halperin and Stasheff [19] in the framework of commutative dgas. They constructed a series of obstruction groups whose vanishing implies intrinsic formality. Later D. Tanré [52] identified these obstruction groups as Harrison cohomology groups. To the best of our knowledge, the noncommutative version, in which Hochschild cohomology replaces Harrison cohomology, is due to T. Kadeishvili [23], who also realized the importance of A∞ -algebras in this context. A general survey of A∞ -algebras and applications is [27]. It is difficult to find a concrete counterexample, but apparently Theorem 4.7 is not true without the augmentedness assumption. This is related to the problem that a general A∞ -algebra with unit might not be quasi-isomorphic to a dga with unit. (We do not know if this question has been settled, but the construction of X below would not work.) Let A be an augmented graded algebra, and let B = (B, dB ) be a dga. An A∞ -morphism γ : A → B is a sequence of maps of graded R-bimodules γq ∈ HomR−R ((A+ )⊗R q , B[1 − q]), q ≥ 1, satisfying the equations q−1

& dB γq a1 , . . . , aq = (−1)Mi γq−1 a1 , . . . , ai ai+1 , . . . , aq i=1

(Eq )

− γi a1 , . . . , ai γq−i (ai+1 , . . . , aq . The Mi are as in the definition of H H ∗ (A, M) above. The first two of these equations are (E1 ) dB γ1 (a1 ) = 0, dB γ2 a1 , a2 = (−1)deg(a1 )−1 γ1 (a1 a2 ) − γ1 (a1 )γ1 (a2 ) . (E2 ) This means that γ1 , which need not be a homomorphism of algebras, nevertheless induces a multiplicative map (γ1 )∗ : A+ → H (B ). In a sense, the nonmultiplicativity of γ1 is corrected by the higher order maps γq , so that A∞ -morphisms are “approximately multiplicative maps.” From a more classical point of view, one can see A∞ -morphisms simply as a convenient way of encoding dga homomorphisms from a certain large dga canonically associated to A, a kind of “thickening of A.” Consider V = A+ [1] as a graded R bimodule, and let T + V = q≥1 V ⊗R q be its tensor algebra, without unit. We write a1 , . . . , aq ∈ T + V instead of a1 ⊗ · · · ⊗ aq . Now consider W = T + V [−1] as a graded R-bimodule in its own right, and form its tensor algebra with unit T W = R ⊕ r≥1 V ⊗R r . The elements of T W (apart from R ⊂ T W ) are linear combinations of expressions of the form

x = a11 , a12 , . . . , a1,q1 ⊗ · · · ⊗ ar1 , . . . , ar,qr with r > 0, q1 , . . . , qr > 0, and aij ∈ A+ . The degree of such an expression is

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degT W (x) = ij degA (aij ) − i qi + r. One defines a dga X = (X, dX ) by taking X = T W with the tensor multiplication and taking dX to be the derivation that acts on elements of W as follows: q−1

&

dX a1 , . . . , aq = (−1)Mi a1 , . . . , ai ai+1 , . . . , aq i=1

− a1 , . . . , ai ⊗ ai+1 , . . . , aq .

The passage from A to X is usually written as a composition of the bar and cobar functors, which go from augmented dg algebras to dg coalgebras and back (see, e.g., [35]). We can now make the above-mentioned connection with A∞ -morphisms. lemma 4.8 For any A∞ -morphism γ : A → B , one can define a dga homomorphism G : X → B by setting G | R to be the unit map ιB , and G(a1 , . . . , aq ) = γq (a1 , . . . , aq ). G is a quasi-isomorphism if and only if ιB ⊕ γ1 induces an isomorphism between R ⊕ A+ ∼ = A and H (B ). Proof The first part follows immediately from comparing the equations (Eq ) with the definition of the differential dX . As for the second part, a classical computation due to J. Moore [34, Théorème 6.2], [35] shows that the inclusion R ⊕ A+ @→ ker dX induces an isomorphism R ⊕ A+ ∼ = H (X ). This implies the desired result. As a trivial example, let A = (A, 0) be the dga given by A with zero differential, and take the A∞ -morphism γ : A → A given by γ1 = id : A+ → A, γq = 0 for all q ≥ 2. Then Lemma 4.8 shows that the corresponding map G : X → A is a quasi-isomorphism of dgas. The next lemma is an instance of “homological perturbation theory” (see, e.g., [18]). Let A be an augmented graded algebra, let B be a dga, and let φ : A → H (B ) be a homomorphism of graded algebras. This makes the cohomology H (B ) into a graded A-bimodule. lemma 4.9 Assume that H H q (A, H (B )[2 − q]) = 0 for all q > 2. Then there is an A∞ morphism γ : A → B such that the induced map (γ1 )∗ : A+ → H (B ) is equal to φ|A+ . Proof Choose a map of graded R-bimodules γ1 : A+ → ker dB ⊂ B which induces φ|A+ .

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Since γ1 is multiplicative on cohomology, we can find a map γ2 such that (E2 ) is satisfied. From here onwards the construction is inductive. Suppose that γ1 , . . . , γq−1 , for some q ≥ 3, are maps such that (E1 ), . . . , (Eq−1 ) hold. Denote the right-hand side of equation (Eq ) for these maps by ψ : (A+ )⊗R q → B[2 − q]. One can compute directly that (4.3) dB ψ a1 , . . . , aq = 0 for all a1 , . . . , aq ∈ A+ and that q

& γ1 (a1 )ψ a2 , . . . , aq+1 + (−1)Mi ψ a1 , . . . , ai ai+1 , . . . , aq+1 i=1

− (−1)Mq ψ a1 , . . . , aq γ1 aq+1 ' q ( & Mi (−1) γi a1 , . . . , ai γq+1−i ai+1 , . . . , aq+1 . = dB

(4.4)

i=1

By (4.3), ψ induces a map ψ¯ : (A+ )⊗R q → H (B )[2 − q], which is just an element of the Hochschild chain group C q (A, H (B )[2 − q]). Equation (4.4) says that ψ¯ is a Hochschild cocycle. By assumption, there is an η¯ ∈ C q−1 (A, H (B )[2 − q]) ¯ Choose any map of graded R-bimodules η : (A+ )⊗R q−1 → such that ∂ q−1 η¯ = ψ. new = γ ¯ and set γq−1 (ker dB )[1 − q] which induces η, q−1 − η. The equations (E1 ), . . . , new (Eq−1 ) continue to hold if one replaces γq−1 by γq−1 . Moreover, if ψ new denotes the right-hand side of (Eq ) after this replacement, one computes that ψ − ψ new a1 , . . . , aq = (−1)deg(a1 ) γ1 (a1 )η a2 , . . . , aq−1 q−1 & + (−1)Mi η a1 , . . . , ai ai+1 , . . . , aq

(4.5)

i=1

− (−1)Mq η a1 , . . . , aq−1 γ1 (aq ). This means that ψ¯ new = ψ¯ − ∂ q−1 η¯ = 0. Clearly, the vanishing of ψ¯ new ensures that new by a map γ such that (E ) holds. one can extend the sequence γ1 , . . . , γq−2 , γq−1 q q This completes the induction step. Note that in the qth step only the (q − 1)st of the given maps γi is changed. Therefore the sequence that we construct does indeed converge to an A∞ -morphism γ .

Proof of Theorem 4.7 Let B be a dga whose cohomology algebra is isomorphic to A. Choose an isomorphism φ : A → H (B ). By Lemma 4.9, there is an A∞ -morphism γ : A → B such that γ1 induces φ | A+ . This obviously means that (ιB ⊕ γ1 )∗ : R ⊕ A+ → H (B ) is an isomorphism. Hence, by Lemma 4.8, the induced map G : X → B is a

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quasi-isomorphism of dgas. We have already seen that there is a quasi-isomorphism X → A = (A, 0). This shows that B is quasi-isomorphic to A and hence is formal. 4.3. The graded algebras Am,n We assume from now on that m ≥ 2; this assumption is retained throughout this section and the following one. In addition, choose an n ≥ 1. Let G be a quiver (an oriented graph) with vertices numbered 1, . . . , m, and with a “degree” (an integer label) attached to each edge. One can associate to it a graded algebra k[G], the path algebra, as follows. As a k-vector space, k[G] is freely generated by the set of all paths (not necessarily closed, of arbitrary length greater than or equal to zero) in G. The degree of a path is the sum of all “degrees” of the edges along which it runs. The product of two paths is their composition if the endpoint of the first one coincides with the starting point of the second one, and zero otherwise. The map ιk[G] : R → (k[G])0 maps ei to the path of length zero at the ith vertex. d1 1

n − d1

d2 2

n − d2

3

m

Figure 4

The example we are interested in is the quiver Gm,n shown in Figure 4. Paths of length l ≥ 0 in this quiver correspond to (l + 1)-tuples (i0 | · · · |il ) with iν ∈ {1, . . . , m} and |iν+1 − iν | = 1. The product of two paths in k[Gm,n ] is given by (i0 | · · · |il )(i0 | · · · |il ) = (i0 | · · · |il |i1 | · · · |il ) if il = i0 , or zero otherwise. The grading is deg(i) = 0, deg(i|i + 1) = di , deg(i + 1|i) = n − di , where we set  1   if n is even,  n 2 (4.6) di =  1  i  n + (−1) if n is odd. 2 We introduce a 2-sided homogeneous ideal Jm,n ⊂ k[Gm,n ] as follows. If m ≥ 3, then Jm,n is generated by (i|i − 1|i) − (i|i + 1|i), (i − 1|i|i + 1), and (i + 1|i|i − 1) for all i = 2, . . . , m − 1; in the remaining case m = 2, Jm,n is generated by (1|2|1|2) and (2|1|2|1). Now define Am,n = k[Gm,n ]/Jm,n . This is again a graded algebra. It is finite-dimensional over k; an explicit basis is given by the (4m − 2) elements

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        

(1), . . . , (m), (1|2), . . . , (m − 1|m),

(2|1), . . . , (m|m − 1),     (1|2|1), (2|3|2) = (2|1|2), . . . , m − 1|m|m − 1     = m − 1|m − 2|m − 1 , m|m − 1|m .

(4.7)

Here we have used the same notation for elements of k[Gm,n ] and their images in Am,n . We will continue to do so in the future; in particular, (i|i ± 1|i) is used to denote the image of both (i|i + 1|i) and (i|i − 1|i) in Am,n . We now explain why these algebras are relevant to our problem. Let K be a category as in Definition 2.2, and let E1 , . . . , Em ∈ K be an (Am )-configuration of n-spherical objects. lemma 4.10 Suppose that for each i = 1, . . . , m − 1 the 1-dimensional space Hom∗ (Ei+1 , Ei ) is concentrated in degree di . Then the cohomology algebra of the dga end(E) is isomorphic to Am,n . We should say that the assumption on Hom∗ (Ei+1 , Ei ) is not really restrictive since, given an arbitrary (Am )-configuration, it can always be achieved by shifting each Ei suitably. Proof Since each Ei is n-spherical, the pairings Hom∗ Ei+1 , Ei ⊗ Hom∗ Ei , Ei+1 −→ Homn Ei , Ei ∼ = k, ∗ ∗ n Hom Ei , Ei+1 ⊗ Hom Ei+1 , Ei −→ Hom Ei+1 , Ei+1 ∼ =k

(4.8)

are nondegenerate for i = 1, . . . , m − 1. Hence Hom∗ (Ei , Ei+1 ) ∼ = k is concentrated in degree n − di . Choose nonzero elements αi ∈ Hom∗ (Ei+1 , Ei ) and βi ∈ Hom∗ (Ei , Ei+1 ). Then, again because of the nondegeneracy of (4.8), one has (4.9) αi βi = ci βi−1 αi−1 in Hom∗ (E, E) for some nonzero constants c2 , . . . , cm−1 ∈ k. Without changing notation, we multiply each βi with c2 c3 · · · ci ; then the same equations (4.9) hold with all ci equal to 1. Since Hom∗ (Ei , Ej ) = 0 for all |i − j | ≥ 2, we also have βi βi−1 = 0, αi−1 αi = 0 for all i = 2, . . . , m − 1. If m ≥ 3, then this shows that there is a homomorphism of graded algebras Am,n → Hom∗ (E, E) which maps (i) to idEi , (i|i + 1) to αi , and (i + 1|i) to βi . One sees easily that this is an isomorphism. In the remaining case m = 2, one has to consider

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β1 α1 β1 ∈ Hom2n−d1 E1 , E2 ,

α1 β1 α1 ∈ Homn+d1 E2 , E1 .

(4.10)

By assumption Hom∗ (E1 , E2 ) is concentrated in degree n − d1 < 2n − d1 , and Hom∗ (E2 , E1 ) is concentrated in degree d1 < n + d1 . Hence both elements in (4.10) are zero, which allows one to define Am,n → Hom∗ (E, E) as before. The proof that this is an isomorphism is again straightforward. An inspection of the preceding proof shows that the result remains true for any other choice of numbers di in the definition of Am,n . Our particular choice (4.6) makes the algebra as “highly connected” as possible; Am,n /R · 1 is concentrated in degrees greater than or equal to [n/2]. This is useful in the Hochschild cohomology computations of Section 4.5. Let Am,n be the dga given by Am,n with zero differential. We now consider the properties of the functors ti on the category D(Am,n ). lemma 4.11 The functors ti : D(Am,n ) → D(Am,n ), 1 ≤ i ≤ m, are exact equivalences. Proof This is closely related to the parallel statements in [28] and in Section 2.2. The strategy, as in Proposition 2.10, is to introduce a left adjoint ti of ti , and then to prove that the canonical natural transformations Id → ti ti , ti ti → Id are isomorphisms. Set A = Am,n and Qi = Pi [n] ∈ Dgm(A ). Define functors ti (1 ≤ i ≤ m) from Dgm(A ) to itself by ηi ti (M ) = M −−→ M ei ⊗k Qi , where M is placed in degree zero, and ηi (x) = x(i|i ± 1|i) ⊗ (i) + x(i + 1|i) ⊗ (i|i + 1) + x(i − 1|i) ⊗ (i|i − 1) + x(i) ⊗ (i|i ± 1|i). (In this formula, the second term should be omitted for i = m and the third term for i = 1; the same convention is used again later on.) To understand why ηi is a module homomorphism, it is sufficient to notice that the element (i|i ± 1|i) ⊗ (i) + (i + 1|i) ⊗ (i|i + 1) + (i − 1|i) ⊗ (i|i − 1) + (i) ⊗ (i|i ± 1|i) ∈ Aei ⊗ ei A

(4.11)

is central, in the sense that left and right multiplication (with respect to the obvious A-bimodule structure of Aei ⊗ ei A) with any a ∈ A have the same effect on it. The same argument as for ti shows that ti descends to exact functors on K(A ) and D(A ). For any M ∈ Dgm(A ), consider the complex of dgms δ0 δ−1 C−1 = M ei ⊗ Pi −−−→ C0 = M ⊕ M ei ⊗ ei Aei ⊗ Qi −−→ C1 = M ei ⊗ Qi ,

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where δ−1 (x ⊗ a) = (xa, x ⊗ a(i|i ± 1|i) ⊗ (i) + x ⊗ a(i + 1|i) ⊗ (i|i + 1) + x ⊗ a(i − 1|i)⊗(i|i −1)+x ⊗a(i)⊗(i|i ±1|i)) = (xa, x ⊗(i)⊗(i|i ±1|i)a +x ⊗(i|i ±1|i)⊗a) and δ0 (x, y ⊗a⊗b) = (ηi (x)−ya⊗b). The reason why the second expression for δ−1 is equal to the first one is again that the element (4.11) is central. A straightforward computation (including some tedious sign checking) shows that the dgm C obtained by collapsing this complex is equal to ti ti (M ). The algebra ei Aei = k(i)⊕k(i|i ±1|i) is simply a 2-dimensional graded k-vector space, nontrivial in degrees zero and n. Take the homomorphism of dgms C0 = M ⊕ M ei ⊗ ei Aei ⊗ Qi −→ M , (4.12) x, y1 ⊗ (i) ⊗ b1 + y2 ⊗ (i|i ± 1|i) ⊗ b2 −→ x − y2 b2 . Extending this by zero to C−1 , C1 yields a dgm homomorphism ψM : C = ti ti (M ) → M because (4.12) vanishes on the image of δ−1 . This homomorphism is surjective for any M , and a computation similar to that in Proposition 2.10 shows that the kernel is always an acyclic dgm. Since ψM is natural in M , we have indeed provided an isomorphism ti ti ∼ = IdD(A ) . The proof that ti ti ∼ = IdD(A ) is parallel. lemma 4.12 The functors ti on D(Am,n ) satisfy the braid relations (up to graded natural isomorphism) ti ti+1 ti ∼ = ti+1 ti ti+1 for i = 1, . . . , m − 1, ti tj ∼ = tj ti for |i − j | ≥ 2. Proof The second relation is easy (it follows immediately from the fact that ei Am,n ej = 0 for |i − j | ≥), and we therefore concentrate on the first one. Moreover, we only explain the salient points of the argument. (A different version of it is described in [28] with full details.) Note that the approach taken in Proposition 2.13 cannot be adapted directly to the present case since we have not developed a general theory of twist functors on derived categories of dgms. Set A = Am,n and Ri = Pi [−n]. For any M ∈ Dgm(A ), consider the complex of dgms δ−3

δ−2

δ−1

C−3 −−−→ C−2 −−−→ C−1 −−−→ C0 ,

where C−3 = M ei ⊗ Ri , C−2 = M ei ⊗ ei Aei ⊗ Pi ⊕ M ei ⊗ ei Aei+1 ⊗ Pi+1 ⊕ M ei+1 ⊗ ei+1 Aei ⊗ Pi ,

(4.13)

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C−1 = M ei ⊗ Pi ⊕ M ei+1 ⊗ Pi+1 ⊕ M ei ⊗ Pi , C0 = M ,

and

δ−2



 −x ⊗ (i|i + 1|i) ⊗ a δ−3 : (x ⊗ a) −→ x ⊗ (i|i + 1) ⊗ (i + 1|i)a  , x(i|i + 1) ⊗ (i + 1|i) ⊗ a     x1 ⊗ a 1 ⊗ b 1 x1 ⊗ a1 b1 + x2 ⊗ (i|i + 1)b2 : x2 ⊗ (i|i + 1) ⊗ b2  −→ −x2 (i|i + 1) ⊗ b2 + x3 ⊗ (i + 1|i)b3  , x3 ⊗ (i + 1|i) ⊗ b3 −x1 a ⊗ b1 − x3 (i + 1|i) ⊗ b3   x1 ⊗ a1 δ−1 : x2 ⊗ a2  −→ x1 a1 + x2 a2 + x3 a3 . x3 ⊗ x3

As in the proof of Lemma 4.11, one can contract this complex to a single dgm, which is in fact canonically isomorphic to ti ti+1 ti (M ). Now, one can map the whole complex (4.13) surjectively to an acyclic complex (concentrated in degrees −3 and −2) id

M ei ⊗ Ri −−→ M ei ⊗ Ri .

This is done by taking the identity map on C−3 together with the homomorphism C−2 ⊃ M ei ⊗ ei Aei ⊗ Pi → M ei ⊗ Ri , m1 ⊗ (i) ⊗ b1 + m2 ⊗ (i|i + 1|i) ⊗ b2 → −m2 ⊗ b2 and extending this by zero to the other summands of C−2 and to C−1 , C0 . The kernel of the dgm homomorphism defined in this way is a certain subcomplex of (4.13). When writing this down explicitly (which we do not do here), one notices that it contains an acyclic subcomplex isomorphic to id

M ei ⊗ Pi −−→ M ei ⊗ Pi ,

located in degrees −2 and −1. If one divides out this acyclic subcomplex, what remains is the complex M ei ⊗ ei Aei+1 ⊗ Pi+1 ⊕ M ei+1 ⊗ ei+1 Aei ⊗ Pi (4.14) δ−1 δ0 −−−→ M ei ⊗ Pi ⊕ M ei+1 ⊗ Pi+1 −−→ M , (x ⊗ (i|i + 1) ⊗ b , x ⊗ (i + 1|i) ⊗ b ) = (x ⊗ (i|i + 1)b − x (i + 1|i) ⊗ with δ−1 1 1 2 2 1 1 2 b2 , −x1 (i|i + 1) ⊗ b1 + x2 ⊗ (i + 1|i)b2 ), δ0 (x1 ⊗ a1 , x2 ⊗ a2 ) = x1 a1 + x2 a2 . The remarkable fact about (4.14) is that it is symmetric with respect to exchanging i and i + 1. Indeed, one can arrive at the same complex by starting with ti+1 ti ti+1 (M ) and removing acyclic parts. This shows that ti+1 ti ti+1 (M ) and ti ti+1 ti (M ) are quasi-

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isomorphic for all M . We leave it to the reader to verify that the argument provides a chain of exact functors and graded natural isomorphisms between them, with ti ti+1 ti and ti+1 ti ti+1 at the two ends of the chain. 4.4. Geometric intersection numbers Consider the weak braid group action ρm,n : Bm+1 → Auteq(D(Am,n )) generated by t1 , . . . , tm . The aim of this section is prove a strong form of faithfulness for it. theorem 4.13 g g Let Rm,n be a functor representing ρm,n (g) for some g ∈ Bm+1 . If Rm,n (Pj ) ∼ = Pj for all j , then g must be the identity element. We begin by looking at the center of Bm+1 . It is infinite cyclic and generated by an element that, in terms of the standard generators g1 , . . . , gm , can be written as (g1 g2 · · · gm )m+1 . lemma 4.14 For any 1 ≤ j ≤ m, (t1 t2 · · · tm )m+1 (Pj ) is isomorphic to Pj [2m − (m + 1)n] in D(Am,n ). Proof For each 1 ≤ j ≤ m there is a short exact sequence of dgms α

multiplication

0 −→ Pj [−n] −→ ej Am,n ej ⊗k Pj −−−−−−−→ Pj −→ 0, where α(x) = (j |j ± 1|j ) ⊗ x − (j ) ⊗ (j |j ± 1|j )x. This implies that the cone of the multiplication map, which is tj (Pj ), is isomorphic to Pj [1 − n] in D(Am,n ). Note also that ti (Pj ) ∼ = Pj whenever |i − j | ≥ 2. Consider the m + 1 differential graded modules ) * +, M0 = P1 [n−1] −→ P2 [2n−1−d1 ] −→ · · · −→ Pm mn −1− d1 − · · · − dm−1 , M1 = P1 , M2 = P2 [1 − d1 ], M3 = P3 [2 − d1 − d2 ],

...

* + Mm = Pm m − 1 − d1 − · · · − dm−1 . The definition of M0 is given by collapsing the complex of dgms in which P1 [n−1] is placed in degree zero, and where the maps are given by left multiplication with (i + 1|i). We prove that

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 ∼   t1 t2 · · · tm M0 = M1 , t 1 t2 · · · tm M i ∼ = Mi+1 for 1 ≤ i < m,  * +   t t · · · t M ∼ 1 2 m m = M0 2m − (m + 1)n ,

(4.15)

which clearly implies the desired result. By the definitions of ti and ti , the second of which is given in the proof of Lemma 4.11, one has ti+1 (Pi ) = Pi+1 [−di ] −→ Pi ∼ = ti Pi+1 [1 − di ], where Pi is placed in degree zero and the arrow is left multiplication with (i|i + 1). This shows that ti ti+1 (Pi ) ∼ = Pi+1 [1 − di ], and since ti (Pj ) ∼ = Pj whenever |i − j | ≥ 2, it proves the second equation in (4.15). To verify the other two equations one computes t1 t2 · · · tm Pm [n − 1] ∼ = t1 t2 · · · tm−1 Pm * + ∼ = t1 t2 · · · tm−2 Pm−1 − n + dm−1 −→ Pm ∼ = t1 t2 · · · tm−3 * + * + · Pm−2 − 2n + dm−2 + dm−1 −→ Pm−1 − n + dm−1 −→ Pm * + ∼ = ··· ∼ = M0 m(1 − n) + d1 + · · · + dm−1 and

tm · · · t2 t1 P1 ∼ · · · t2 P1 [n − 1] = tm * + ∼ · · · t3 P1 [n − 1] −→ P2 2n − 1 − d1 = tm ∼ · · · t4 = tm * + * + · P1 [n − 1] −→ P2 2n − 1 − d1 −→ P3 3n − 1 − d1 − d2 ∼ = ··· ∼ = M0 .

It seems likely that (t1 t2 · · · tm )m+1 is in fact isomorphic to the translation functor [2m − (m + 1)n], but we have not checked this. Before proceeding further, we need to recall some basic notions from the topology of curves on surfaces. Let D be a closed disc, and let ⊂ D \ ∂D be a set of m + 1 marked points. Diff(D, ∂D; ) denotes the group of diffeomorphisms f : D → D which satisfy f |∂D = id and f ( ) = . We write f0 ' f1 for isotopy within this group. By a curve in (D, ), we mean a subset c ⊂ D \ ∂D which can be represented as the image of a smooth embedding γ : [0; 1] → D such that γ −1 ( ) = {0; 1}. In other words, c is an unoriented embedded path in D \ ∂D whose endpoints lie

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in , and which does not meet anywhere else. There is an obvious notion of isotopy for curves, denoted again by c0 ' c1 . For any two curves c0 , c1 there is a geometric intersection number I (c0 , c1 ) ≥ 0, which is defined by I (c0 , c1 ) = |(c0 ∩ c1 ) \ | + (1/2)|(c0 ∩ c1 ) ∩ | for some c0 ' c0 which has minimal intersection with c1 . (This means, roughly speaking, that c0 is obtained from c0 by removing all unnecessary intersection points with c1 .) We refer to [28, Section 2a] for the proof that this is well defined. Once one has shown this, the following properties are fairly obvious: (I1) I (c0 , c1 ) depends only on the isotopy classes of c0 and c1 ; (I2) I (c0 , c1 ) = I (f (c0 ), f (c1 )) for all f ∈ Diff(D, ∂D; ); (I3) I (c0 , c1 ) = I (c1 , c0 ). Note that in general I (c0 , c1 ) is only a half-integer because of the weight 1/2 which the common endpoints of c0 and c1 contribute. The next lemma, whose proof we omit, is a modified version of [13, Proposition III.16]. lemma 4.15 Let c0 , c1 be two curves in (D, ) such that I (d, c0 ) = I (d, c1 ) for all d. Then c0 ' c1 .

b1

b2

bm

points of Figure 5

From now on, fix a collection of curves b1 , . . . , bm as in Figure 5, as well as an orientation of D. Then one can identify π0 (Diff(D, ∂D; )) with the braid group by mapping the standard generators g1 , . . . , gm ∈ Bm+1 to positive half-twists along b1 , . . . , bm . lemma 4.16 Let f ∈ Diff(D, ∂D; ) be a diffeomorphism that satisfies f (bj ) ' bj for all 1 ≤ j ≤ m. The corresponding element g ∈ Bm+1 must be of the form g = (g1 g2 · · · gm )ν(m+1) for some ν ∈ Z.

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Proof Since f (bj ) ' bj , f commutes up to isotopy with the half-twist along bj and hence with any element of Diff(D, ∂D; ). This implies that g is central. The next lemma, which is far more substantial than the previous ones, establishes a relationship between the topology of curves in (D, ) and the algebraically defined braid group action ρm,n . lemma 4.17 For g ∈ Bm+1 , let f ∈ Diff(D, ∂D; ) be a diffeomorphism in the isotopy class g corresponding to g, and let Rm,n be a functor that represents ρm,n (g). Then & g dimk HomD(Am,n ) Pi , Rm,n (Pj )[r] = 2I bi , f (bj ) r∈Z

for all 1 ≤ i, j ≤ m. A statement of the same kind, concerning a category and braid group action slightly different from ours, has been proved in [28, Theorem 1.1]. In principle, the proof given there can be adapted to our situation, but verifying all the details is a rather tedious business. For this reason we take a slightly different approach, which is to derive the result as stated here from its counterpart in [28]. To do this, we first need to recall the situation considered in that paper. In order to avoid confusion, objects that belong to the setup of [28] are denoted by overlined symbols.

0

1

0

0

0 1

1

2

1

m

Figure 6

Consider the quiver G m in Figure 6 with vertices numbered 0, . . . , m and whose edges are labeled with “degrees” zero or 1. Paths of length l in G m are described by (l + 1)-tuples of numbers i0 , . . . , il ∈ {0, . . . , m}; we use the notation (i0 | · · · |il ) for them. The path algebra k[G m ] is a graded algebra, whose ground ring is R = k m+1 . Let J m be the homogeneous 2-sided ideal in it, generated by the elements (i − 1|i|i + 1), (i + 1|i|i − 1), (i|i + 1|i) − (i|i − 1|i) (1 ≤ i ≤ m − 1), and (0|1|0). The quotient Am = k[G m ]/J m is a finite-dimensional graded algebra; a concrete basis is given by the 4m + 1 elements

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SEIDEL AND THOMAS

      

(0), . . . , (m), (0|1), . . . , (m−1|m) of degree zero, and (1|0), . . . , (m|m − 1), (1|2|1) = (1|0|1), . . . , (m − 1|m − 2|m − 1)

(4.16)

= (m − 1|m|m − 1), (m|m−1|m) of degree 1.

Am is evidently a close cousin of our algebras Am,n . We now make the relationship precise on the level of categories. Let Am -mod be the abelian category of finitely generated graded right modules over Am , and let D b (Am -mod) be its bounded derived category. (In contrast to the situation in Section 4.1, this is the derived category in the ordinary sense, not in the differential graded one.) There is an automorphism {1} that shifts the grading of a module up by 1. This descends to an automorphism of D b (Am -mod), which is not the same as the translation functor. In particular, for any X, Y ∈ D b (Am -mod) there is a bigraded vector space HomD b (Am -mod) X, Y {r1 }[r2 ] . r1 ,r2

We denote by P i ∈ Am -mod the projective modules (i)Am for 0 ≤ i ≤ m. Let P ⊂ Am -mod be the full subcategory whose objects are direct sums of P i {r} for i = 1, . . . , m and r ∈ Z; the important thing is that P 0 is not allowed. We write K b (P) for the full subcategory of K b (Am -mod) whose objects are finite complexes in P. This is an abuse of notation since P is not an abelian category; however, K b (P) is still a triangulated category because it contains the cone of any homomorphism. lemma 4.18 There is an exact functor T : K b (P) → D(Am,n ) with the following properties: (1) T(P i ) is isomorphic to Pi up to some shift; (2) there is a canonical isomorphism of functors T ◦ {1} ∼ = [−n] ◦ T; (3) the natural map, which exists in view of property 2, HomK b (P) X, Y {r1 }[r2 ] −→ HomD(Am,n ) T(X), T(Y ) , r2 = nr1

is an isomorphism for all X, Y ∈ K b (P). Proof As a first step, consider the functor T : P → Dgm(Am,n ) defined as follows. The object P i {r} goes to the dgm Pi [σi −nr], where σi = −d1 −d2 −· · ·−di−1 , and this is d extended to direct sums in the obvious way. Let Am be the space of elements of degree d in Am . Homomorphisms of graded modules P i {r} → P j {s} correspond in a natural r−s way to elements of (j )Am (i). On the other hand, dgm homomorphisms between Pi [σi − nr] and Pj [σj − ns] correspond to elements of degree σj − σi − n(s − r)

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in (j )Am,n (i). There is an obvious isomorphism, for any 1 ≤ i, j ≤ m and d ∈ Z, σj −σi +n d d (i) (j )Am (i) ∼ = (j )Am,n

(4.17)

which sends any basis element in (4.16) of the form (i0 | · · · |iν ) to the corresponding element (i0 | · · · |iν ) ∈ Am,n ; one needs to check, case by case, that the degrees turn out right. We use (4.17) to define T on morphisms; this is obviously compatible with composition, so that the outcome is indeed a functor. Note that T ◦ {1} ∼ = [−n] ◦ T . Now take a finite chain complex in P. Applying T to each object in the complex yields a chain complex in Dgm(Am,n ), which one can then collapse into a single dgm. This procedure yields a functor K b (P) → K(Am,n ), which is exact since it carries cones to cones. We define T to be the composition of this with the quotient functor K(Am,n ) → D(Am,n ). Properties (1) and (2) are now obvious from the definition of T . The remaining property (3) can be reduced, by repeated use of the five lemma, to the case when X = P i {r}, Y = P i {s}; then it comes down to the fact that (4.17) is an isomorphism. Define exact functors t¯1 , . . . , t¯m from D b (Am -mod) to itself by t¯i (X) = X(i) ⊗k P i −→ X .

(4.18)

Here X(i) is considered as a complex of graded k-vector spaces; tensoring with P i over k makes this into a complex of graded Am -modules, and the arrow is the multiplication map. We now state the results of [28]. lemma 4.19 The functors t¯1 , . . . , t¯m are exact equivalences and generate a weak braid group action ρ¯m : Bm+1 → Auteq(D b (Am -mod)). lemma 4.20 For g ∈ Bm+1 , let f ∈ Diff(D, ∂D; ) be a diffeomorphism in the isotopy class g corresponding to g, and let R m be a functor that represents ρ¯m (g). Then & g dimk HomD b (Am -mod) P i , R m P j {r1 }[r2 ] = 2I bi , f (bj ) r1 ,r2

for all 1 ≤ i, j ≤ m. Lemma 4.19 essentially summarizes the contents of [28, Section 3], and Lemma 4.20 is [28, Theorem 1.1]. The notation here is slightly different. (Our Am , P i , and t¯i are the Am , Pi , and Ri of that paper.) We have also modified the definitions very slightly; namely, we use right modules instead of left modules as in [28], and the coefficients

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are k instead of Z. These changes do not affect the results at all. (A very conscientious reader might want to check that inversion of paths defines an isomorphism between Am and its opposite, and that a result similar to Lemma 4.18 can be proved for an algebra Am defined over Z.) Proof of Lemma 4.17 Since the modules P i are projective, the obvious exact functorK b(P) →D b (Am -mod) is full and faithful. To save notation, we consider K b (P) simply as a subcategory of D b (Am -mod). An inspection of (4.18) shows that the t¯i preserve this subcategory, and the same is true of their inverses, defined in [28]. In other words, the weak braid group action ρ¯m restricts to one on K b (P). It follows from the definition of T that g g T ◦ t¯i |K b (P) ∼ = ti ◦ T. Hence, if R m and Rm,n are functors representing ρ¯m (g), respectively, ρm,n (g), the diagram g

Rm

K b (P)

K b (P)

T

D Am,n

T

g

Rm,n

D Am,n

commutes up to isomorphism. Using this, Lemma 4.18(3), and Lemma 4.20, one sees that & g dimk HomD(Am,n ) Pi , Rm,n Pj [r] r

=

& r

=

& r1 ,r2

g dimk HomD(Am,n ) T P i , TR m P j [r] g dimk HomD b (Am -mod) P i , R m P j {r1 }[r2 ]

= 2I bi , f (bj ) . Proof of Theorem 4.13 g For g ∈ Bm+1 , choose f and Rm,n as in Lemma 4.17. Take also another element g g ∈ Bm+1 and, correspondingly, f and Rm,n . Applying Lemma 4.17 to (g )−1 g shows that I f (bi ), f (bj ) = I bi , (f )−1 f (bj ) g −1 g 1& dimk Hom Pi , Rm,n Rm,n Pj = 2 r g

and, assuming that Rm,n (Pj ) ∼ = Pj for all j ,

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g −1 1& dimk Hom Pi , Rm,n Pj 2 r = I bi , (f )−1 (bj ) = I f (bi ), bj . =

Since i and f can be chosen arbitrarily, it follows from Lemma 4.15 that f (bj ) ' bj for all j . Hence, by Lemma 4.16, g = (g1 g2 · · · gm )ν(m+1) for some ν ∈ Z. But then g Rm,n (Pj ) ∼ = Pj [ν(2m − (m + 1)n)] by Lemma 4.14. In view of the assumption that g Rm,n (Pj ) ∼ = Pj , this implies that ν = 0 and hence that g = 1. 4.5. Conclusion The graded algebras Am,n are always augmented. For n ≥ 2 they are even connected, so that there is only one choice of augmentation map. This makes it possible to apply Theorem 4.7. lemma 4.21 Am,n is intrinsically formal for all m, n ≥ 2. The proof is by a straight computation of Hochschild cohomology. (It would be nice to have a more conceptual explanation of the result.) Its difficulty depends strongly on the parameter n. The easy case is when n > 2, since then already the relevant Hochschild cochain groups are zero; this is no longer true for n = 2. At first sight the computation may appear to rely on our specific choice (4.6) of degrees di , but in fact this only serves to simplify the bookkeeping; the Hochschild cohomology remains the same for any other choice. Throughout, we write G, A instead of Gm,n , Am,n . Proof for n > 2 Note that the “degree” label on any edge of G is greater than or equal to [n/2]. Moreover, the labels on any two consecutive edges add up to n. These two facts imply that the degree of any nonzero path (i0 | · · · |il ) of length l in k[G] is greater than or equal to [(nl)/2]. Now, any element of (A+ )⊗R q can be written as a sum of expressions of the form c = i1,0 | · · · |i1,l1 ⊗ i2,0 | · · · |i2,l2 ⊗ · · · ⊗ iq,0 | · · · |iq,lq , with all lq > 0. Because the tensor product is over R, such a c can be nonzero only if the paths (iν,0 | · · · |iν,lν ) match up, in the sense that iν,lν = iν+1,0 . Then, using the observation made above, one finds that - . l1 + · · · + lq deg(c) = deg i1,0 | · · · |i1,l1 |i2,1 | · · · |i2,l2 |i3,1 | · · · |iq,lq ≥ n . 2 Hence (A+ )⊗q is concentrated in degrees greater than or equal to [(nq)/2]. On the

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other hand, A[2 − q] is concentrated in degrees less than or equal to n + q − 2, which implies that C q A, A[2 − q] = HomR−R (A+ )⊗R q , A[2 − q] = 0 if n ≥ 4 or q ≥ 4. We now focus on the remaining case (n, q) = (3, 3). Then (A+ )⊗R 3 is concentrated in degrees greater than or equal to 4, while A[−1] is concentrated in degrees less than or equal to 4. The degree 4 part of (A+ )⊗R 3 is spanned by elements c = (i0 |i1 ) ⊗ (i1 |i2 ) ⊗ (i2 |i3 ), which obviously satisfy i3 = i0 . It follows that as an R-bimodule, the degree 4 part satisfies ei ((A+ )⊗R 3 )4 ei = 0. On the other hand, the degree 4 part of A[−1] is spanned by the elements (i|i ± 1|i), so it satisfies ei A[−1]4 ej = 0 for all i = j . This implies that there can be no nonzero R-bimodule maps between (A+ )⊗R 3 and A[−1] and hence that C 3 (A, A[−1]) is after all trivial. Proof for n = 2 Consider the relevant piece of the Hochschild complex, ∂ q−1 ∂q C q−1 A, A[2 − q] −−−→ C q A, A[2 − q] −−→ C q+1 A, A[2 − q] ; C q+1 (A, A[2 − q]) is zero for degree reasons. In fact, since all edges in G have “degree” labels 1, paths are now graded by their length, so that (A+ )⊗R q+1 is concentrated in degrees greater than or equal to q + 1, while A[2 − q] is concentrated in degrees less than or equal to q. In contrast, C q (A, A[2 − q]) is nonzero for all even q. To give a more precise description of this group, we use the basis of A from (4.7) and the basis of (A+ )⊗R q derived from that. Let (i0 | · · · |iq ), iq = i0 , be a closed path of length q in G. Define φi0 ,...,im ∈ C q (A, A[2 − q]) by setting i0 |i0 ± 1|i0 if c = i0 |i1 ⊗ · · · ⊗ iq−1 |iq , φi0 ,...,iq (c) = 0 on all other basis elements c. We claim that the elements defined in this way, with (i0 | · · · |iq ) ranging over all closed paths, form a basis of C q (A, A[2 − q]). To prove this, note that there is only one degree, which is q, where both (A+ )⊗q and A[2 − q] are nonzero. The degree q part of (A+ )⊗q is spanned by expressions c = (i0 |i1 ) ⊗ · · · ⊗ (iq−1 |iq ), with iq not necessarily equal to i0 . The degree q part of A[2 − q] is spanned by elements (i|i ± 1|i). Hence, an argument using the R-bimodule structure shows that if iq = i0 , then φ(c) = 0 for all φ ∈ C q (A, A[2 − q]). This essentially implies what we have claimed. We now turn to C q−1 (A, A[2 − q]); for this group we do not need a complete description, but only some sample elements. Given a closed path (i0 | · · · |iq ) as before in G, we define φ ∈ C q−1 (A, A[2 − q]) by setting φ (c) = (i0 |iq−1 ) if c = (i0 |i1 ) ⊗ · · · ⊗ (iq−2 |iq−1 ), and zero on all other basis elements c. A simple computa-

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tion shows that δ q−1 (φ ) = −φi0 ,...,iq − φiq−1 ,i0 ,i1 ,...,iq−1 . Also, for any closed path (i0 | · · · |iq ) with i2 = i0 and i1 = i0 + 1, define φ ∈ C q−1 (A, A[2 − q]) by setting φ (c) = (i0 |i0 ± 1|i0 ) for c = (i0 |i1 |i2 ) ⊗ (i2 |i3 ) ⊗ · · · ⊗ (iq−1 |iq ), and again zero for all other basis elements c. Then δ q−1 (φ ) is equal to −φi0 ,i1 ,...,iq − φi0 ,i1 −2,i2 ,...,iq for i0 > 1, and to −φi0 ,i1 ,...,iq for i0 = 1. To summarize, we have now established that the following relations hold in H H q (A, A[2 − q]): (1) [φi0 ,...,iq ] = −[φiq−1 ,iq ,i1 ,...,iq−1 ] for all closed paths (i0 | · · · |iq ) in the quiver G; (2) [φi0 ,...,iq ] = −[φi0 ,i1 −2,i2 ,...,iq ] whenever i0 = i2 ≥ 2 and i1 = i0 + 1; (3) [φi0 ,...,iq ] = 0 whenever i0 = i2 = 1 and i1 = 2. Take an arbitrary element φi0 ,...,iq . By applying (1) repeatedly, one can find another element φi0 ,...,iq that represents the same Hochschild cohomology class, up to a sign, and such that i1 is maximal among all iν . This implies that i0 = i2 = i1 − 1. If i1 = 2, then we can apply (3) to show that our Hochschild cohomology class is zero. Otherwise, pass to φi0 ,i1 −2,...,iq , which represents the same Hochschild cohomology class up to sign due to (2), and repeat the argument. The iteration terminates after finitely many moves because the sum of the iν decreases by 2 in each step. Hence H H q (A, A[2 − q]) is zero for all q ≥ 1. Proof of Theorem 2.18 We first need to dispose of the trivial case m = 1. In that case, choose a resolution F1 ∈ K of E1 . Pick a nonzero morphism φ : F1 → F1 [n]. This, together with idF1 , determines an isomorphism of graded vector spaces Hom∗ (F1 , F1 ) ∼ = k ⊕ k[−n] and ∗ hence an isomorphism in K between F1 ⊕ F1 [−n] and Hom (F1 , F1 ) ⊗ F1 . Consider the commutative diagram Hom∗ (F1 , F1 ) ⊗ F1

TF1 (F1 )[−1]

ev

∼ =

F1 [−n]

(−φ,id)

F1 ⊕ F1 [−n]

F1 id

(id,φ)

F1

The upper row is a piece of the exact triangle which comes from the definition of TF1 as a cone, and the lower row is obviously also a piece of an exact triangle. By the axioms of a triangulated category, the diagram can be filled in with an isomorphism between F1 [−n] and TF1 (F1 )[−1]. Transporting the result to D b (S ) yields TE1 (E1 ) ∼ = E1 [1 − n]. Since n ≥ 2 by assumption, it follows that TEr 1 (E1 ) ∼ = E1 unless r = 0. From now on, suppose that m ≥ 2. After shifting each Ei by some amount, we may assume that Hom∗ (Ei+1 , Ei ) is concentrated in degree di for i = 1, . . . , m − 1. (Shifting does not affect the statement because TEi [j ] is isomorphic to TEi for any

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∈ K for E , . . . , E . Lemma 4.10 shows that j ∈ Z.) Choose resolutions E1 , . . . , Em 1 m the endomorphism dga end(E ) has H (end(E )) ∼ = Am,n . By Lemma 4.21, end(E ) must be quasi-isomorphic to Am,n . Define an exact functor . to be the composition

∼ ∼ .E = = D b (S ) ←−− K −−−→ D end(E ) −−→ D Am,n . The first arrow is the standard equivalence, and the last one is the equivalence induced by some sequence of dgas and quasi-isomorphisms. By construction, .(Ei ) ∼ = Pi for i = 1, . . . , m. In the diagram D b (S )

∼ =

.E

TE

TEi

D b (S )

K

∼ =

ti

i

∼ =

D end(E )

K

.E

D end(E )

D Am,n ti

∼ =

D Am,n

the first square commutes because that is the definition of TEi , the second square by Lemma 4.3, and the third one by Lemma 4.2. Now let g be an element of Bm+1 , g R g : D b (S ) → D b (S ), a functor that represents ρ(g), and Rm,n : D(Am,n ) → D(Am,n ), a functor that represents ρm,n (g). By applying the previous diagram several times, one sees that g Rm,n ◦ . ∼ = . ◦ Rg . g g Assume that Rg (Ei ) ∼ = Ei for all i; then also Rm,n (Pi ) = Rm,n .(Ei ) ∼ = .R g (Ei ) ∼ = ∼ .(Ei ) = Pi . By Theorem 4.13, it follows that g must be the identity.

We have not tried to compute the Hochschild cohomology of Am,n for n = 1. However, an indirect argument using the nonfaithful B4 -action of Section 3.4 shows that A3,1 cannot be intrinsically formal. More explicitly, if one takes the sheaves Ox , O , Oy used in that example and if one chooses injective resolutions by quasicoherent sheaves for them, then the resulting dga end(E ) is not formal. One can give a more direct proof of the same fact by using essentially the same Massey product computation as Polishchuk in [44, p. 3]. Acknowledgments. Although he does not figure as an author, the paper was originally conceived jointly with Mikhail Khovanov, and several of the basic ideas are his. At an early stage of this work, we had a stimulating conversation with Maxim Kontsevich. We would also like to thank Mark Gross for discussions about mirror symmetry and singularities, and Brian Conrad, Umar Salam, and Balazs Szendroi for helpful comments. As mentioned earlier, Kontsevich, Bridgeland, and Maciocia also knew about the invertibility of the twist functors.

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Addendum. The results here were first announced at the Harvard Winter School on Mirror Symmetry in January of 1999 (see [53]). In the meantime, a preprint by R. Horja [22] has appeared which is inspired by similar mirror symmetry considerations. While there is little actual overlap ([22] does not operate in the derived category), Horja uses monodromy calculations to predict corresponding conjectural mirror Fourier-Mukai transforms that ought to be connected to our work, linking it to the toric construction of mirror manifolds.

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Seidel ´ Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau Cedex, France; [email protected] Thomas Mathematical Institute, University of Oxford, 24-29 Saint Giles’, Oxford OX1 3LB, United Kingdom; [email protected]

FINITENESS THEOREMS FOR NONNEGATIVELY CURVED VECTOR BUNDLES IGOR BELEGRADEK and VITALI KAPOVITCH Abstract We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of “bounded geometry.” 1. Introduction Much of the recent work in Riemannian geometry is centered around finiteness and precompactness theorems for various classes of Riemannian manifolds. Some versions of precompactness results typically work for compact domains in Riemannian manifolds. The main point of this paper is that one can sometimes get diffeomorphism finiteness for ambient Riemannian manifolds, provided their topology is concentrated on compact domains of “bounded geometry.” We postpone the discussion of our main technical results until Section 5 and concentrate on applications to nonnegative curvature. Recall that, according to the soul theorem of J. Cheeger and D. Gromoll, a complete open manifold of nonnegative sectional curvature is diffeomorphic to the total space of the normal bundle of a compact totally geodesic submanifold which is called the soul. One of the harder questions in the subject is what kind of normal bundles can occur. See [Che], [Rig], [Yan], [GZ1], and [GZ2] for examples of open nonneg¨ atively curved manifolds, and see [OW], [BK2], and [BK1] for known obstructions. Theorem 1.1 is our first result. theorem 1.1 Given a closed Riemannian manifold S with sec(S) ≥ 0 and positive D, r, v, n, there exists a finite collection of vector bundles over S such that, for every complete open Riemannian n-manifolds N with sec(N) ≥ 0 and an isometric embedding e : S → N of S onto a soul of N, the normal bundle νe is isomorphic to a bundle of the collection, provided e is homotopic to a map f such that diam(f (S)) ≤ D and volN (B(p, r)) ≥ v for some p ∈ f (S). DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 1, Received 3 February 2000. 2000 Mathematics Subject Classification. Primary 53C20.

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There is also a “variable base” version of Theorem 1.1. We say that two vector bundles ξ , ξ over different bases B, B are topologically equivalent if there is a homeomorphism h : B → B such that h# ξ is isomorphic to ξ . Moreover, if h is a diffeomorphism, we say that ξ and ξ are smoothly equivalent. For bundles over manifolds of dimension less than or equal to 3, topological equivalence implies smooth equivalence because in this case any homeomorphism is homotopic to a (nearby) diffeomorphism (see [Mun], [Moi]). The diffeomorphism type of the total space of a vector bundle (of positive rank over a closed manifold) is determined up to finite ambiguity by the topological equivalence class of the bundle (see [HM], [KS] if dimension of the total space is greater than or equal to 5, and see [Mun], [Moi] otherwise). In the appendix we discuss the extent to which the total space determines a vector bundle and give an example of infinitely many pairwise topologically nonequivalent vector bundles over a closed manifold with diffeomorphic total spaces. The following result can be thought of as a generalization of the Grove-PetersenWu finiteness theorem. theorem 1.2 Given positive D, r, v, v , n, there exists a finite collection of vector bundles such that, for any open complete Riemannian n-manifold N with sec(N) ≥ 0 and a soul S ⊂ N , the normal bundle to the soul is topologically equivalent to a bundle of the collection, provided vol(S) ≥ v and the inclusion S → N is homotopic to a map f such that diam(f (S)) ≤ D and volN (B(p, r)) ≥ v for some p ∈ f (S). We suspect that in Theorem 1.2 the lower volume bound on the soul follows from the rest of the assumptions and thus can be omitted. For example, it follows from [HK] that the lower volume bound for a soul S comes from a lower volume bound on an ambient manifold N; that is, volN B(p, r) ≥ v implies vol(S) ≥ v , provided the distance from p to S is uniformly bounded. (The latter can also be forced by purely topological assumptions on S, as we show in Corollary 6.5.) Thus we deduce the following corollary. corollary 1.3 Given positive D, r, v, n, there exists a finite collection of vector bundles such that, for any open complete Riemannian n-manifold N with sec(N) ≥ 0 and a soul S ⊂ N, the normal bundle to the soul is topologically equivalent to a bundle of the collection, provided diam(S) ≤ D and volN (B(p, r)) ≥ v for some p ∈ N such that the distance from S to p is uniformly bounded. The finiteness of homeomorphism types of total spaces in Corollary 1.3 can be easily obtained from the parametrized version of Perelman’s stability theorem in [Per2]

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111

and the regularity properties of the distance function; however, the conclusion of Corollary 1.3 is strictly stronger (cf. Example A.1). There is a version of the above corollary for totally geodesic submanifolds in Riemannian manifolds with a lower sectional curvature bound. Another version works when ambient manifolds have a lower bound on Ricci curvature and injectivity radius. G. Perelman [Per1] proved that the distance-nonincreasing retraction onto the soul introduced in [Sha] is a C 1,1 -Riemannian submersion. We observe that a local bound on the vertical curvatures of this submersion gives a lower volume bound for the ambient nonnegatively curved manifold. In particular, we deduce the following corollary. corollary 1.4 Given positive n, r, K, D and a closed nonnegatively curved Riemannian manifold S, there is a finite collection of vector bundles over S such that, for any open complete Riemannian n-manifold N with sec(N) ≥ 0 and an isometric embedding e : S → N of S onto a soul of N , the normal bundle νe is isomorphic to a bundle of the collection, provided diam(S) ≤ D and there is a point p ∈ e(S) such that all the vertical curvatures at the points of B(p, r) are bounded above by K. Note that Corollary 1.4 generalizes the main result of [GW] (cf. [Tap]), where the same statement is proved for a soul isometric to the round sphere. Again, there is a “variable base” version of Corollary 1.4 when souls vary in the Grove-Petersen-Wu class. Similarly, Corollary 1.4 holds when the soul varies in Cheeger-Andersen class (see [AC]): diam(S) ≤ D and injrad(S) ≥ i0 , where the conclusion of finiteness up to topological equivalence gets improved to the finiteness up to smooth equivalence. There is a counterpart of Theorem 1.2 in nonpositive curvature. Let sec(N) ≤ 0, and let e : M → N be a totally geodesic embedding that is a homotopy equivalence. Then the orthogonal projection N → e(M) is distance nonincreasing. Moreover, inj(N ) = inj(M), and we obtain the following corollary (which also has a “fixed base” version). corollary 1.5 Given positive D, r, v, n, K, there exists a finite collection of vector bundles such that, for any totally geodesic embedding e : M → N of a closed Riemannian manifold M into an open complete Riemannian n-manifold N with sec(N) ≤ 0, the normal bundle νe is topologically equivalent to a bundle of the collection, provided sec(M) ≥ −1, vol(M) ≥ v, and e is a homotopy equivalence homotopic to a map f with diam(f (M)) ≤ D such that the sectional curvature at any point of the rneighborhood of f (M) is greater than or equal to −K.

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Normal bundles to totally geodesic embeddings in nonpositively curved manifolds can be fairly arbitrary, as the following example shows. M. Anderson proved that the total space E(ξ ) of any vector bundle ξ over a closed nonpositively curved manifold M carries a complete metric with −1 ≤ sec ≤ 0 [And1]. Let M be a closed locally symmetric manifold of nonpositive curvature and rank(M) ≥ 2 such that no finite cover of M splits as a Riemannian product. Let ξ be an orientable vector bundle over M with nonzero Euler class. Then according to [SY, p. 326], the zero section M → E(ξ ) is homotopic to a harmonic map that, by the harmonic map superrigidity (see [MSY]), is a totally geodesic embedding (after rescaling the metric on E(ξ )). The normal bundle to this totally geodesic embedding is stably isomorphic to ξ , and, furthermore, it has the same Euler class as ξ . One may wonder when there are infinitely many vector bundles of rank m over a given base M. For example, if m ≥ dim(M), this happens whenever M has nonzero Betti number in a dimension divisible by 4 (e.g., if M is a closed orientable manifold of dimension divisible by 4). The reason is that the Pontrjagin character defines an ⊗ Q, where K(M) is the group of stable isomorphism of ⊕i>0 H 4i (M, Q) and K(M) equivalence classes of vector bundles over M. Furthermore, the Euler class defines a one-to-one correspondence between the set of isomorphism classes of oriented rank 2 bundles over M and H 2 (M, Z). Also, if M is closed, orientable, and 2n-dimensional, then there are infinitely many rank 2n bundles over M obtained as pullbacks of T S 2n via maps M → S 2n of nonzero degree. The structure of the paper is as follows. Section 2 reviews some well-known results on homotopy count of maps in equicontinuous families. Section 3 discusses local versions of precompactness theorems in [AC] and [Per2]. Section 4 provides a background in characteristic classes and related invariants of maps. The main technical results are proved in Section 5. In Section 6 we prove applications to nonnegatively/nonpositively curved manifolds. In the appendix we explain the extent to which a vector bundle is determined by its total space.

2. Equicontinuity and homotopy count of maps Definition 2.1 A family of maps of metric spaces fα : Xα → Yα is called -equicontinuous if there exist δ > 0 such that dYα (fα (x), fα (x )) < for any α and any x, x ∈ Xα with dXα (x, x ) < δ. A family fα is called equicontinuous if it is -equicontinuous for every . A family fα is called almost equicontinuous if for any there exists a finite subset S ⊂ {fα } whose complement is -equicontinuous.


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Example 2.2 Assume that fα : Xα → Yα is a family of maps of metric spaces. (1) If each fα is (α, L)-Hölder (i.e., dYα (fα (x), f (x )) ≤ L dXα (x, x )α ), then {fα } is equicontinuous. (2) If each fα is an α -Hausdorff approximation (or more generally, if fα satisfies dY (fα (x), fα (x )) − dX (x, x ) ≤ α α α for any x, x ∈ Xα ) and α → 0, then {fα } is almost equicontinuous. (3) If {fα } is almost equicontinuous and gα is α -close to fα with α → 0, then {gα } is almost equicontinuous. (4) If {fα } where fα : Xα → Yα is almost equicontinuous and {gα } with gα : Yα → Zα is almost equicontinuous, then {gα ◦ fα } is almost equicontinuous. The importance of the following result in Riemannian geometry was first observed by M. Gromov [Gro]. proposition 2.3 Let Y be a compact metric space such that there exists an = (Y ) with the property that any two 4-close continuous maps of a compact metric space into Y are homotopic. Then, given a compact metric space X, any -equicontinuous family of maps fα : X → Y falls into finitely many homotopy classes. Proof Fix δ > 0 such that dYα (fα (x), fα (x )) < for any k and any x, x ∈ Xα with dXα (x, x ) < δ. Find a finite δ/2-net NX in X and a finite -net NY in Y . Any map f : X → Y produces a (nonunique) map fˆ : NX → NY defined so that fˆ(x) is a point of NY whose distance to f (x) is less than or equal to . Now if f and g are -equicontinuous maps with fˆ = g, ˆ then f and g are 4-close and hence homotopic. In particular, {fα } fall into at most card(NY )card(NX ) homotopy classes. Remark 2.4 Such an (Y ) exists if, for example, the compact metric spaces X and Y are separable, finite-dimensional ANR (see [Pet1]). Note that, for compact, separable, finitedimensional metric spaces, being ANR is equivalent to being locally contractible (see [Bor, Corollary V.10.4]) any such space is homotopy equivalent to a finite cell complex (see [Wes]). 3. Local convergence results In this section we discuss local versions of the C α -precompactness theorem of

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M. Anderson and Cheeger [AC] and of Perelman’s stability theorem in [Per2]. The results provide sufficient conditions under which a sequence of compact domains in Riemannian manifolds has uniformly bounded geometry in the sense defined below. In fact, the theorems in [AC] and [Per2] are stated in a local form, so we just give details needed for the “compact domains” version. Let Uα be a family of compact domains (i.e., compact codimension zero topological submanifolds) of Riemannian n-manifolds Nα . We say that {Uα } has uniformly bounded geometry if any sequence of domains in the family has a subsequence {Uk } such that there exist a metric space V and homeomorphisms hk : V → Vk of V onto compact domains Vk ⊃ Uk such that both {hk } and {h−1 k } are almost equicontinuous. In case ∂Uk = ∅, we necessarily have Uk = Vk = Nk . Throughout the paper we always denote the closed -neighborhood of a subspace S by S . theorem 3.1 [AC] Given > 0, let Uk be a sequence of compact domains with smooth boundaries in Riemannian n-manifolds Nk such that the closed -neighborhood Uk of Uk is compact. Assume that, for some positive H , V , i0 , the following holds: Ric(Uk ) ≥ /2 −(n−1)H , vol(Uk ) ≤ V , and injNk (x) ≥ i0 for any x ∈ Uk . Then, after passing to /2

a subsequence, there are compact domains Vk with Uk ⊂ Vk ⊂ Uk , a manifold V , and C ∞ -diffeomorphisms hk : V → Vk such that the pullback metrics h∗k gk converge in a C α -topology to a C α -Riemannian metric on the interior of V .

Proof For the reader’s convenience we review the argument in [AC], emphasizing its local /2 nature. It is proved in [AC, pp. 269–270] that any domain Uk as above has an atlas of harmonic coordinate charts Fν : B(x, rh ) → Rn where B(x, rh ) is a metric ball at /2 x ∈ Uk whose radius rh ≤ /10 depends only on the initial data. Further, the metric tensor coefficients in the charts Fν are controlled in C α -topology. An elliptic estimate then shows that the transition functions Fµ ◦ Fν−1 are controlled in C 1,α -topology. All these results are stated and proved locally. Next, the relative volume comparison implies that one can choose a finite subatlas so that there is a uniform bound on the multiplicities of intersections of the coordinate /2 charts and the balls B(x, rh /2) still cover Uk . (This argument involves only small balls and hence is local.) The lower injectivity radius bound gives a lower bound for the volume of any small ball that depends only on the radius of the ball (see [Cro]). This, together with an upper bound on vol(Uk ), implies an upper bound on the number of coordinate charts. Finally, following Cheeger’s thesis (as outlined in [AC, pp. 266–267]), one can


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“glue the charts together,” which proves the theorem. Alternatively, one can follow (almost word by word) the argument in [And2, pp. 464–466], where a “compact domain” version of the Cheeger-Gromov convergence theorem is proved. Remark 3.2 There are many other convergence theorems, notably those involving integral curvature bounds (see [Pet2]). For example, T. Hiroshima [Hir] generalized [AC], replacing a lower Ricci curvature bound by an integral bound on an eigenvalue of the Ricci curvature. Hiroshima’s proof is given for complete manifolds; however, a local version of [Hir] is likely to hold. We leave this matter for an interested reader to clarify. Before starting the proof of Theorem 3.5, we need a local version of the packing lemma that ensures Gromov-Hausdorff convergence. We say that a metric space (X, d) is locally interior if for any point x ∈ X there exists an > 0 such that for any y, z ∈ B(x, ) we have d(y, z) = inf γ L(γ ) where the infimum is taken over all paths γ connecting y and z. For example, all Riemannian manifolds are locally interior. Remark 3.3 Notice that for locally compact metric spaces the property of being locally interior is easily seen to be equivalent to the following one. For any point x ∈ X there exists an > 0 such that for any y, z ∈ B(x, ) there exists a sequence pn ∈ B(x, 2) such that d(pn , y) → d(y, z)/2 and d(pn , z) → d(y, z)/2. Here is how locally interior spaces arise in this paper. Let Vk be a sequence of compact domains in Riemannian n-manifolds. Equip Vk with induced Riemannian metrics, and assume that Vk converges in Gromov-Hausdorff topology to a compact metric space V . Consider fk : Vk → R defined by fi (x) = dist(x, ∂Vi ). Then each fi is 1Lipschitz and by the Arzela-Ascoli theorem this sequence converges to 1-Lipschitz function f : V → R. We call the open set {x ∈ U : f (x) > 0} the interior of U . Then it is easy to show that the interior of U is a locally interior space. lemma 3.4 Given > 0, let Uk be a sequence of compact connected domains with smooth boundaries in Riemannian n-manifolds Nk such that the closed -neighborhood Uk of Uk is compact. Assume that, for some positive H , V , v0 , r0 < /10, the following holds: /2 Ric(Uk ) ≥ −(n − 1)H , vol(Uk ) ≤ V , and vol(B(x, r0 )) ≥ v0 for any x ∈ Uk . /2 Then, after passing to a subsequence, the compact domains Uk converge in GromovHausdorff topology to a compact metric space U whose interior is a locally interior metric space.

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Proof Take an arbitrary r < r0 . To prove precompactness in Gromov-Hausdorff topology, it /2 is enough to show that the number of elements in a maximal r-net in Uk is bounded above by some N(r) independent of k. /2 Fix maximal r-separated nets Nk in Uk so that r-balls with centers in Nk are disjoint and 2r-balls cover Uk . The relative volume comparison gives a uniform lower /2 bound for the volume of the r-ball centered at any point of Uk , say, vol(B(x, r) ≥ v. /2 Then #Nk ≤ V /v and Uk converge in the Gromov-Hausdorff topology to a compact metric space U . As we explained above, the interior of U is necessarily locally interior. theorem 3.5 [Per2] Given > 0, let Uk be a sequence of compact connected domains with smooth boundaries in Riemannian n-manifolds Nk such that the closed -neighborhood of Uk , denoted by Uk , is compact. Assume that for some positive K, V , v, r < /10, /2 sec(Uk ) ≥ −K, vol(Uk ) ≤ V , and vol(B(x, r)) ≥ v for any x ∈ Uk . Then, /4

after passing to a subsequence, there are compact domains Vk with Uk ⊂ Vk ⊂ /2 Uk , a manifold V , and homeomorphisms hk : V → Vk which are k -Hausdorff approximations with k → 0 as k → ∞. Proof /2 By Lemma 3.4, Uk subconverges in the Gromov-Hausdorff topology to a compact metric space U whose interior int(U ) is a locally interior metric space. We are in position to apply Perelman’s stability theorem in [Per2], which asserts that int(U ) is a topological manifold, and, moreover, that any compact subset of int(U ) lies in a compact domain V ⊂ int(U ) such that there are topological embeddings hk : /2 V → Uk . Furthermore, hk induce Hausdorff approximations that become arbitrary /2 close to the given Hausdorff approximations between U and Uk . Choosing V large /4 enough, one can ensure that hk (V ) ⊃ Uk , as promised. Remark 3.6 Let Uk be a sequence of compact domains with smooth boundaries in Riemannian nmanifolds Nk such that each Uk is contained in a compact metric ball B(pk , R) ⊂ Nk for some R > > 0. Assume that Ric(B(pk , R)) ≥ −(n − 1)H for some H > 0. Then the absolute volume comparison implies that vol(Uk ) is uniformly bounded above by B H (R). Now if vol(B(xk , /2)) is uniformly bounded below for some xk ∈ B(pk , R − ), then the relative volume comparison ensures that vol(B(x, r)) is uniformly bounded below for any x ∈ B(pk , R − ) and any r < /2.


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In particular, if each Nk is complete and sec(Nk ) ≥ −1, then any sequence of compact domains Uk has uniformly bounded geometry, provided diam(Uk ) ≤ D and there are points xk ∈ Uk with vol(B(xk , r)) ≥ v. Remark 3.7 Let Uk be a sequence of compact connected domains with smooth boundaries in Riemannian n-manifolds Nk such that Uk is compact. Assume that Ric(Uk ) ≥ −(n− 1)H for some H > 0. Then the following two conditions are equivalent: /2 (i) vol(Uk ) ≤ V , and vol(B(x, r)) ≥ v for any x ∈ Uk ; /2 /2 (ii) there is a point x0 ∈ Uk such that vol(B(x0 , r) ≥ v and diamint (Uk ) ≤ D, where the diameter is taken with respect to the intrinsic distance induced by the /2 Riemannian metric on Uk . Indeed, let us show that (i) ⇒ (ii). Using the relative volume comparison, we can make r and v slightly smaller so that r < /4. Fix δ < r/100, and find a path in /2 /2 /2 Uk of length between the numbers diamint (Uk ) and diamint (Uk ) + δ. This path is almost length minimizing with the error less than or equal to δ. Hence, one can find /2 N = [diamint (Uk )/3r] points on the path such that r-balls centered at the points are /2 disjoint. Thus, V ≥ vol(Uk ) ≥ Nv and we get a uniform bound on diamint (Uk ). Conversely, let us prove (ii) ⇒ (i). Fix δ < /10. First, show that there is a /2 uniform lower bound for vol(B(x, δ)) for any x ∈ Uk . Take an arbitrary point int x ∈ Uk /2. Since diam (Uk ) ≤ D, there is a sequence of points xi ∈ Uk , i = 0, . . . , N, where N = [D/δ] + 1, which starts at x0 , ends at xN = x, and satisfies d(xi , xi+1 ) ≤ δ. Let vn (r, H ) denote the volume of the ball of radius r in a complete simply connected n-dimensional space of constant sectional curvature equal to H . Using induction on i and the relative volume comparison, one can show that, for every i, vn (δ, H ) i v. vol(Bδ (pi )) ≥ vn (2δ, H ) In particular, there is a uniform lower bound for vol(B(x, δ)). /2 Now fix a finite covering of Uk by δ-balls. As before, the relative volume comparison gives a uniform upper bound Nloc (δ) on the multiplicities of intersections in this covering. (The argument involves only small balls, so it works because the balls are far enough from the boundary.) By the absolute volume comparison, the volume of each δ-ball is uniformly bounded above, and hence a bound vol(Uk ) ≤ V would follow from a bound on the number of balls in the covering. Set rj = j δ, j = 0, . . . , m with m = [D/δ] + 1. Let Nj be the number of balls in the covering whose centers are in the rj -ball around x0 . (As before, the ball is taken with respect to the induced Riemannian metric on /2 Uk .) Since multiplicities are bounded by Nloc , for each j we have that Nj +1 ≤

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(δ)

Nj + Nj loc . This gives a uniform bound on the number of balls in the covering and hence on vol(Uk ).

4. Invariants of maps Definition 4.1 Let B be a topological space, and let S(B) be a set. Given a smooth manifold N, denote by C(B, N) the set of all continuous maps from B to N. Suppose that for any smooth N we have a map ι : C(B, N) → S(B). Then we call ι an S(B)-valued invariant of maps B if the two following conditions hold: (1) homotopic maps f1 : B → N and f2 : B → N have the same invariant; (2) if h : N → L is a homeomorphism of N onto an open subset of L, then, for any continuous map f : B → N, the maps f : B → N and h ◦ f : B → L have the same invariant. There is a variation of this definition for maps into oriented manifolds. Namely, we require that the target manifold be oriented and that the homeomorphism h preserve orientation. In that case we say that ι is an invariant of maps into oriented manifolds. Example 4.2 (Pontrjagin classes) As usual, the total (rational) Pontrjagin class of a bundle ξ is denoted by p(ξ ). Given a continuous map of smooth manifolds f : B → N, set p(f ) to be the solution of f ∗ p(T N ) = p(T B) ∪ p(f ). (A total Pontrjagin class is a unit, so there exists a unique solution.) The fact that p(f ) is an H ∗ (B, Q)-valued invariant follows from topological invariance of rational Pontrjagin classes (see [Nov]). When f is a smooth immersion, p(f ) is the the total Pontrjagin class of the normal bundle to f . Finally, note that Stiefel-Whitney classes are preserved by homeomorphism (see [Spa]); hence they also give rise to invariants of maps. Remark 4.3 The isomorphism class of the pullback of the tangent bundle to N under f would be an invariant (for paracompact B) if we only required that invariants be preserved by diffeomorphisms. In general, homeomorphisms do not preserve tangent bundles. However, a tangent bundle (and, in fact, any vector bundle over a finite cell complex) is recovered up to finitely many possibilities by the total Pontrjagin class and the Euler class of its orientable (1-or 2-fold) cover. (See [Bel] for a proof of this folklore result.)


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Example 4.4 (Intersection number in oriented n-manifolds) Assume B is a compact space, and fix two homology classes α ∈ Hm (B, Q) and β ∈ Hn−m (B, Q). (Unless stated otherwise, we always use singular (co)homology with rational coefficients.) Let f : B → N be a continuous map of a compact topological space B into an oriented n-manifold N. Set In,α,β (f ) to be the intersection number of f∗ α and f∗ β in N . It is easy to see that In,α,β is a Q-valued invariant of maps into oriented manifolds. Example 4.5 (Generalized Euler class) Let B be a closed oriented m-manifold, and let f : B → N be a map of B into an oriented n-manifold N . Define the rational Euler class χ(f ) by requiring that χ (f ), α = In,α,[B] . This is an H n−m (B, Q)-valued invariant for maps into oriented manifolds. If f is a smooth embedding, χ(f ) is the Euler class of the normal bundle νf . Note that when the orientation is changed on B or N, the invariant χ(f ) may change sign. More generally, if B and N are not assumed to be orientable, one can define a (generalized) Euler class of a continuous map as follows. Recall that a smooth manifold L is orientable if and only if the first StiefelWhitney class w1 (T L) ∈ H 1 (L, Z/2Z) vanishes. Note that H 1 L, Z/2Z ∼ = Hom H1 (L), Z/2Z ∼ = Hom π1 (L), Z/2Z , so elements of H 1 (L, Z/2Z) correspond to subgroups of index less than or equal to 2 in π1 (L) which are the kernels of homomorphisms in Hom(π1 (L), Z/2Z). Let Kf be the intersection of the subgroups corresponding to w1 (B) and f ∗ w1 → B be a covering associated to Kf , and let N → N be a covering (N ). Let B ˜ associated with f∗ (Kf ). Then f lifts to a map f : B → N of orientable manifolds. Define the generalized Euler class χ(f ˜ ) as a pair (Kf , ±χ(f˜)). (Note that χ(f˜) and N, so it is only well defined up to sign.) depends on the choice of orientations in B It is easy to see that χ˜ (f ) is an invariant because homotopies and homeomorphisms lift to covering spaces and because Stiefel-Whitney classes are topological invariants (see [Spa]). → B and two cohomology Thus, for our purposes, χ˜ (f ) is a regular covering B Q). For a map of orientable manifolds f : B → M, classes χ (f˜), −χ (f˜) in H n−m (B, ˜ ) generalizes χ(f ). χ˜ (f ) = (π1 (B), ±χ (f )), so, up to sign, χ(f If f is a smooth embedding of nonorientable manifolds with orientable normal bundle νf , then the Euler class of νf is taken to ±χ(f˜) by the map H n−m (B, Q) → Q) induced by the covering. H n−m (B,

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proposition 4.6 Let ek : B → Nk be a sequence of smooth embeddings of a closed manifold B into ˜ k ) are independent of k. Then manifolds Nk . Assume that invariants p(ek ) and χ(e the set of isomorphism classes of normal bundles νek is finite. Proof It is well known to experts that a vector bundle over a finite cell complex is recovered up to finitely many possibilities by the total Pontrjagin class and the Euler class of its orientable (1- or 2-fold) cover (see [Bel] for a proof). We now reduce to this result. In what follows we use the notation of Example 4.5. Since χ(e ˜ k ) = (Kek , ±χ (e˜k )) is independent of k, there is a covering B → B associated with Kek ≤ π1 (B) k → Nk associated with ek∗ (Kek ). The embedding ek and, for each k, a covering N k of orientable manifolds. lifts to an embedding e˜k : B → N 1 Using that H (B, Z/2Z) is a finite group, we can partition νek into finitely many subsequences, each having the same first Stiefel-Whitney class. It suffices to show that any such subsequence falls into finitely many isomorphism classes, so we can assume that w1 (νek ), and hence ek∗ w1 (T Nk ) = w1 (νek ) + w1 (T B), is independent of k. Let B → B be a covering associated to the subgroup of π1 (B) which corresponds to w1 (νek ). This subgroup lies in Kek , so B is an intermediate covering space between and B; that is, we have coverings q˜ : B → B and q¯ : B → B. Also, let N k → Nk B be a covering associated to ek∗ q¯∗ (π1 (B)). The embedding ek lifts to an embedding e¯k : B → N. Now the normal bundle νe¯k is orientable, so its Euler class is well defined (up to sign since there is no canonical choice of orientations). Note that q˜ ∗ takes the Euler class of νe¯k to ±χ(e˜k ). It is a general fact that finite covers induce injective maps in rational cohomology. (The point is that the transfer map goes the other way, and precomposing the transfer with the homomorphism induced by the covering is multiplication by the order of the covering.) Also, the Pontrjagin class of νe¯k is q¯ ∗ -image of p(ek ). Thus, given χ (e˜k ) and p(ek ), one can uniquely recover the (rational) Euler and Pontrjagin class of νe¯k . As we mentioned above, these classes determine νek up to finitely many possibilities. Therefore νek are determined up to finitely many possibilities by χ˜ (ek ) and p(ek ), as desired. Remark 4.7 The above proof actually gives a slightly more general result that is useful in our applications. Namely, instead of assuming that ek ’s are smooth embeddings, it suffices to assume that each ek is a topological embedding such that ek (B) is a smooth submanifold of Nk . Then ek (B) has a normal bundle in Nk whose pullback via ek is still denoted νek .


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5. Main technical results proposition 5.1 Let fk : M → Nk be a sequence of continuous maps of a closed Riemannian manifold M into (possibly incomplete) Riemannian n-manifolds. Assume that, for each k, there exists a compact domain Uk ⊃ fk (M) such that {Uk } has uniformly bounded geometry. Assume that either (i) {fk } is almost equicontinuous, or (ii) fk is a homotopy equivalence with a homotopy inverse gk : Nk → M such that {gk } is almost equicontinuous. Then, for any S(M)-valued invariant of maps ι, the subset {ι(fk )} of S(M) is finite. Proof Since {Uk } has uniformly bounded geometry, there exist a metric space V and homeomorphisms hk : V → Vk of V onto compact domains Vk ⊃ Uk such that both {hk } and {h−1 k } are almost equicontinuous. If (i) holds, then {h−1 k ◦fk } is an almost equicontinuous sequence of maps from M into V . Thus, Proposition 2.3 implies that the maps h−1 k ◦ fk ’s fall into finitely many homotopy classes. Now we are done by definition of an invariant since h−1 k ’s are homeomorphisms. If (ii) holds, then {gk ◦ hk } is an almost equicontinuous sequence of maps from V into M. Again, by Proposition 2.3 there are only finitely many homotopy classes of maps among gk ◦hk . It suffices to show that, whenever gk ◦hk is homotopic to gm ◦hm , the maps fk and fm have the same invariants. Let G : V ×[0, 1] → M be a homotopy that connects gk ◦ hk and gm ◦ hm . The homotopy F : M × [0, 1] → V defined as −1 −1 F (x, t) = G(h−1 k (fk (x)), t) connects gm ◦ hm ◦ hk ◦ fk with gk ◦ hk ◦ hk ◦ fk = gk ◦ fk ∼ idM . Thus, fm is homotopic to −1 −1 fm ◦ gm ◦ hm ◦ h−1 k ◦ fk ∼ idNm ◦hm ◦ hk ◦ fk ∼ hm ◦ hk ◦ fk .

Since hm ◦ h−1 k is a homeomorphism, fk and fm have the same invariants, as desired.

Remark 5.2 In the above proposition, M can be chosen as in Remark 2.4. Note that the spaces mentioned in Remark 2.4 are homotopy equivalent to finite cell complexes (see [Wes]); in particular, characteristic classes determine a vector bundle over such a space up to finitely many possibilities. Also, instead of assuming that {gk } is almost equicontinuous, it is enough to assume that gk |Vk is almost equicontinuous.

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There is a version of the theorem for invariants of maps into oriented manifolds. First of all, by pulling back the orientation from V , one can define orientations on Nk so that hk preserve orientations. In general, change of orientation on N may lead to an unknown change of an invariant of a map into N . However, if ι = In,α,β , then change of orientation on N may only lead to the sign change for the intersection number. Thus, for ι = In,α,β , Proposition 5.1 holds. corollary 5.3 Let M be a closed Riemannian manifold, and let ek : M → Nk be a sequence of topological embeddings of M into Riemannian n-manifolds Nk such that ek (M) ⊂ Nk is a smooth submanifold. Assume that, for each k, ek is homotopic to fk : M → Nk and that there exists a compact domain Uk ⊃ fk (M) such that {Uk } has uniformly bounded geometry. Assume that either (i) {fk } is almost equicontinuous or (ii) fk is a homotopy equivalence with a homotopy inverse gk : Nk → M such that {gk } is almost equicontinuous. Then the set of isomorphism classes of normal bundles νek is finite. Proof For any invariant ι, ι(fk ) = ι(ek ). In particular, this is true for the rational Pontrjagin class and the generalized Euler class. The result now follows from Proposition 5.1 combined with Proposition 4.6 and Remark 4.7. Remark 5.4 Note that Corollary 5.3 also holds when ek ’s are only immersions, provided dim(Nk )− dim(M) is either odd or greater than dim(M). Indeed, under these codimension assumptions the rational Euler class of νek vanishes while the total Pontrjagin class of νek is equal to p(ek ). corollary 5.5 Let ek : Mk → Nk be a sequence of topological embeddings of closed Riemannian manifolds Mk into Riemannian n-manifolds Nk such that ek (Mk ) ⊂ Nk is a smooth submanifold and {Mk } has uniformly bounded geometry. Assume that ek is homotopic to fk : Mk → Nk , and assume that there exists a compact domain Uk ⊃ fk (Mk ) such that {Uk } has uniformly bounded geometry. Assume that either (i) {fk } is almost equicontinuous or (ii) fk is a homotopy equivalence with a homotopy inverse gk : Nk → Mk such that {gk } is almost equicontinuous. Then the set of topological equivalence classes of normal bundles νek is finite.


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Proof Since Mk has bounded geometry, there exist M and homeomorphisms hk : M → Mk such that both {hk } and {h−1 k } are almost equicontinuous. Note that ek (hk (M)) = ek (Mk ) is a smooth submanifold of Nk . If {fk } is almost equicontinuous, then so is {fk ◦ hk }. Similarly, if {gk } is almost equicontinuous, then so is {h−1 k ◦ gk }. Thus, Corollary 5.3 implies that the set of isomorphism classes of normal bundles νek ◦hk is finite. In particular, the set of topological equivalence classes of normal bundles νek is finite. Remark 5.6 For future applications we note that if hk ’s are diffeomorphisms, then the conclusion of Corollary 5.5 can clearly be improved to “the set of smooth equivalence classes of normal bundles νek is finite.” 6. Geometric applications This section contains proofs of the various finiteness theorems that follow from Section 5. corollary 6.1 Let eα : M → Nα be an almost equicontinuous family of smooth embeddings of a closed Riemannian manifold M into complete Riemannian n-manifolds Nα with sec(Nα ) ≥ −1. Assume that for each α there is a point pα ∈ Nα such that vol(B(pα , 1)) is uniformly bounded below and distNα (pα , eα (M)) is uniformly bounded above. Then the set of isomorphism classes of normal bundles νeα is finite. Proof Since {eα } is almost equicontinuous, diam(eα (M)) is uniformly bounded above. The result now follows from Corollary 5.3 and Section 3. Proof of Theorem 1.1 Since diam(f (S)) ≤ D, we can find a compact domain U ⊃ f (S) with diam(U ) ≤ 2D. By results of Section 3, any family of such domains U has bounded geometry; hence the conclusion follows from Corollary 5.3. Proof of Theorem 1.2 Let Nα be a family of nonnegatively curved manifolds satisfying conditions of Theorem 1.2. For any α, let Sα ⊂ Nα be a soul of Nα . First, we show that {Sα } has uniformly bounded geometry. By assumption, Sα has lower volume bound. Lower sectional curvature bound follows because souls are totally geodesic. Since diam(fα (Sα )) ≤ D

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and there is a distance-nonincreasing retraction of rα : Nα → Sα (see [Sha]), the diameter of rα (fα (Sα )) is at most D. The map rα ◦ fα : Sα → Sα is a homotopy equivalence; in particular, it has nonzero degree, and hence it is surjective. We conclude that diam(Sα ) ≤ D. Thus, {Sα } has uniformly bounded geometry. Since diam(fα (Sα )) ≤ D, we can find compact domains Uα ⊃ fα (Sα ) with diam(Uα ) ≤ 2D. Again, {Uα } has uniformly bounded geometry, and the conclusion follows from Corollary 5.5. We now prove a theorem that, in particular, implies Corollary 1.3. theorem 6.2 Given n, K, D, r, v, there is a finite collection of vector bundles such that, for any totally geodesic embedding of a closed Riemannian manifold M into a complete Riemannian manifold N , the normal bundle of M is topologically equivalent to a bundle of the collection, provided diam(M) ≤ D, sec(N) ≥ −1, and provided there exist positive r, v, and a point p ∈ e(M) such that volN B(p, r) ≥ v. Proof Start with an arbitrary family of totally geodesic embeddings eα : Mα → Nα as above. First, we show that {Mα } has uniformly bounded geometry where Mα is equipped with the induced Riemannian metric. By a result of E. Heintze and H. Karcher [HK], volNα B(pα , r) ≥ v implies a lower volume bound on Mα . Since Mα is totally geodesic, sec(Mα ) ≥ −1, and by assumption diam(Mα ) ≤ D. Thus, Perelman’s stability theorem implies that {Mα } has uniformly bounded geometry (see Theorem 3.5) Note that diam(eα (Mα )) ≤ diam(Mα ) ≤ D; hence, Theorem 3.5 implies that there is a compact domain Wα ⊃ eα (Mα ) such that {Wα } has uniformly bounded geometry. The result now follows from Corollary 5.5(ii) because totally geodesic embeddings {eα } are 1-Lipschitz; in particular, {eα } is equicontinuous. The same proof gives the following theorem. theorem 6.3 Given positive n, H , D, i0 , and , there is a finite collection of vector bundles such that, for any totally geodesic embeddings of a closed Riemannian manifold M into a complete Riemannian manifold N, the normal bundle of M is topologically equivalent to a bundle of the collection, provided diam(M) ≤ D, Ric(N) ≥ −1, and inj(x) ≥ i0 for any x in the -neighborhood of image of M.


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Remark 6.4 There are obvious “fixed base” modifications of Theorems 6.2 and 6.3. corollary 6.5 Given positive D, r, v, n and a closed manifold M with ⊕i>0 H 4i (M, Q) = 0, there exists a finite collection of vector bundles over M such that, for any open complete Riemannian n-manifold N with sec(N) ≥ 0 and a soul S ⊂ N, the normal bundle to S is topologically equivalent to a bundle of the collection, provided S is homeomorphic to M, and provided the inclusion S → N is homotopic to a map f such that diam(f (S)) ≤ D and volN (B(p, r)) ≥ v for some p ∈ f (S). Proof First, note that, up to topological equivalence, only finitely many of the bundles νS can have zero Euler class. (Otherwise, there is a sequence of pairwise topologically inequivalent bundles νSk with zero Euler class). Use homeomorphisms M → Sk to pull the bundles back to M. These pullback bundles clearly have zero Euler class as well as zero rational Pontrjagin classes since ⊕i>0 H 4i (M, Q) = 0. Thus the bundles belong to finitely many isomorphism classes (which implies that νSk belong to finitely many topological equivalence classes). Now if the Euler class of the normal bundle to S is nonzero, then f (S) ∩ S # = ∅; hence the distance from p to S is ≤ D, and the result follows from Theorem 6.2. Proof of Corollary 1.4 Let N n and p ∈ S ⊂ N be chosen to satisfy the assumptions of Corollary 1.4. According to Theorem 1.1, we only have to show that under our assumptions we √ have a uniform lower bound on vol B(p, r). Let r0 = min{r/2, π/(2 K)}. Let l , g ) be a round sphere of constant curvature K, and let l = codim S − 1, let (SK can l → Sl . p¯ be any point on this sphere. Consider the exponential map expK : Tp¯ SK K Denote by v(l, K, t) the volume of the ball of radius t centered at p. ¯ First of all, notice that, by the triangle inequality, B(p, r) contains a tubular neighborhood U (p, r0 ) consisting of all points x ∈ N such that d(x, S) ≤ r0 and d(p, Sh(x)) ≤ r0 . Here Sh stands for the Sharafutdinov retraction Sh : N → S. For any x ∈ S, denote B ⊥ (x, t) = y ∈ N | d(y, S) ≤ t and Sh(y) = x . Since Sharafutdinov retraction is a C 1 -Riemannian submersion (see [Per1]), we can apply Fubini’s theorem to see that vol U (p, r0 ) = vol B ⊥ (x, r0 ) dvol(x). (1) BS (p,r0 )

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Here BS (p, r0 ) stands for the ball of radius r0 around p in S. It suffices to show that for each x ∈ BS (p, r0 ) we have vol B ⊥ (x, r0 ) ≥ v(l, K, r0 ). (Indeed, it would imply that vol(B(p, r)) ≥ vol(U (p, r0 )) ≥ vol(BS (p, r0 )) · v(l, K, r0 ). Finally, by volume comparison, vol(BS (p, r0 )) is bounded below in terms of D and v, and we are done.) ⊥ Fix an x ∈ BS (p, r0 ), and consider the normal exponential map exp⊥ x : Tx S → ⊥ N. It follows from [Per1] that this map sends the ball BTx⊥ (0, r0 ) onto B (x, r0 ). l , and use it to equip B (0, r ) with Choose a linear isometry between Tx⊥ and Tp¯ SK 0 Tx⊥ the metric gK of constant curvature K. Let gx be the induced Riemannian metric on the Sharafutdinov fiber over x. To finish the proof it is enough to establish the following lemma saying that “reverse Toponogov comparison” holds on B ⊥ (x, r0 ). lemma 6.6 ⊥ The surjection exp⊥ x : (BTx⊥ (0, r0 ), gK ) → (B (x, r0 ), gx ) is a distance-nondecreasing diffeomorphism. Let v be a unit vector in Tx⊥ , and let γ (t) = exp(tv) be the normal geodesic in direction v. We now show that, for any t ≤ r0 and any X ∈ Tγ (t) with |X| = 1 and X, γ (t), we have K(X, γ (t)) ≤ K. Write X = Xh + Xv as a sum of its horizontal and vertical components. Then K X, γ (t) = R γ (t), Xh + X v γ (t), Xh + X v

= R γ (t), Xv γ (t), Xv + R γ (t), Xh γ (t), Xh

+ R γ (t), Xh γ (t), Xv + R γ (t), Xv γ (t), Xh . The first term on the right-hand side is less than or equal to K by assumption and also because |X v | ≤ |X| = 1. By [Per1], R(γ (t), Xh )γ (t) = 0, and therefore

R γ (t), Xh γ (t), Xv = R γ (t), Xh γ (t), Xh = 0. By the symmetry of the curvature tensor, the fourth term is equal to the third one and h h hence is also equal to zero. √ Thus, K(X, γ (t) = R(γ (t), X )γ (t), X ≤ K. Now since r0 < π/ K and all the two planes along γ (t) containing γ (t) have curvature less than or equal to K, we can apply the Rauch comparison theorem to conclude that the differential of exp⊥ x does not decrease the lengths of tangent vectors ⊥ : (B (0, r ), g and thus that exp⊥ ⊥ 0 K ) → (B (x, r0 ), gx ) is a local diffeomorphism Tx x that does not decrease lengths of curves. It remains to show that this map is one-to-one. Suppose it is not. Then the injectivity radius rx of B ⊥ (x, r0 ) at x is strictly less than r0 . Let v, u ∈ Tx⊥ be such that |u| = |v| = rx and exp(v) = exp(u). Denote q = exp(v) = exp(u). Notice that geodesics γv (t) = exp(tv) and γu (t) = exp(tu) : [0, 1] → N connecting x and q are obviously distance minimizing. By [CE,


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Lemma 5.6.5] these geodesics must form a geodesic loop at x (i.e., γu (1) = −γv (1)). This is impossible since, according to [CG], the soul S is totally convex and x lies in S.

corollary 6.7 Given positive D, n, v, r, K, there exists a finite family of vector bundles such that, for any complete open Riemannian n-manifold N with sec(N) ≥ 0 and a soul S ⊂ N , the normal bundle to S is topologically equivalent to a bundle of the collection, provided diam(S) ≤ D, vol(S) ≥ v, and there is a point p ∈ S such that all the vertical curvatures at the points of B(p, r) are bounded above by K. Proof The proof of Corollary 1.4 gives a uniform lower bound on vol B(p, r), so the result follows from Theorem 1.2. corollary 6.8 Given positive D, r, v, n, K and a closed Riemannian manifold M with sec(M) ∈ [−1, 0], there exists a finite collection of vector bundles over M such that, for any totally geodesic embedding e : M → N of M into an open complete Riemannian n-manifold N with sec(N ) ≤ 0, the normal bundle νe is isomorphic to a bundle of the collection, provided vol(M) ≥ v and e is a homotopy equivalence homotopic to a map f with diam(f (M)) ≤ D such that the sectional curvature at any point of the r-neighborhood of f (M) is greater than or equal to −K. Proof Start with an arbitrary family of totally geodesic embeddings eα : M → Nα as above. First, note that for any α the injectivity radius of Nα satisfies inj(Nα ) ≥ inj(M) = inj(eα (M)). If not, there is a point p ∈ Nα with injNα (p) < inj(M). Since Nα and M have nonpositive sectional curvatures, the injectivity radius at any point is half the length of the shortest geodesic loop at this point (see [CE, Lemma 5.6.5]). Take a geodesic loop at p of length less than inj(M)/2, and project it to eα (M) by the closest point retraction rα : Nα → eα (M) (see [BGS]). The retraction is a distancenonincreasing homotopy equivalence. (In fact, since sec(Nα ) ≤ 0, the normal exponential map identifies Nα with the normal bundle to eα (M) where rα corresponds to the bundle projection.) Thus, we get a homotopically nontrivial curve of length less than inj(M)/2 in eα (M). Since eα is an isometric embedding, it preserves lengths of curves. Therefore, we obtain a homotopically nontrivial curve of length less than inj(M)/2 in M, which is impossible because loops of length less than inj(M)/2 lift to loops in the universal cover.

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Find compact domains Uα ⊃ fα (M) that lie in the r-neighborhood of fα (M). Since diam(fα (M)) ≤ D, the diameter of Uα is less than or equal to D + 2r. Also, sec(Uα ) ∈ [−K, 0] and inj(Uα ) ≥ inj(M); hence Theorem 3.1 implies that {Uα } has bounded geometry, and the conclusion follows from Corollary 5.3. Proof of Corollary 1.5 Since sec(Mα ) is bounded in absolute value, a lower bound on volume implies a lower bound on the injectivity radius. Thus, the result follows from Corollary 5.5 exactly as in the proof of Corollary 6.8. Remark 6.9 In the statement of Corollary 1.5 one can replace “vol(M) ≥ v” by “π1 (M) has no normal virtually abelian subgroups” or, equivalently, by “the universal cover of M has no Euclidean de Rham factor” (see, e.g., [Fuk, pp. 395–396]). Remark 6.10 Given a closed Riemannian manifold M, there are only finitely many isomorphism classes of normal bundles of isometric immersions f : M → N into Riemannian manifolds N such that | sec(N)| and the second fundamental form of f are uniformly bounded. Indeed, these bounds imply a uniform bound on the curvature form of the normal bundle to νf . Then by Chern-Weil theory we get bounds on Euler and Pontrjagin classes which determine a vector bundle up to finitely many possibilities. An alternative proof was recently found by K. Tapp [Tap]. Instead of getting bounds on the characteristic classes, Tapp estimates the number of homotopy classes of maps into the classifying space.

Appendix. Vector bundles with diffeomorphic total spaces The purpose of this appendix is to discuss the extent to which a vector bundle is determined by its total space. We got interested in this problem when we noticed that, under the assumptions of Corollary 1.3, the homeomorphism finiteness for the total spaces can be easily obtained from the parametrized version of Perelman’s stability theorem in [Per2]. Let ηk be an infinite sequence of vector bundles over a closed smooth manifold M such that the total spaces E(ηk ) are homeomorphic. Assume that the natural homomorphism Homeo(M) → E (M) of the homeomorphism group of M into the group of (free) homotopy classes of self-homotopy equivalences of M has finite cokernel. Then the homeomorphism type of the total space determines (the topological equivalence class of) a vector bundle, up to a finite ambiguity. (Indeed, the homeomorphism E(ηi ) → E(η1 ) induces a self-homotopy equivalence gi of M. Passing to


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a subsequence, we can assume that gi−1 ◦ gj is homotopic to a homeomorphism for any i, j . Let si be the zero section of ηi . Then the maps sj and si ◦ gi−1 ◦ gj have equal invariants in the sense of Section 4. Hence, for a fixed j , Proposition 4.6 implies that the bundles (gi−1 ◦ gj )# ηi fall into finitely many isomorphism classes. Therefore, the bundles ηk fall into finitely many topological equivalence classes.) For example, if the group E (M) is finite, then the homeomorphism type of the total space determines (the isomorphism class of) a vector bundle, up to a finite ambiguity. In fact, since a vector bundle is determined, up to a finite ambiguity, by its characteristic classes, it suffices to assume that the natural action of E (M) on the cohomology groups that contain these classes is an action of a finite group. For instance, up to isomorphism, there are only finitely many vector bundles over CP n with homeomorphic total spaces because E (CPn ) is finite (see [Rut, Chapter 12, Section 18.3] for many more examples; also see [DW] for the case when M is a sphere). Also, if the image of the diffeomorphism group Diffeo(M) in E (M) has finite index, then the homeomorphism type of the total space determines (the smooth equivalence class of) a vector bundle, up to finite ambiguity. If dim(M) ≥ 6 and π1 (M) = 1, then the image of D (M) in E (M) is commensurable to the isotropy subgroup of the total rational Pontrjagin class of T M (see [Sul, pp. 322–323]). In particular, the image of D (M) has finite index in E (M) when p(T M) = 1. For instance, up to smooth equivalence, there are only finitely many vector bundles with homeomorphic total spaces over the direct product of finitely many spheres. Now we give an example of an infinite sequence of pairwise topologically inequivalent vector bundles with diffeomorphic total spaces. The base manifold is homotopy equivalent to S 3 × S 3 × S 5 . We are thankful to Shmuel Weinberger for providing a key idea for this example. (As usual, the authors assume all responsibility for possible mistakes.) Example A.1 Consider a manifold N = S 3 × S 3 × S 5 , and let t be a nonzero element of H 8 (N, Z). One can always find a positive integer q and a vector bundle τ over N of rank greater than or equal to dim(N ) such that qt = p2 (τ ) ∈ H 12 (N, Z), where p2 is the second integral Pontrjagin class. (Indeed, the Pontrjagin character ph defines an isomorphism of vector spaces ph : K(N) ⊗ Q −→ ⊕i>0 H 4i (N, Q). Consider a finite sum i [ηi ]⊗pi /qi , where pi , qi are integers and ηi are bundles over N such that ph( i [ηi ] ⊗ pi /qi ) = t ⊗ 1. Reducing to a common denominator and using that pi [ηi ] is the class of pi -fold Whitney power of ηi , we get t ⊗1 = ph(ξ )/q for some positive integer q and a bundle ξ over N; one can choose the rank of ξ to

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be any number greater than or equal to 15. Since the first Pontrjagin class lives in the zero group, the formula for the ph2 , the part of the Pontrjagin character that lives in 8th cohomology, reduces to ph2 = −p2 /6, and we are done.) Replacing τ with its Whitney power, we can assume that τ is fiber homotopy equivalent to the trivial bundle T N. (Note that p2 (τ ) is still an integral multiple of t.) Since dim(N ) is odd and greater than or equal to 5, and since N is simply connected, a result of W. Browder and S. Novikov [Bor, II.3.1] implies that there is a closed smooth manifold M and a homotopy equivalence f : M → N such that f # τ is stably isomorphic to T M. It follows from [Sie] that any automorphism of H3 (S 3 × S 3 , Z) ∼ = Z2 is induced 3 3 by a self-homotopy equivalence. Since the inclusion S × S → N induces an isomorphism of the third integral homology groups, any automorphism of H3 (N, Z) is induced by a self-homotopy equivalence. The same is therefore true for M. Furthermore, any automorphism of H 8 (M, Z) is induced by a self-homotopy equivalence. (Indeed, start with φ ∈ Aut(H 8 (N, Z)) and conjugate it to φ ∈ Aut(H3 (N, Z)) via the Poincaré duality isomorphism. If f is a self-homotopy equivalence of M inducing φ , then φ can be identified with the transfer map for f . The transfer map is the inverse to f ∗ ; hence f −1 induces φ.) Note that Aut(H 8 (M, Z)) ∼ = GL(2, Z). Recall that any vector (a, b) ∈ Z2 is GL(2, Z)-equivalent to (k, 0), where k = gcd(a, b). The vectors vm = (k, km) are GL(2, Z)-equivalent to (k, 0) and lie in different orbits of the GL(2, Z)-stabilizer of (k, 0). Thus, one can find an infinite sequence of elements wm ∈ H 8 (M, Z) ∼ = Z2 that lie in different orbits of the stabilizer of p2 (T M) in the group GL(2, Z) and are GL(2, Z)-equivalent to p2 (T M). Find self-homotopy equivalences gm of M such ∗ p (T M); let g = id. Let η be a bundle over M with [η ] = that wm = gm 2 1 m m # T M] − [T M]; one can choose the rank of η to be any number greater than or [gm m equal to 15. Now if ηi is topologically equivalent to ηj , then there is a self-homeomorphism h of M that takes p2 (ηi ) = p2 (gi# T M) − p2 (T M) to p2 (ηj ) = p2 (gj# T M) − p2 (T M) (p2 is additive because p1 = 0). Since homeomorphisms preserve rational Pontrjagin classes, we get h∗ gi∗ p2 (T M) = gj∗ p2 (T M), up to elements of finite order. The group H 8 (M, Z) is torsion free; hence the above equality holds exactly. So h∗ wi = wj , which contradicts the definition of wm . Thus, ηm are pairwise topologically inequivalent. The map gm induces a homotopy equivalence from the total space E(ηm ) to # T M] − [T M] + [T M] = [g # T M]. E(η1 ). Now [T E(ηm )] = [ηm ] + [T M] = [gm m Thus, the homotopy equivalence E(ηm ) → E(η1 ) is tangential and hence is homotopic to a diffeomorphism (see [LS, pp. 226–228]).


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Remark A.2 We now briefly describe a generalization of the above example. First of all, instead of S 3 × S 3 , one can start with a product P of an arbitrary number of 1-connected homology m-spheres with odd m > 1. (Any odd-dimensional sphere is rationally homotopy equivalent to a K(Z, m) space, so the product P is rationally homotopy equivalent to K(Zk , m), where k is the number of factors. This implies that the natural action E (P ) → Aut(Hm (P , Z)) ∼ = GL(k, Z) has finite cokernel. Thus, “almost” every automorphism is induced by a homotopy equivalence that turns out to be enough for us.) By taking a product with a suitable 1-connected manifold S (which was S 5 in the example), we can shift dimensions so that dim(P × S) is odd and E (P × S) −→ Aut H 4i (P × S, Z) ∼ = GL(k, Z) has finite cokernel for some i. Using [Bro, Theorem II.3.1], we replace P × S by a homotopy equivalent manifold M with a nonzero Pontrjagin class pi . (The freedom in the choice of τ gives infinitely many possibilities for the homeomorphism type of M.) Now it is not hard to cook up an infinite sequence of pairwise topologically inequivalent bundles ηj over M with diffeomorphic total spaces. One can also get nonsimply connected examples by taking products of E(ηj )×L, where L is a suitable closed manifold. Acknowledgments. We are grateful to M. Anderson for an illuminating communication on the local version of [AC] and to S. Weinberger for the idea of Example A.1. I. Belegradek is thankful to A. Nicas and I. Hambleton for several helpful discussions on self-equivalences of manifolds. V. Kapovitch is grateful to Kris Tapp for bringing to his attention the idea of bounding homotopy types of maps using equicontinuity and for many helpful conversations on nonnegatively curved manifolds.

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Belegradek Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125, USA; [email protected] Kapovitch Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA; fax (215)-573-4063; [email protected]

DYNAMICAL QUANTUM GROUPS AT ROOTS OF 1 PAVEL ETINGOF and DMITRI NIKSHYCH

Abstract Given a dynamical twist for a finite-dimensional Hopf algebra, we construct two weak Hopf algebras, using methods of P. Xu and of P. Etingof and A. Varchenko, and we show that they are dual to each other. We generalize the theory of dynamical quantum groups to the case when the quantum parameter q is a root of unity. These objects turn out to be self-dual—which is a fundamentally new property, not satisfied by the usual Drinfeld-Jimbo quantum groups.

1. Introduction 1.1. The notion of a dynamical quantum group was first suggested by G. Felder [Fe] in 1994. Namely, Felder considered the quantum dynamical Yang-Baxter equation (also known as the Gervais-Neveu equation), which is a generalization of the usual quantum Yang-Baxter equation, and used the Faddeev-Reshetikhin-Takhtajan method to associate to a solution R of this equation a certain algebra—the dynamical quantum group FR . Felder also considered representations of FR and showed that, although FR is not a Hopf algebra, its representations form a tensor category. In 1991 O. Babelon [Ba] generalized V. Drinfeld’s twisting method to the dynamical case, introducing the notion of a dynamical twist (see also [BBB]). Given a dynamical twist in a quasi-triangular Hopf algebra U , one can define a solution of the dynamical Yang-Baxter equation acting on the tensor square of any representation of U . In 1997 it was shown independently in [ABRR], [JOKS], and [EV2] that one can naturally construct a dynamical twist in the universal enveloping algebra of any simple Lie algebra or its q-deformation. Using the method of [BBB], one can obtain from these twists the solutions of the quantum dynamical Yang-Baxter equation from [Fe]. DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 1, Received 28 March 2000. 2000 Mathematics Subject Classification. Primary 17B37, 81R50; Secondary 16W30. Etingof’s work supported by National Science Foundation grant number DMS-9700477. Nikshych’s work supported in part by a University of California at Los Angeles research assistantship.

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At the same time, it was shown in [EV2] that, to any dynamical twist J , one can associate a Hopf algebroid F (J ). In the cases of [Fe], this Hopf algebroid coincides with FR as an algebra. In particular (as was shown already in [EV1]), FR has the structure of a Hopf algebroid. This explains the existence of a tensor product on the category of representations of FR . In 1999 P. Xu [Xu2] associated another Hopf algebroid U (J ) to a dynamical twist J on a Hopf algebra U , obtained by twisting U by means of J . This Hopf algebroid U (J ) is closely related to the quasi-Hopf algebra associated to (U, J ) in [BBB] and [JOKS] (see [Xu1]). Xu suggested that U (J ) should be dual, in an appropriate sense, to F (J ). (This is motivated by the duality of their classical limits.) However, it is not very convenient to formulate such a statement precisely because of difficulties with the notion of a dual Hopf algebroid. Moreover, it was shown in [EV2] that, for dynamical twists J constructed in [ABRR], [JOKS], and [EV2], the category of representations of F (J ) is essentially the same (as a tensor category) as that of U (g) or Uq (g). Thus, it is essentially the same as that of U (J ), which suggests that U (J ) and F (J ) should be not only dual to each other but also isomorphic. In other words, U (J ) and F (J ) should be selfdual. Note that this would be a fundamentally new property, not satisfied by the usual Drinfeld-Jimbo quantum groups. 1.2. This paper has the following two main goals: (1) to make the above picture precise, and (2) to generalize the theory of dynamical quantum groups to the case when the quantum parameter q is a root of unity. Our first step is that we replace the notion of a Hopf algebroid with the recently introduced notion of a weak Hopf algebra (see [BNSz], [BSz]). Roughly speaking, a weak Hopf algebra is an algebra and a coalgebra such that is a homomorphism of algebras but is allowed to map 1 to some idempotent not equal to 1 ⊗ 1 (see Section 2 for a precise definition). Every weak Hopf algebra has a natural structure of a Hopf algebroid, but not vice versa. However, it turns out that dynamical quantum groups (at roots of unity) are a nice class of Hopf algebroids that do come from weak Hopf algebras. Moreover, regarding dynamical quantum groups as weak Hopf algebras rather than Hopf algebroids is convenient for two reasons: first, the definition of a weak Hopf algebra is much simpler, and second, it is naturally self-dual. Our main results can be summarized as follows. (1) We generalize Drinfeld’s twisting theory to weak Hopf algebras (see Section 3). In particular, we show that twisting of a quasi-triangular weak Hopf algebra (defined in [NTV]) gives another quasi-triangular weak Hopf algebra.

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(2) For every dynamical twist J : T → U ⊗2 of a Hopf algebra U , we define two weak Hopf algebras H and D: the first by analogy with the construction of [Xu2], and the second by analogy with the construction of [EV2]. (They correspond to the Hopf algebroids of [Xu2], [EV2]; see Section 4.) We show that H is isomorphic to D ∗ with opposite comultiplication. We consider the special case when U is quasi-triangular, and we analyze the homomorphism H ∗ op = D → H defined by the quasi-triangular structure on H . We give a criterion when this homomorphism is an isomorphism. (3) We take U to be quantum group Uq (g) (for a finite-dimensional simply laced Lie algebra g) when q is a root of unity (more precisely, the finite-dimensional version considered by G. Lusztig [L]; see Section 5). We show that the known methods of producing dynamical twists for generic q (see [ABRR], [ESS]) can also be used to produce dynamical twists when q is a root of unity. In particular, for every generalized Belavin-Drinfeld triple ( 1 , 2 , T ) for g, we construct (following [ESS]) a family of dynamical twists for Uq (g) which depends on | 1 | parameters. With appropriate modifications, this construction can be carried out in the nonsimply laced case as well. (4) We show that if T is an automorphism of the Dynkin diagram of g (in particular, if T = 1), then the weak Hopf algebras H and D associated to the corresponding twists are isomorphic (via the R-matrix of H ). In particular, D is self-dual (isomorphic to D ∗ op ), as was expected (for T = 1) in the case of generic q. In particular, this implies that in this case the category of representations of the algebra D = DT corresponding to T is equivalent (as a tensor category) to Rep(U ) and thus to Rep(DT ) for any other automorphism T . In particular, Rep(DT ) is equivalent to Rep(D1 ). Note that an analog of the latter result (when q is generic, g is the affine Lie algebra sln , and T is the rotation of the Dynkin diagram of g by the angle 2πk/n, (k, n) = 1) is proved in [ES1]. 2. Weak Hopf algebras and Hopf algebroids Throughout this paper we work over an algebraically closed field k and use M. Sweedler’s notation for comultiplication, writing (h) = h(1) ⊗ h(2) . 2.1. Weak Hopf algebras Definition 2.1.1 [BNSz], [BSz] A weak bialgebra is a k-vector space H that has structures of an algebra (H, m, 1) and a coalgebra (H, , ε) such that the following axioms hold: (1) is a (not necessarily unit-preserving) homomorphism (hg) = (h) (g); (2) the unit and counit satisfy the identities

(1)

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ε(hgf ) = ε(hg(1) )ε(g(2) f ) = ε(hg(2) )ε(g(1) f ), ( ⊗ id) (1) = (1) ⊗ 1 1 ⊗ (1) = 1 ⊗ (1) (1) ⊗ 1 ,

(2) (3)

for all h, g, f ∈ H . Definition 2.1.2 [BNSz], [BSz] A weak Hopf algebra H is a weak bialgebra equipped with a linear map S : H → H , called an antipode, satisfying the following axioms: m(id ⊗S) (h) = (ε ⊗ id) (1)(h ⊗ 1) , (4) m(S ⊗ id) (h) = (id ⊗ε) (1 ⊗ h) (1) , (5) S(h(1) )h(2) S(h(3) ) = S(h),

(6)

for all h ∈ H . Here axioms (2) and (3) of Definition 2.1.1 are analogous to the bialgebra axioms of ε being an algebra homomorphism and of being a unit-preserving map; axioms (4) and (5) of Definition 2.1.2 generalize the properties of the antipode with respect to the counit. Also, it is possible to show that, given (1)–(5), axiom (6) is equivalent to S being both an antialgebra and an anticoalgebra map. A morphism of weak Hopf algebras is a map between them which is both an algebra and a coalgebra morphism commuting with the antipode. The image of a morphism is clearly a weak Hopf algebra. The tensor product of two weak Hopf algebras is defined in an obvious way. We summarize below the basic properties of weak Hopf algebras (see [BNSz] for the proofs). The antipode S of a weak Hopf algebra H is unique; if H is finite-dimensional, then it is bijective (see [BNSz]). The right-hand sides of formulas (4) and (5) are called the target and source counital maps and denoted εt , εs , respectively: εt (h) = (ε ⊗ id) (1)(h ⊗ 1) , (7) (8) εs (h) = (id ⊗ε) (1 ⊗ h) (1) . The counital maps εt and εs are idempotents in Endk (H ), and they satisfy relations S ◦ εt = εs ◦ S and S ◦ εs = εt ◦ S. The main difference between weak and usual Hopf algebras is that the images of the counital maps are not necessarily equal to k. They turn out to be subalgebras of H called target and source counital subalgebras or bases because they generalize the notion of a base of a groupoid (cf. examples below):

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Ht = h ∈ H | εt (h) = h = (φ ⊗ id) (1) | φ ∈ H ∗ , Hs = h ∈ H | εs (h) = h = (id ⊗φ) (1) | φ ∈ H ∗ .

(9) (10)

The counital subalgebras commute, and the restriction of the antipode gives an antiisomorphism between Ht and Hs . Any morphism between weak Hopf algebras preserves counital subalgebras; that is, if : H → H is a morphism, then its restrictions on the counital subalgebras are isomorphisms: |Ht : Ht ∼ = Ht and |Hs : Hs ∼ = Hs . The algebra Ht (resp., Hs ) is separable (and, therefore, semisimple (see [P])) with the separability idempotent et = (S ⊗ id) (1) (resp., es = (id ⊗S) (1)); that is, we have m(et ) = m(es ) = 1, and (z1 ⊗ 1)et (z2 ⊗ 1) = (1 ⊗ z2 )et (1 ⊗ z1 ),

z1 , z2 ∈ Ht ,

(y1 ⊗ 1)es (y2 ⊗ 1) = (1 ⊗ y2 )es (1 ⊗ y1 ),

y1 , y2 ∈ Hs .

As a consequence of this fact and of being a homomorphism, we have the following useful identities: (1 ⊗ z2 ) (h)(1 ⊗ z1 ) = S(z1 ) ⊗ 1 (h) S(z2 ) ⊗ 1 , z1 , z2 ∈ Ht , (11) (12) (y1 ⊗ 1) (h)(y2 ⊗ 1) = 1 ⊗ S(y2 ) (h) 1 ⊗ S(y1 ) , y1 , y2 ∈ Hs . Note that H is an ordinary Hopf algebra if and only if (1) = 1 ⊗ 1, if and only if ε is a homomorphism, and if and only if Ht = Hs = k. The dual vector space H ∗ has a natural structure of a weak Hopf algebra with the structure operations dual to those of H : φψ, h = φ ⊗ ψ, (h) , (13) (φ), h ⊗ g = φ, hg , (14) S(φ), h = φ, S(h) , (15) for all φ, ψ ∈ H ∗ , h, g ∈ H . The unit of H ∗ is ε, and the counit is φ → φ, 1. One can check that if S is invertible, then the opposite algebra H op is a weak Hopf algebra with the same coalgebra structure and the antipode S −1 . Similarly, the coopposite coalgebra H cop is a weak Hopf algebra with the same algebra structure and the antipode S −1 . It was shown in [NTV] that modules over any weak Hopf algebra H form a monoidal category, called the representation category and denoted Rep(H ) with the product of the modules V and W being equal to (1)(V ⊗ W ) and with the unit object given by Ht , which is an H -module via h · z = εt (hz), h ∈ H , z ∈ Ht . Example 2.1.3 Let G be a groupoid over a finite base (i.e., a category with finitely many objects

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such that each morphism is invertible); then the groupoid algebra kG is generated by morphisms g ∈ G with the unit 1 = X idX , where the sum is taken over all objects X of G, and the product of two morphisms is equal to their composition if the latter is defined and zero otherwise. It becomes a weak Hopf algebra via (g) = g ⊗ g,

ε(g) = 1,

S(g) = g −1 ,

g ∈ G.

(16)

The counital maps are given by εt (g) = gg −1 = idtarget(g) and εs (g) = g −1 g = idsource(g) . If G is finite, then the dual weak Hopf algebra (kG)∗ is generated by idempotents pg , g ∈ G, such that pg ph = δg,h pg and (pg ) = pu ⊗ pv , ε(pg ) = δg,gg −1 = δg,g −1 g , S(pg ) = pg −1 . (17) uv=g

It is known that any group action on a set gives rise to a finite groupoid. Similarly, in the noncommutative situation, one can associate a weak Hopf algebra with every action of a usual Hopf algebra on a separable algebra (see [NTV] for details). 2.2. Hopf algebroids The following notions were introduced in [Lu] (see also [Xu2]). Definition 2.2.1 An algebra H is called a total algebra with a base algebra R if there exist a target map α : R → H which is an algebra homomorphism and a source map β : R → H which is an algebra antihomomorphism such that the images α(R) and β(R) commute in H , that is, α(a)β(b) = β(b)α(a), ∀a, b ∈ R. (18) If H is a total algebra, then the above maps define a natural (R−R)-bimodule structure on H via a · h · b = α(a)β(b)h, h ∈ H, a, b ∈ R. Note that the bimodule tensor products H ⊗R H , H ⊗R H ⊗R H, . . . are (R − R)bimodules in an obvious way. Definition 2.2.2 A comultiplication on a total algebra H is an (R − R)-bimodule map : H → H ⊗R H satisfying (1) = 1 ⊗R 1, ⊗R id = id ⊗R : H −→ H ⊗R H ⊗R H, (19) which is compatible with the maps α, β and multiplication in the sense that

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(h) β(a) ⊗ 1 − 1 ⊗ α(a) = 0, (hg) = (h) (g),

h, g ∈ H, a ∈ R.

(20) (21)

Note that the right-hand side of equation (21) is well defined in H ⊗R H because of condition (20). Definition 2.2.3 A counit for H is an (R − R)-bimodule map - : H → R (where R is an (R − R)bimodule via multiplication) such that -(1H ) = 1R and - ⊗ idH = idH ⊗- = idH , (22) where we identify R ⊗R H ∼ = H ⊗R R ∼ = H. Definition 2.2.4 A bialgebroid is a total algebra H that possesses a comultiplication and counit. Definition 2.2.5 An antipode for a bialgebroid H with a base R is a map τ : H → H which is an algebra antihomomorphism such that τ ◦ β = α and satisfies the following properties: (1) m(τ ⊗ id) = β ◦ ε ◦ τ ; (2) there exists a linear map γ : H ⊗R H → H ⊗ H which is a right inverse for the natural projection H ⊗ H → H ⊗R H such that m(id ⊗τ )γ = α ◦ ε. Note that m(τ ⊗ id) is well defined on H ⊗R H but m(id ⊗τ ) is not; this is why it is necessary to assume the existence of γ in Definition 2.2.5. Definition 2.2.6 A Hopf algebroid H is a bialgebroid with an antipode. A (base-preserving) morphism between Hopf algebroids H = (H, R, . . . ) and H = (H , R , . . . ) is a pair (ψ, 0), where ψ : R → R is an algebra isomorphism and 0 : H → H is an algebra homomorphism that is also an (R − R)-bimodule map such that (0 ⊗ 0) = ◦ 0 and α ◦ ψ = 0 ◦ α,

β ◦ ψ = 0 ◦ β,

ε ◦ 0 = ψ ◦ ε,

τ ◦ 0 = 0 ◦ τ.

2.3. The Hopf algebroid corresponding to a weak Hopf algebra It turns out that weak Hopf algebras form a proper subclass of Hopf algebroids.

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proposition 2.3.1 Any weak Hopf algebra H (not necessarily finite-dimensional) has a natural structure of a Hopf algebroid (i.e., this assignment defines a functor). Proof Define the base algebra to be the target subalgebra of H , that is, R = Ht , and let α = idHt , β = S −1 |Ht . Then, clearly, α(R) = Ht , β(R) = Hs , so that images of α and β commute. The comultiplication regarded as a map to H ⊗R H is coassociative and compatible with the multiplication. It is a bimodule map since (α(a)h) = α(a) ⊗ 1 (h), (β(a)h) = 1 ⊗ β(a) (h), (23) for all h ∈ H and a ∈ R. We also have the compatibility condition (h) β(a) ⊗ 1 − 1 ⊗ α(a) = h(1) S −1 (a) ⊗ h(2) − h(1) ⊗ h(2) a = 0. Next, let - = εt ; then we have -(1) = 1 and -(a · h · b) = aεt (h)b = a-(h)b, for all h ∈ H , a, b ∈ R; that is, - is an (R − R)-module map; also, -(h(1) ) · h(2) = εt (h(1) )h(2) = ε 1(1) h(1) 1(2) h(2) = h, h(1) · -(h(2) ) = S −1 -(h(2) ) h(1) = 1(1) h(1) ε S(1(2) )h(2) = h, where we used the antipode axioms of a weak Hopf algebra. Thus, - satisfies the counit axiom. The antipode τ = S is an antialgebra morphism such that τ ◦ β = idHt = α. The section γ : H ⊗R H → H ⊗ H is given by γ (h ⊗R g) = h · S(1(1) ) ⊗ (1(2) · g) = (1)(h ⊗ g). Finally, we verify the antipode properties m(τ ⊗R id) = εs = S −1 ◦ εt ◦ S = β ◦ - ◦ τ, m(id ⊗R τ )γ = εt = α ◦ -. −1 Thus, (H, Ht , idHt , SH , , εt , S) is a Hopf algebroid. t If 0 : H → H is a morphism between weak Hopf algebras, then it is clear from our definitions that the pair (0|Ht , 0) gives a morphism between the corresponding Hopf algebroids.

Remark 2.3.2 The converse of Proposition 2.3.1 is false even when H is finite-dimensional. Indeed, the base algebra R = Ht of H is necessarily separable; on the other hand, for any

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algebra A, the space A ⊗ Aop has a structure of a Hopf algebroid with the base A (see [Lu, Examples 3.1 and 4.4]). In the case when A is not separable, this Hopf algebroid is not a weak Hopf algebra.

3. Weak Hopf algebras coming from twisting 3.1. Twisting We describe the procedure of constructing new weak Hopf algebras by twisting a comultiplication. Twisting of Hopf algebroids without an antipode was developed in [Xu2], and a special case of twisting of weak Hopf ∗-algebras was considered in [NV]. Definition 3.1.1 ¯ with A twist for a weak Hopf algebra H is a pair (2, 2), 2 ∈ (1)(H ⊗ H ),

¯ ∈ (H ⊗ H ) (1), 2

and

¯ = (1) 22

(24)

satisfying the following axioms: ¯ = (id ⊗ε)2 ¯ = 1, (ε ⊗ id)2 = (id ⊗ε)2 = (ε ⊗ id)2

(25)

( ⊗ id)(2)(2 ⊗ 1) = (id ⊗ )(2)(1 ⊗ 2), ¯ ⊗ 1)( ⊗ id)(2) ¯ = (1 ⊗ 2)(id ¯ ¯ (2 ⊗ )(2),

(26)

¯ ¯ ( ⊗ id)(2)(id ⊗ )(2) = (2 ⊗ 1)(1 ⊗ 2), ¯ ¯ ⊗ 1). (id ⊗ )(2)( ⊗ id)(2) = (1 ⊗ 2)(2

(27) (28) (29)

For ordinary Hopf algebras this notion coincides with the usual notion of twist, and ¯ are, each of the four conditions (26)–(29) implies the other three. But since 2 and 2 in general, not invertible, we need to impose all of them. The next proposition extends Drinfeld’s twisting construction to the weak case. proposition 3.1.2 ¯ be a twist for a weak Hopf algebra H . Then there is a weak Hopf algebra Let (2, 2) H2 having the same algebra structure and counit as H with a comultiplication and antipode given by ¯ 2 (h) = 2 (h)2,

S2 (h) = v −1 S(h)v,

(30)

for all h ∈ H2 , where v = m(S ⊗ id)2 is invertible in H2 . Proof Clearly, 2 is an algebra homomorphism. Its coassociativity follows from axioms (26) and (27).

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Observe that the relations between 2 and ε yield the following identities (recall that S is invertible when restricted on the counital subalgebras since the latter are finite-dimensional): S −1 εt 2(2) 2(1) = 1, (31) εt 2(1) 2(2) = 1, (2) (1) (1) (2) −1 ¯ ¯ ¯ ¯ = 1, 2 S εs 2 = 1. (32) 2 εs 2 ¯ =2 ¯ (1) ⊗ 2 ¯ (2) . Here and in what follows we write 2 = 2(1) ⊗ 2(2) and 2 Using these properties and equation (11), we check that ε is still a counit for (H, 2 ): (2) (1) ¯ h(2) 2(2) ¯ h(1) 2(1) 2 (ε ⊗ id) 2 (h) = ε 2 ¯ (2) S −1 εs 2 ¯ (1) hεt 2(1) 2(2) = h, =2 (2) ¯ (1) h(1) 2(1) ε 2 ¯ h(2) 2(2) (id ⊗ε) 2 (h) = 2 (2) −1 (2) (1) ¯ ¯ (1) εs 2 hS εt 2 2 = h. =2 We proceed to verify the rest of the axioms of a weak Hopf algebra, writing 2 (h) = h2(1) ⊗ h2(2) :

(2) ¯ (1) h(1) 2(1) ε 2 ¯ h(2) 2(2) f ε gh2(1) ε h2(2) f = ε g 2 (1) (2) ¯ ¯ εs 2 h(1) ε h(2) εt 2(1) 2(2) f = ε g2 = ε gh(1) ε h(2) f = ε(ghf ), (2) (1) ¯ h(2) 2(2) ε 2 ¯ h(1) 2(1) f ε gh2(2) ε h2(1) f = ε g 2 (2) −1 (1) ¯ ¯ S εs 2 h(2) ε h(1) S −1 εt 2(2) 2(1) f = ε g2 = ε gh(2) ε h(1) f = ε(ghf ), for all g, h, f ∈ H . The axioms involving 2 (1) are checked using identities (28) and (29) of Definition 3.1.1: ¯ ¯ ⊗1 1 ⊗ 2 (1) 2 (1) ⊗ 1 = 1 ⊗ 22 22 ¯ (id ⊗ )2 ¯ ⊗ id 2(2 ⊗ 1) = 1⊗2 = ( 2 ⊗ id) 2 (1) ¯ ⊗ 1)( ⊗ id)2(id ¯ = (2 ⊗ )2(1 ⊗ 2) ¯ ⊗ 1 1 ⊗ 22 ¯ = 22 = 2 (1) ⊗ 1 1 ⊗ 2 (1) . We define a new antipode by


(2) ¯ (1) S 2 ¯ S2 (h) = 2 S(h)S 2(1) 2(2)

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(33)

¯ , and so on, for additional copies of 2 and 2) ¯ and compute (writing 2 , 2 (2) (2) ¯ (1) S 2 ¯ (1) h(1) 2(1) 2 ¯ S 2 m(id ⊗S2 ) 2 (h) = 2 (2) (1) (2) ¯ × S h(2) S 2 S 2 2 ¯ (1) h(1) S h(2) S 2 ¯ (2) S 2 (1) 2 (2) =2 ¯ (1) εt (h)S 2(1) 2 ¯ 2 2(2) =2 2 (2) ¯ (1) 2(1) (1) εt (h)S 2(1) (2) εs 2 ¯ 2 =2 ¯ (1) εs 2 ¯ 2 εt 2(1) h 2(2) =2 = εt 2(1) h 2(2) = ε 1(1) 2(1) h 1(2) 2(2) (1) (2) ¯ 1(1) 2(1) h 2 ¯ 1(2) 2(2) =ε 2 = (ε ⊗ id) 2 (1)(h ⊗ 1) , ¯ and properties of the counital maps. Note that for where we used axioms of (2, 2) h = 1 we get the identity (2) (1) (2) ¯ ¯ (1) S 2 · S 2 2 = 1; 2 ¯ (2) ) is the inverse of v = S(2(1) )2(2) . It can be proved similarly that ¯ (1) S(2 that is, 2 m(S2 ⊗ id) 2 (h) = (id ⊗ε) (1 ⊗ h) 2 (1) , h ∈ H. Finally, let us write ¯ (1) h(1) 2(1) ⊗ 2 ¯ (2) h(2) 2(2) ⊗ 2 ¯ (3) h(3) 2(3) ( 2 ⊗ id) 2 (h) = 2 and compute m(m ⊗ id) S2 ⊗ id ⊗S2 2 (h) (1) (1) (2) (2) ¯ ¯ S 2 2 2 = v −1 S 2(1) S h(1) S 2 (2) (3) (3) ¯ (1) S 2 ¯ ¯ × h(2) 2(2) 2 S 2 S h(3) S 2 v (2) ¯ (1) h(2) εt 2(2) S h(2) S 2 ¯ v = v −1 S 2(1) S h(1) εs 2 ¯ (1) h(1) εt 2(2) S h(2) S 2 ¯ (2) v = v −1 S 2(1) εs 2 ¯ (1) h(1) S h(2) S 2 ¯ (2) v = v −1 εs 2

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(1) (2) ¯ ¯ = v −1 S h(1) εs 2 εt h(2) S 2 v = v −1 S(h)v = S2 (h), whence H2 is a weak Hopf algebra. Remark 3.1.3 Note that it follows from the proof of Proposition 3.1.2 that the counital maps of the twisted weak Hopf algebra H2 are given by (εt )2 (h) = ε 2(1) h 2(2) , (34) (2) (1) ¯ ε h2 ¯ , (35) (εs )2 (h) = 2 so the counital subalgebras are also getting deformed in general. Remark 3.1.4 ¯ is a twist for H and if x ∈ H is an invertible element such that εt (x) = If (2, 2) ¯ x ), where εs (x) = 1, then (2x , 2 ¯ x = x −1 ⊗ x −1 2 (x), ¯ and 2 2x = (x)−1 2(x ⊗ x) ¯ x ) are called gauge equivalent, ¯ and (2x , 2 is also a twist for H . The twists (2, 2) and x is called a gauge transformation. Given such an x, the map h → x −1 hx is an isomorphism between weak Hopf algebras H2 and H2x . Remark 3.1.5 If 2 is a twist for a weak Hopf algebra H , then it also defines a twist of the corresponding Hopf algebroid constructed as in Proposition 3.1.2 (cf. [Xu2]). 3.2. Quasi-triangular weak Hopf algebras The notion of a quasi-triangular weak Hopf algebra was introduced and studied in [NTV]. It is defined as a triple (H, R , R¯ ), where H is a weak Hopf algebra with R ∈ op (1)(H ⊗ H ) (1), R R¯ = op (1), R¯ R = (1),

R¯ ∈ (1)(H ⊗ H ) op (1),

and

(h)R = R (h), op

(36) (37)

for all h ∈ H , where op denotes the comultiplication opposite to , and such that R obeys the following conditions: (id ⊗ )R = R13 R12 ,

( ⊗ id)R = R13 R23 ,

(38)

where R12 = R ⊗ 1, and so on, as usual. The existence of a quasi-triangular structure R on H is equivalent to Rep(H )

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being a braided category, and any quasi-triangular structure R is a solution of the quantum Yang-Baxter equation R12 R13 R23 = R23 R13 R12 .

(39)

An example of a quasi-triangular weak Hopf algebra is given by the Drinfeld double of a finite-dimensional weak Hopf algebra (see [NTV]). As in the case of ordinary Hopf algebras, a quasi-triangular structure R on H defines two homomorphisms of weak Hopf algebras, ρ1 : H ∗ φ −→ (id ⊗φ)(R ) ∈ H op , ∗

ρ2 : H φ −→ (φ ⊗ id)(R ) ∈ H

cop

(40) ;

(41)

in particular, when R has a maximal (equal to dim H ) rank, then it defines isomorphisms H ∗ ∼ = H op ∼ = H cop . A twisting of a quasi-triangular weak Hopf algebra is again quasi-triangular. ¯ is a twist and if R is a quasi-triangular structure for H , then Namely, if (2, 2) ¯ 21 R 2, 2 ¯ R¯ 221 ), where 2 ¯ 21 = the quasi-triangular structure for H2 is given by (2 (2) (1) ¯ . The proof of this fact is exactly the same as for ordinary Hopf algebras. ¯ ⊗2 2 4. Weak Hopf algebras arising from dynamical twists We describe two methods of constructing weak Hopf algebras, which are finitedimensional modifications of constructions proposed in [Xu2] and [EV2]. It turns out that the resulting weak Hopf algebras are dual to each other. 4.1. Dynamical twists on Hopf algebras Dynamical twists first appeared in the work of Babelon [Ba] (see also [BBB]). Let U be a Hopf algebra, and let A = Map(T, k) be a commutative and cocommutative Hopf algebra of functions on a finite Abelian group T that is a Hopf subalgebra of U . Let Pµ , µ ∈ T, be the minimal idempotents in A. We fix this notation for the rest of the paper. Remark 4.1.1 In [EV2] the role of the group T is played by the dual space of a Cartan subalgebra of a simple Lie algebra. Definition 4.1.2 We say that an element x in U ⊗n , n ≥ 1, has zero weight if x commutes with n (a), for all a ∈ A, where n : A → A⊗n is the iterated comultiplication. Definition 4.1.3 An invertible, zero-weight U ⊗2 -valued function J (λ) on T is called a dynamical twist

148


for U if it satisfies the following functional equations: ( ⊗ id)J (λ) J λ + h(3) ⊗ 1 = (id ⊗ )J (λ)(1 ⊗ J (λ)), (ε ⊗ id)J (λ) = (id ⊗ε)J (λ) = 1.

(42) (43)

Here and in what follows, the notation λ + h(i) means that the argument λ is shifted by the weight of the ith component; for example, J (λ+h(3) ) = µ J (λ+µ)⊗Pµ ∈ U ⊗2 ⊗ A. Remark 4.1.4 If J (λ) is a dynamical twist for U and if x(λ) is an invertible, zero-weight U -valued function on T such that ε(x(λ)) ≡ 1, then J x (λ) = x(λ)−1 J (λ) x λ + h(2) ⊗ x(λ) is also a dynamical twist for U , gauge equivalent to J (λ). Note that for every fixed λ ∈ T the element J (λ) ∈ U ⊗ U does not have to be a twist for U in the sense of Drinfeld. It turns out that an appropriate object for which J defines a twisting is a certain weak Hopf algebra that we describe next. 4.2. Twisted weak Hopf algebra (Endk (A) ⊗ U )J (cf. [Xu2]) Observe that the simple algebra Endk (A) has a natural structure of a weak Hopf algebra, given as follows. Let {Eλµ }λ,µ∈T be a basis of Endk (A) such that Eλµ f (ν) = δµν f (λ), f ∈ A, λ, µ, ν ∈ T; (44) then the comultiplication, counit, and antipode of Endk (A) are given by Eλµ = Eλµ ⊗ Eλµ , ε Eλµ = 1, S Eλµ = Eµλ .

(45)

Remark 4.2.1 With the above operations, Endk (A) is a cocommutative weak Hopf algebra associated to a groupoid with |T| objects such that there is a unique morphism Eλµ between any two of them (cf. Example 2.1.3). Define the tensor product weak Hopf algebra H = Endk (A) ⊗ U . proposition 4.2.2 The elements 2= Eλλ+µ ⊗ Eλλ Pµ λµ

define a twist for H .

and

¯ = 2

λµ

Eλ+µλ ⊗ Eλλ Pµ

(46)


149

Proof ¯ = 2 (1), ¯ ¯ = (1). The relations between Clearly, we have 2 = (1)2, 2 and 22 ¯ 2, 2 and counit are straightforward. We also compute ( ⊗ id)(2)(2 ⊗ 1) = (id ⊗ )(2)(1 ⊗ 2) Eλλ+µ+ν ⊗ Eλλ+µ Pν ⊗ Eλλ Pµ , = λµν

¯ ⊗ 1)( ⊗ id)(2) ¯ = (1 ⊗ 2)(id ¯ ¯ (2 ⊗ )(2) Eλ+µ+νλ ⊗ Eλ+µλ Pν ⊗ Eλλ Pν . = λµν

One verifies the identities (28) and (29) of Definition 3.1.1 in a similar way. Thus, according to Proposition 3.1.2, H2 = (Endk (A) ⊗ U )2 becomes a weak Hopf algebra. It is noncommutative, noncocommutative, and not a Hopf algebra if |T| > 1. Following [Xu2], we show that it can be further twisted by means of a dynamical twist J (λ) on U . lemma 4.2.3 Let J (λ) ∈ A ⊗ U ⊗2 be an invertible, zero-weight U ⊗2 -valued function on T. Then the following identities hold in H ⊗3 : (47) ( ⊗ id)(2) J (λ) ⊗ 1 = J λ + h(3) ⊗ 1 ( ⊗ id)(2), (id ⊗ )(2) 1 ⊗ J (λ) = 1 ⊗ J (λ) (id ⊗ )(2), (48) −1 −1 (3) ¯ ¯ λ+h ⊗1 , (49) J (λ) ⊗ 1 ( ⊗ id)(2) = ( ⊗ id)(2) J −1 −1 ¯ = (id ⊗ )(2) ¯ 1 ⊗ J (λ) , 1 ⊗ J (λ) (id ⊗ )(2) (50) where J (λ) = J (1) (λ) ⊗ J (2) (λ) is embedded in H ⊗ H as Eλλ J (1) (λ) ⊗ Eλλ J (2) (λ). J (λ) =

(51)

λ

Proof We first check two identities, leaving the rest as an exercise to the reader. Using the ¯ we compute formulas for 2 and 2, Eλλ+µ Eνν J (1) (ν) ⊗ Eλλ+µ Eνν J (2) (ν) ⊗ Eλλ Pµ ( ⊗ id)(2) J (λ) ⊗ 1 = λµν

=

λµ

J (1) (λ + µ)Eλλ+µ ⊗ J (2) (λ + µ)Eλλ+µ ⊗ Eλλ Pµ

150


= J λ + h(3) ⊗ 1 ( ⊗ id)(2), Eλλ+µ ⊗ Eλλ Pµ (1) J (1) (λ) ⊗ Eλλ Pµ (2) J (2) (λ) (id ⊗ )(2) 1 ⊗ J (λ) = λµ

=

λµ

Eλλ+µ ⊗ J (1) (λ)Eλλ Pµ (1) ⊗ J (2) (λ)Eλλ Pµ (2)

= 1 ⊗ J (λ) (id ⊗ )(2), where we used the zero-weight property of J (λ). proposition 4.2.4 If J (λ) is a dynamical twist for U , then the pair (F (λ), F¯ (λ)), where F (λ) = J (λ)2

and

¯ −1 (λ), F¯ (λ) = 2J

defines a twist for H = Endk (A) ⊗ U . Proof The twist relations involving counit are obvious; for the rest we have, using identities from Lemma 4.2.3, ( ⊗ id)(F (λ)) F (λ) ⊗ 1 = ( ⊗ id)(J (λ)2)(J (λ)2 ⊗ 1) = ( ⊗ id)(J (λ))J λ + h(3) ( ⊗ id)(2)(2 ⊗ 1) = (id ⊗ )(J (λ)) 1 ⊗ J (λ) (id ⊗ )(2)(1 ⊗ 2) = (id ⊗ )(J (λ)2) 1 ⊗ J (λ)2 , −1 −1 ¯ ¯ F¯ (λ) ⊗ 1 ( ⊗ id) F¯ (λ) = 2J (λ) ⊗ 1 ( ⊗ id) 2J (λ) ¯ ⊗ 1)( ⊗ id)(2) ¯ = (2 −1 (3) λ+h ⊗ 1 ( ⊗ id) J −1 (λ) × J ¯ ¯ 1 ⊗ J −1 (λ) (id ⊗ ) J −1 (λ) = (1 ⊗ 2)(id ⊗ )(2) = 1 ⊗ F¯ (λ) (id ⊗ ) F¯ (λ) , and one checks other identities similarly. Thus, every dynamical twist J (λ) for a Hopf algebra U gives rise to a weak Hopf algebra HJ = HJ (λ)2 . Remark 4.2.5 According to Proposition 3.1.2, the antipode SJ of HJ is given by SJ (h) = v −1 S(h)v, for all h ∈ HJ , where S is the antipode of H and where


v=

151

S J (1) J (2) (λ)Pmu . λµ

Remark 4.2.6 If J x (λ) is a dynamical twist for U gauge equivalent to J (λ) by means of some x(λ) as in Remark 4.1.4, then x = x(λ)Eλλ is a gauge transformation of H establishing a ¯ x )−1 (λ)). ¯ −1 (λ)) and (J x (λ)2, 2(J gauge equivalence between the twists (J (λ)2, 2J 4.3. Dynamical quantum groups of [EV2] Suppose that dim U < ∞. We introduce a weak Hopf algebra DJ on a vector space D = Map(T × T, k) ⊗ U ∗ by adapting the construction of [EV2] to the finitedimensional case, and we show that this weak Hopf algebra is in fact dual to the twisted weak Hopf algebra HJ . Let {Ui } and {Li } be dual bases in U and U ∗ ; then the element L = i Ui ⊗ Li does not depend on the choice of the bases. Define the coalgebra structure on DJ as dual to the algebra structure of H : (id ⊗ )L = L12 L13 , ( (f )) λ1 , λ2 , µ1 , µ2 = f λ1 + µ1 , λ2 + µ2 ,

(52)

(id ⊗ε)(L) = 1, f (λ, λ), ε(f ) =

(54)

(53)

(55)

λ

for all f ∈ Map(T × T, k) and λ1 , λ2 , µ1 , µ2 ∈ T. The algebra structure of DJ is given as follows. Observe that the vector space ∗ U is bigraded:

U ∗ = ⊕U ∗ α 1 , α 2 , where U ∗ α 1 , α 2 = Homk Pα 1 U Pα 2 , k . Let us set f λ1 , λ2 g λ1 , λ2 = g λ1 , λ2 f λ1 , λ2 , f λ1 , λ2 Lα 1 α 2 = Lα 1 α 2 f λ1 + α 1 , λ2 + α 2 , −1 1 L23 L13 =: J12 λ ( ⊗ id)(L)J12 λ2 :,

(56) (57) (58)

where f (λ1 , λ2 ), g(λ1 , λ2 ) ∈ Map(T × T, k) and where Lα 1 α 2 ∈ U ∗ [α 1 , α 2 ]. Equation (58) is in U ⊗ DJ , and the sign :: (normal ordering) means that the matrix elements of L should be put on the right of the elements of J −1 (λ1 ), J (λ2 ). The unit of DJ is defined in an obvious way: 1 = (ε ⊗ id)L.

(59)

152


Let us consider two U -valued functions on T: ¯ K(λ) = m(id ⊗S)J −1 (λ),

K(λ) = m(S ⊗ id)J (λ − h).

(60)

¯ Note that K(λ) and K(λ) are inverses of each other and commute with A since J (λ) is of zero weight. Define the antipode of DJ by setting (Sf ) λ1 , λ2 = f λ2 , λ1 , (61) 2 −1 1 (id ⊗S)(L) =: K¯ λ S ⊗ id (L)K λ :, (62) for all f ∈ Map(T × T, k) and λ1 , λ2 ∈ T and by extending S to an algebra antihomomorphism. Next, we introduce a basis {Iλ1 λ2 } of delta-functions on T × T, that is, Iλ1 λ2 α 1 , α 2 = δλ1 α 1 δλ2 α 2 . (63) Define a duality form between DJ = Map(T × T, k) ⊗ U ∗ = A ⊗ A ⊗ U ∗ and H = Endk (A) ⊗ U by Iλ1 λ2 x, Eα 1 α 2 u = δλ1 α 1 δλ2 α 2 x, u, (64) for all x ∈ U ∗ and u ∈ U . Then, in terms of the homogeneous elements Lα 1 α 2 ∈ U ∗ [α 1 , α 2 ], the relations (58) and (62) defining the multiplication and antipode of DJ can be rewritten as Lα 1 α 2 Lβ 1 β 2 , Eν 1 ν 2 u = Lβ 1 β 2 ⊗ Lα 1 α 2 , J −1 ν 1 (u)J ν 2 , (65) 2 −1 1 S Lα 1 α 2 , Eν 1 ν 2 u = Lα 1 α 2 , K¯ ν S (u)K ν , (66) for all α 1 , α 2 , β 1 , β 2 , ν 1 , ν 2 ∈ T and u ∈ U . theorem 4.3.1 With the above operations, DJ becomes a weak Hopf algebra opposite to HJ∗ . Proof We need to show that the structure operations of DJ are obtained by transposing the corresponding operations of HJ . For the unit and counit, this is obvious. According to (65) and (57), the multiplication on the elements Iλ1 λ2 Lα 1 α 2 of DJ , where Lα 1 α 2 ∈ U ∗ [α 1 , α 2 ], can be found by evaluating the product Iλ1 λ2 Lα 1 ,α 2 Iµ1 µ2 Lβ 1 β 2 against the elements of HJ as follows:


153

Iµ1 µ2 Lβ 1 β 2 , Eν 1 ν 2 u = δλ1 µ1 −α 1 δλ2 µ2 −α 2 Iλ1 λ2 Lα 1 α 2 Lβ 1 β 2 , Eν 1 ν 2 u = δλ1 µ1 −α 1 δλ1 ν 1 δλ2 µ2 −α 2 δλ2 ν 2 Lβ 1 β 2 ⊗ Lα 1 α 2 , J −1 ν 1 (u)J ν 2 .

Iλ1 λ2 Lα 1 α 2

On the other hand, we have Iµ1 µ2 Lβ 1 β 2 ⊗ Iλ1 λ2 Lα 1 α 2 , J Eν 1 ν 2 u = Iµ1 µ2 Lβ 1 β 2 , Eν 1 +η1 ν 2 +η2 J¯(1) ν 1 u(1) J (1) ν 2 η1 ,η2

× Iλ1 λ2 Lα 1 α 2 , Eν 1 ν 2 Pη1 J¯(2) ν 1 u(2) J (2) ν 2 Pη2 = δλ1 ν 1 δλ2 ν 2 Lβ 1 β 2 , J¯(1) ν 1 u(1) J (1) ν 2 × Lα 1 α 2 , Pµ1 −ν 1 J¯(2) ν 1 u(2) J (2) ν 2 Pµ2 −ν 2 = δλ1 ν 1 δλ2 ν 2 δα 1 µ1 −ν 1 δα 2 µ2 −ν 2 × Lβ 1 β 2 ⊗ Lα 1 α 2 , J −1 ν 1 (u)J ν 2 , where J −1 = J¯(1) ⊗ J¯(2) . Comparing the last two relations, we conclude that the multiplication in DJ is opposite to the one induced by comultiplication in HJ . Finally, the antipode defined by formula (66) satisfies S Iλ1 λ2 Lα 1 α 2 , Eν 1 ν 2 u = Iλ2 +α 2 λ1 +α 1 S Lα 1 α 2 , Eν 1 ν 2 u = Iλ2 +α 2 λ1 +α 1 , Eν 1 ν 2 Lα 1 α 2 , K¯ ν 2 S −1 (u)K ν 1 = δλ2 +α 2 ν 1 δλ1 +α 1 ν 2 Lα 1 α 2 , K¯ ν 2 S −1 (u)K ν 1 , for all u ∈ U and λ1 , λ2 , α 1 , α 2 , ν 1 , ν 2 ∈ T, while the transpose of SJ−1 gives Iλ1 λ2 Lα 1 α 2 , S −1 Eν 1 ν 2 u Iλ1 λ2 , Eν 2 +µ2 ν 1 −µ1 Lα 1 α 2 , K¯ ν 2 P−µ2 S −1 (u)K ν 1 Pµ1 = µ1 µ2

= Lα 1 α 2 , Pν 2 −λ1 K¯ ν 2 S −1 (u)K ν 1 Pν 1 −λ2 = δα 1 ν 2 −λ1 δα 2 ν 1 −λ2 Lα 1 α 2 , K¯ ν 2 S −1 (u)K ν 1 ,

which completes the proof. 4.4. The case of quasi-triangular U When U has a quasi-triangular structure R , the weak Hopf algebra H = Endk (A)⊗U also has a quasi-triangular structure R (1) = λ (Eλλ ⊗ Eλλ )R . Thus, the twisted weak Hopf algebra HJ is also quasi-triangular by means of the matrix

154


¯ −1 (λ)R J (λ)2 = R (λ) = 2J 21

Eλλ+ν Pµ R J (1) (λ) ⊗ Eλ+µλ R J (2) (λ)Pν , (67)

λµν −1 = J21 (λ)R J (λ). By (40), R (λ) establishes where op ∗ homomorphism HJ → HJ , that is, a homomorphism ρ : DJ

R J (λ)

a weak Hopf algebra → HJ .

Remark 4.4.1 Note that dim(Pµ U ) = dim(U Pν ) = dim U/|T|, for all µ, ν ∈ T, since any finitedimensional Hopf algebra is free over its Hopf subalgebra (see [M]). In particular, dim U/|T| is an integer. proposition 4.4.2 The homomorphism ρ is an isomorphism if and only if the element J Rµν (λ) = Pµ R J (1) (λ) ⊗ R J (2) (λ)Pν ∈ U ⊗ U

(68)

has the maximal possible rank (equal to dim U/|T|) for all fixed λ, µ, ν ∈ T. Proof Since H and D are finite-dimensional, ρ is an isomorphism if and only if its image coincides with H , that is, if and only if J (2) J (1) ∗ Image(ρ) = span Eλλ+ν Pµ R (λ)φ R (λ)Pν λ, µ ∈ T, φ ∈ U =

ν

Eλλ+ν span Pµ R J (1) (λ)φ R J (2) (λ)Pν φ ∈ U ∗ = H,

λµν

which happens precisely when span Pµ R J (1) (λ)φ R J (2) (λ)Pν φ ∈ U ∗ = Pµ U,

(69)

for all λ, µ, ν ∈ T. Since a Hopf algebra U is a free Map(T, k)-module, we see that J (λ) has to be equal to dim(P U ) and, therefore, to dim U/|T|, by the rank of Rµν µ Remark 4.4.1.

5. Dynamical twists for Uq (g) at roots of 1 5.1. Construction of J (λ) Suppose that g is a simple Lie algebra of type A, D, or E, and suppose that q is a primitive @th root of unity in k, where @ ≥ 3 is odd and coprime with the determinant of the Cartan matrix (aij )ij =1,...,m of g. Let U = Uq (g) be the corresponding quantum group that is a finite-dimensional

155


Hopf algebra with generators Ei , Fi , Ki , where i = 1, . . . , m, and with the following relations (see [L]): Ki@ = 1,

Ei@ = 0,

Ki Ej = q aij Ej Ki ,

Ki Kj = Kj Ki ,

Ei Fj − Fj Ei = δij

Fi@ = 0,

Ki Fj = q −aij Fj Ki ,

Ki − Ki−1 , q − q −1

Fi Fj = Fj Fi if aij = 0, Ei Ej = Ej Ei , Ei2 Ej − q + q −1 Ei Ej Ei + Ej Ei2 = 0 if aij = −1, Fi2 Fj − q + q −1 Fi Fj Fi + Fj Fi2 = 0 if aij = −1, with the comultiplication, counit, and antipode given by (Ki ) = Ki ⊗ Ki , S(Ki ) = Ki−1 ,

(Ei ) = Ei ⊗ 1 + Ki ⊗ Ei ,

(Fi ) = Fi ⊗ Ki−1 + 1 ⊗ Fi , S(Ei ) = −Ki−1 Ei ,

ε(Ki ) = 1,

S(Fi ) = −Fi Ki ,

ε(Ei ) = ε(Fi ) = 0.

and

Denote by (·, ·) the bilinear form on Zm defined by the Cartan matrix (aij ). Let T ∼ = (Z/@Z)m be the Abelian group generated by Ki , i = 1, . . . , m. For any γ γ m-tuple of integers γ = (γ1 , . . . , γm ), we write Kγ = K1 1 · · · Kmm ∈ T. Denote by I the set of all integer m-tuples γ = (γ1 , . . . , γm ) with 0 ≤ γi < @, i = 1, . . . , m. Let + be the set of positive roots of g. For each α ∈ + , define Eα , Fα ∈ Uq (g) inductively by setting Eαi = Ei , Fαi = Fi for all simple roots αi , i = 1, . . . , m, and Eα+α = q −1 Eα Eα − Eα Eα

Fα+α = qFα Fα − Fα Fα ,

and

α, α ∈ + .

Let β1 , . . . , βN be the normal ordering of + , and, for every N-tuple of nonnegative integers a = (a1 , . . . , aN ), introduce the monomials Ea = Eβa11 · · · EβaNN

and

Fa = Fβa11 · · · FβaNN .

Then the universal R-matrix of Uq (g) is given by [R] and [T]: @−1 n N 1 −n(n+1)/2 1 − q 2 n n (β,γ ) Eβs ⊗ Fβs R= m q q Kβ ⊗ K γ , @ [n]q ! β,γ ∈I s=1 n=0 (70) where [n]q ! = [1]q [2]q · · · [n]q , [n]q = (q n − q −n )/(q − q −1 ), and 1 (β,γ ) D= m q Kβ ⊗ K γ (71) @ β,γ ∈I

is the “Cartan part” of R .

156


Note that the idempotents Pβ =

1 (β,λ) q Kλ |T| λ∈I

generate a commutative and cocommutative Hopf subalgebra A = Map(T, k) of U . Observe that U is Zm -graded in such a way that a monomial X in Ei , Fi , Ki ) are such belongs to U [β − β ], where β = (β1 , . . . , βm ) and β = (β1 , . . . , βm that Ei appears exactly βi times and Fi appears exactly βi times in X for each i. This induces a Z-grading of the algebra U with deg(Ei ) = 1,

deg(Fi ) = −1,

deg(Ki ) = 0,

i = 1, . . . , m,

(72)

and deg(XY ) = deg(X) + deg(Y ), for all X and Y . Of course, there are only finitely many nonzero components of U since it is finite-dimensional. Let U+ be the subalgebra of U generated by the elements Ei , Ki , i = 1, . . . , m, let U− be the subalgebra generated by Fi , Ki , i = 1, . . . , m, and let I± be the kernels of the projections from U± to the elements of zero degree. For arbitrary nonzero constants F1 , . . . , Fm , define a Hopf algebra automorphism F of U by setting F(Ei ) = Fi Ei , F(Ki ) = Ki ,

F(Fi ) = F−1 i Fi ,

and

for all i = 1, . . . , m. β

(73)

β

If for β = (β1 , . . . , βm ) ∈ Zm we write Fβ = F1 1 · · · Fmm , then F|U [β] = Fβ idU [β] .

(74)

Definition 5.1.1 We say that F = (F1 , . . . , Fm ) is generic if the spectrum of F does not contain @th roots of unity. For every Kλ ∈ T, we introduce the following linear operator on U ⊗ U : −1 AL . 2 (λ)X = Ad Kλ ◦ F ⊗ id R XD

(75)

proposition 5.1.2 For every generic F, there exists a unique element J (λ) ∈ 1 + I+ ⊗ I− that satisfies the following ABRR relation (see [ABRR], [ES2], [ESS]): AL 2 (λ)J (λ) = J (λ).

(76)

Proof Let us write X = j ≥1 X j , where X j is the sum of all terms having the Z-degree j


157

in the first component. Using the structure of the R-matrix of U , we can write (76) as a finite system of linear equations labeled by degree j ≥ 1: (77) Xj = Ad Kλ ◦ F ⊗ id DX j D−1 + · · · , where · · · stands for the terms involving X i for i < j . Thus, equation (76) can be solved recursively, starting with X0 = 1, and the solution is unique, provided that the operator (78) id − Ad Kλ ◦ F ⊗ id ◦ Ad D is invertible in Endk (U ⊗ U ). Let us show that this operator is diagonalizable, and let us compute its eigenvalues. For all Xα ∈ U [α], Xα ∈ U [α ], and Kβ ∈ T, we have Ad Kβ Xα = q (α ,β) Xα , Ad Kβ Xα = q (α,β) Xα , and therefore Ad D Xα ⊗ Xα = q (α,α ) K−α ⊗ K−α Xα ⊗ Xα , whence the eigenvalues of Ad D in U [α] ⊗ U [α ] are the numbers dχ χ = q (α,α ) χ K−α χ K−α , where χ and χ are characters of T. In particular, each dχ χ is an @th root of unity. Clearly, the eigenvalue of the operator (Ad Kλ ◦ F ⊗ id) in U [α] ⊗ U [α ] is Fα q (α,λ) . Putting these numbers together, we conclude that (78) is invertible in U ⊗U if and only if 1 − Fα q (α,λ+α ) dχ χ " = 0, for all λ, α, α , χ , χ , which is the case for generic F. Remark 5.1.3 If J (λ) ∈ 1 + I+ ⊗ I− is a solution of (76), then it has zero weight, by the uniqueness result of Proposition 5.1.2, since (Ad (Kβ ))J (λ) is also a solution of (76) for all Kβ ∈ T. Our goal is to show that the above element J (λ) gives rise to a dynamical twist for Uq (g). Similarly to (75), define the operator (79) A2R (λ)X = id ⊗ Ad K−λ ◦ F−1 R XD−1 , for all X ∈ U ⊗ U .

158


lemma 5.1.4 A2L (λ) and A2R (λ) commute. Proof For all X ∈ U ⊗ U , we have Ad Kλ ◦ F ⊗ id R Ad Kλ ◦ F ⊗ Ad K−λ ◦ F−1 R XD−1 D−1 = id ⊗ Ad K−λ ◦ F−1 R Ad Kλ ◦ F ⊗ Ad K−λ ◦ F−1 R XD−1 D−1 since (F ⊗ id)R = (id ⊗F−1 )R , whence A2L (λ) ◦ A2R (λ) = A2R (λ) ◦ A2L (λ). corollary 5.1.5 J (λ) is the unique solution of the system A2L (λ)X = X

and

A2R (λ)X = X,

(80)

with X ∈ 1 + I+ ⊗ I− . Proof We have A2L (λ) ◦ A2R (λ)J (λ) = A2R (λ) ◦ A2L (λ)J (λ) = A2R (λ)J (λ); hence A2R (λ)J (λ) = J (λ) by the uniqueness of the solution of (76). Following [ESS], consider the 3-component operators −1 A3L (λ)X = Ad Kλ ◦ F ⊗ id ⊗ id R13 R12 XD−1 12 D13 , −1 A3R (λ)X = id ⊗ id ⊗ Ad K−λ ◦ F−1 R13 R23 XD−1 13 D23 . lemma 5.1.6 The operators A3L (λ) and A3R (λ) commute. Proof This statement amounts to showing that Ad Kλ ◦ F ⊗ id ⊗ id R13 R12 Ad Kλ ◦ F ⊗ id ⊗ Ad K−λ ◦ F−1 R13 R23 = id ⊗ id ⊗ Ad K−λ ◦ F−1 R13 R23 × Ad Kλ ◦ F ⊗ id ⊗ Ad K−λ ◦ F−1 R13 R12 . ˆ = (Ad Kλ ◦ F ⊗ id)R = (id ⊗ Ad K−λ ◦ F−1 )R and If we denote R

(81) (82)

159


˜ = Ad Kλ ◦ F ⊗ Ad K−λ ◦ F−1 R , R then the above equality translates to ˆ 12 R ˜ 13 R ˆ 23 = R ˆ 23 R ˜ 13 R ˆ 12 , ˆ 13 R ˆ 13 R R which follows from the quantum Yang-Baxter equation after cancelling the first factor.

lemma 5.1.7 If there exists a solution X of the system A3L (λ)X = A3R (λ)X = X,

(83)

with X ∈ I+ ⊗ U ⊗ U + U ⊗ U ⊗ I− , then it is unique. Proof It is enough to show that such a solution X is unique for the equation A3L A3R X = X. Let us write X k,l , X =1+ k,l≥0;k+l>0

where X k,l is the sum of all elements having Z-degree k in the first component and having −l in the third one. Then the equation A3L (λ)A3R (λ)X = X transforms to the system X k,l = Ad Kλ ◦ F ⊗ id ⊗ Ad K−λ ◦ F−1 W X k,l W −1

+ terms depending on X k ,l , with k + l < k + l, for all k ≥ 0 and l ≥ 0 such that k + l > 0, where W = D12 D23 (D13 )2 . As in Proposition 5.1.2, one can check that the operator id − Ad Kλ ◦ F ⊗ id ⊗ Ad K−λ ◦ F−1 ◦ Ad W

(84)

is invertible for generic F; therefore, the above system can be solved recursively and the solution is unique. theorem 5.1.8 J (λ) satisfies the equations ( ⊗ id)J (λ) J λ + h(3) ⊗ 1 = (id ⊗ )J (λ) 1 ⊗ J λ − h(1) , (ε ⊗ id)J (λ) = (id ⊗ε)J (λ) = 1.

(85) (86)

Proof Let us denote the left-hand side of (85) by YL , and let us denote the right-hand side by YR . We show that both YL and YR are solutions of the system A3L (λ)X =

160


A3R (λ)X = X; then the result follows from Lemma 5.1.7. We have −1 A3R YL = id ⊗ id ⊗ Ad K−λ ◦ F−1 R13 R23 YL D−1 13 D23 = id ⊗ id ⊗ Ad K−λ ◦ F−1 ( ⊗ id)(R J (λ))J λ + h(3) ( ⊗ id)D−1 = ( ⊗ id) id ⊗ Ad K−λ ◦ F−1 R J (λ)D−1 J λ + h(3) ⊗ 1 = ( ⊗ id)J (λ) J λ + h(3) ⊗ 1 = YL , −1 A3L YR = Ad Kλ ◦ F ⊗ id ⊗ id R13 R12 YR D−1 12 D13 = Ad Kλ ◦ F ⊗ id ⊗ id (id ⊗ )(R J (λ)) 1 ⊗ J λ − h(1) (id ⊗ )D−1 = (id ⊗ ) Ad Kλ ◦ F ⊗ id R J (λ)D−1 1 ⊗ J λ − h(1) = (id ⊗ )J (λ) 1 ⊗ J λ − h(1) = YR , where we used that A2L J = A2R J = J , properties of the R-matrix, and that J commutes with Ad Kλ . To establish that A3L YL = YL , note that both A3L YL and YL are solutions of the equation A3R X = X and therefore are determined uniquely by their parts of zero degree in the third component. Thus it suffices to compare these parts: 3 0 AL YL = Ad Kλ ◦ F ⊗ id ⊗ id ◦ Ad D13 R12 J12 λ + h(3) D−1 12 Ad Kλ+β ◦ F ⊗ id R J λ + h(3) D−1 ⊗ Pβ = β = Ad Kλ+h(3) ◦ F ⊗ id R J λ + h(3) )D−1 = J λ + h(3) = YL0 , 3 0 AR YR = id ⊗ id ⊗ Ad K−λ ◦ F−1 ◦ Ad D13 R23 J23 λ − h(1) D−1 23 Pβ ⊗ id ⊗ Ad K−λ+β ◦ F−1 R J λ − h(1) D−1 = β = id ⊗ Ad K−(λ−h(1) ) ◦ F−1 R J λ − h(1) D−1 = J λ − h(1) = YR0 . The relations between J (λ) and ε are obvious. proposition 5.1.9 The element

J (λ) = J 2λ + h(1) + h(2)

(87)

is a dynamical twist in the sense of Definition 4.1.3. Proof We directly compute ( ⊗ id)J (λ) J λ + h(3) ⊗ 1 = ( ⊗ id)J 2λ + h(1) + h(2) + h(3) J 2λ + h(1) + h(2) + 2h(3) ⊗ 1

161


= (id ⊗ )J 2λ + h(1) + h(2) + h(3) 1 ⊗ J 2λ + h(2) + h(3) = (id ⊗ )J (λ) 1 ⊗ J (λ) . The identities (ε ⊗ id)J (λ) = 1 and (id ⊗ε)J (λ) = 1 are clear. Example 5.1.10 Let us give an explicit expression for the twists J (λ), J (λ) in the case g = sl(2). In this case, Uq (g) is generated by E, F, K with the standard relations. The element analogous to J (λ) for generic q was computed in [Ba] (see also [BBB]). If we switch to our conventions, this element takes the form ∞ n 2 n Fq 2λ −n(n+1)/2 1 − q n n . E ⊗F q J (λ) = 2λ+2ν K ⊗ K −1 [n]q ! 1 − Fq n=0 ν=1 It is obvious that the formula for q being a primitive @th root of unity is simply obtained by truncating this formula: @−1 n 2 n Fq 2λ −n(n+1)/2 1 − q n n . J (λ) = E ⊗F q [n]q ! 1 − Fq 2λ+2ν K ⊗ K −1 n=0

ν=1

Therefore, J (λ) =

@−1 n=0

q

−n(n+1)/2

1 − q2 [n]q !

n

En ⊗ F n

n

Fq 4λ K ⊗ K . 1 − Fq 4λ+2ν K 2 ⊗ 1 ν=1

Note that the term of this sum corresponding to n = 1 coincides with the one computed in Section 5.3. 5.2. Dynamical twists arising from generalized Belavin-Drinfeld triples (cf. [ESS]) Definition 5.2.1 A generalized Belavin-Drinfeld triple for a simple Lie algebra g consists of subsets 1 , 2 of the set = (α1 , . . . , αm ) of simple roots of g together with an inner product preserving bijection T : 1 → 2 . We say that T is nilpotent if for any i = 1, . . . , m there exists a positive integer di such that T di (αi ) " ∈ 1 . For nonnilpotent T , we define an order of T (denoted by n(T )) to be the least common multiple of the lengths of orbits of T . Let Q1 , Q2 , Q be the free Abelian groups generated by the root sets 1 , 2 , , respectively, and (88) L = λ ∈ Q | (λ, α) = (λ, T α) ∀α ∈ 1 . Then T extends to the isomorphism between Q1 and Q2 , and one can check as in

162


[ESS, Lemma 3.1] that Q1 ∩ L = {λ ∈ Q1 | T λ = λ} and that Q1 + L and L⊥ + L are finite index subgroups of Q. Let n1 = Q : Q1 + L and n2 = Q : L⊥ + L . (89) We assume that @ is coprime with both n1 and n2 , and we introduce a homomorphism Q1 + L → Q also denoted by T , letting T | Q1 = T ,

T |L = id .

Since T = id on Q1 ∩ L, it follows that T is well defined. Factorization by @Q yields an automorphism of T = Q/@Q: T : Q1 + L /@Q = Q/@Q −→ Q/@Q,

(90)

(91)

which preserves the inner product (·, ·) : T × T → Z/@Z and extends to the algebra homomorphisms T± : U± → U± defined by T+ (Kλ ) = KT λ , T+ (Ei ) = ET (i) T− (Fi ) = FT −1 (i)

T− (Kλ ) = KT −1 λ ,

(92)

if αi ∈ 1 and T+ (Ei ) = 0 otherwise,

(93)

if αi ∈ 2 and T− (Fi ) = 0 otherwise,

(94)

for all λ ∈ T and i = 1, . . . , m, where T (i) denotes the number such that T (αi ) = αT (i) and where T −1 (i) denotes the number such that T −1 (αi ) = αT −1 (i) . ⊥ Let TL and T⊥ L be the images of L and L in T, and let DL = q (β,γ ) Kβ ⊗ Kγ , DL⊥ = q (β,γ ) Kβ ⊗ Kγ . (95) Kβ ,Kγ ∈TL

Kβ ,Kγ ∈T⊥ L

Then T = TL ⊕ T⊥ L and D = DL DL⊥ . We define a modification of the operator A2L (λ) introduced in (75): A2L (λ)X = T+ ◦ Ad Kλ ◦ F ⊗ id R XD−1 (96) λ ∈ TL . L , One can show, using the same argument as in Proposition 5.1.3, that if Fi = FT (i) , for all αi ∈ 1 , and if the spectrum of F (in the case when T is not nilpotent) does not contain roots of unity of order n(T )@, then there exists a unique element JT (λ) ∈ Z + I+ ⊗ I− , where Z = ((id −T )−1 T ⊗ id)DL⊥ , satisfying the modified ABRR relation (cf. [ESS]), A2L (λ)JT (λ) = JT (λ), and commuting with Ad (Kλ ), for all λ ∈ TL . Next, if we define A2R (λ)X = id ⊗T− ◦ Ad Kλ ◦ F R XD−1 λ ∈ TL , L , then Lemma 5.1.4 and Corollary 5.1.5 are still valid because of the identity

(97)

(98)


163

T+ ⊗ id DL = id ⊗T− DL that follows from the inner product preserving property of T . Modifying definitions of A3L (λ) and A3R (λ) as in [ESS], it is possible to repeat the proofs of Lemmas 5.1.6 and 5.1.7 and of Theorem 5.1.8, showing that JT (λ) satisfies (85) and hence that JT (λ), constructed as in Proposition 5.1.9, is a dynamical twist for U . 5.3. Nondegeneracy of the twisted R-matrix and self-duality We show that for the dynamical twist J (λ) for a quantum group Uq (g) the corresponding weak Hopf algebras described in Sections 4.2 and 4.3 are isomorphic, that is, that the twisted R-matrix R (λ) = Eλλ+ν Pµ R J (1) (λ) ⊗ Eλ+µλ R J (2) (λ)Pν , (99) λµν −1 (λ)R J (λ), establishes an isomorphism between weak Hopf where R J (λ) = J21 algebras DJ = Map(T × T, k) ⊗ Uq (g)∗ and HJ = (End(A) ⊗ Uq (g))J (λ)2 = ∗ op DJ .

proposition 5.3.1 ∗ → H op defined by φ → (id ⊗φ)R (λ) be the homomorphism of weak Let ρ : HJ J

Hopf algebras given by the R-matrix R (λ) of HJ . Then the elements Eλµ , Ki±1 , Ei , Fi , λ, µ ∈ T, i = 1, . . . , m, belong to Image(ρ).

Proof It follows from the explicit formula (70) for the universal R-matrix R of Uq (g), defining equation (76) of J (λ), and the expression (67) for R (λ) that R (λ) = Ra,b (λ), where Ra,b (λ) =

a,b

Eλλ+ν ⊗ Eλ+µλ Fa Eb ⊗ Ea Fb (Pµ ⊗ 1)Ca,b (λ)(1 ⊗ Pν ),

λ,µ,ν

the “coefficients” Ca,b (λ) are (kT)⊗2 -valued functions on T, and a, b run over Ntuples of nonnegative integers. Note that the terms of R (λ) occurring in R0,0 (λ) are linearly independent from the rest, and so are those occurring in Rδi ,0 (λ) and R0,δi (λ), where δi is the N-tuple with 1 in the position corresponding to the single root αi , i = 1, . . . , m, and zeros elsewhere. Hence, the subspaces

164


∗ Va,b = (id ⊗φ)Rab (λ) | φ ∈ HJ ,

where (a, b) = (0, 0), (0, δi ), or (δi , 0),

belong to the image of ρ. In all of the three above cases, we show that Ca,b (λ) is invertible in (kT)⊗2 (and, therefore, (Pµ ⊗ 1)Ca,b (λ)(1 ⊗ Pν ) are nonzero elements for all µ, ν) and that the generators of HJ lie in the algebra generated by the above subspaces Va,b . Clearly, C0,0 (λ) = D is invertible, whence V0,0 is spanned by the elements Eλλ+ν Pµ ; that is, Eλµ ∈ Image(ρ), for all λ, µ, and Ki ∈ Image(ρ) for i = 1, . . . , m. Next, C0,δi (λ) is the coefficient with Ei ⊗ Fi in RJ (λ). To determine it, note that by (70) and (76) we have Ei ⊗ Fi bi (λ) + · · · , bi (λ) ∈ (kT)⊗2 , J (λ) = 1 + i

R = 1 + q −1 − q (Ei ⊗ Fi ) + · · · D, i

where · · · stand for the terms of degree greater than 1 in the first component. We use the recursive relation (77) to find bi (λ), i = 1, . . . , m, Ei ⊗ Fi bi (λ) = Ad Kλ ◦ F ⊗ id ◦ Ad D Ei ⊗ Fi bi (λ) + q −1 − q Ei ⊗ Fi = Fi q (λ,αi ) Ei ⊗ Fi bi (λ)q 2 Ki ⊗ Ki−1 + q −1 − q , from which we obtain

Fi q (λ,αi ) q −1 − q bi (λ) = , 1 − Fi q (λ,αi )+2 Ki ⊗ Ki−1

and, consequently,

C0,δi (λ) = q 2 Ki ⊗ Ki−1 bi 2λ + h(1) + h(2) D + q −1 − q D −1 Fi q 2(λ,αi )+2 Ki2 ⊗ 1 +1 D = q −q 1 − Fi q 2(λ,αi )+2 Ki2 ⊗ 1 −1 q −q D , = 1 − Fi q 2(λ,αi )+2 Ki2 ⊗ 1

which is invertible for all generic F. It follows that V0,δi is spanned by the elements Eλλ+ν Pµ Ei , whence Ei ∈ Image(ρ). Finally, −1 (λ) = 1 − Fi ⊗ Ei bi (λ)21 + · · · ; J21 therefore, Cδi ,0 (λ) = −bi (2λ+h(1) +h(2) )21 D is invertible and Fi ∈ Image(ρ). As a corollary, we obtain the following theorem.


165

theorem 5.3.2 The R-matrix R (λ) defines an isomorphism between weak Hopf algebras HJ and ∗ op DJ ∼ = HJ . Proof ∗ → H op contains all the generWe saw in Proposition 5.3.1 that the image of ρ : HJ J

ators of the algebra H , and therefore Image(ρ) = H . Since H is finite-dimensional, ρ is an isomorphism.

5.4. Nondegeneracy of the twisted R-matrix in the case when T is an automorphism of the Dynkin diagram of g We extend the result of Section 5.3 to the case when 1 = 2 = and T " = id. The dynamical twist JT (λ) constructed in Section 5.2 then has the form ⊗2 JT (λ) = Z + (100) Ei ⊗ Fj bij (λ) + · · · , bij (λ) ∈ kT , ij

where bij (λ) ≡ 0 if αi and αj belong to different orbits of T and where · · · stands for the terms having the Z-degree greater than 1 in the first component. Similarly, one has ⊗2 (101) Fi ⊗ Ej b˜ij (λ) + · · · , b˜ij (λ) ∈ kT . (JT )−1 (λ) = Z −1 + 21

21

ij

As in Proposition 5.3.1, it is possible to find the coefficients bij (λ) explicitly: −1 q − q q (λ,αi )+(λ+αj ,sij ) Ksij ⊗ Ks−1 Z Fn−k i ij bij (λ) = , −1 1 − Fni q (λ+αj ,s) Ks ⊗ Ks where n is the length of the corresponding orbit, k is the unique number such that T k (αi ) = αj (0 ≤ k < n), sij = T (αj ) + · · · + T n−k−1 (αj ), and s = αi + · · · + T n−1 (αi ) is the orbit sum. Note that, since Fi is independent from i within an orbit and λ ∈ TL , the denominator in the right-hand side of the above formula does not depend on i and j . The dynamical R-matrix R JT (λ) of Uq (g) has a form −1 R JT (λ) = DZZ21 + Fi ⊗ Ej b˜ij (λ)DZ +

ij

ij

−1 Ei Z21

⊗ Fj q aij Kj ⊗ Ki−1 bij (λ) + δij q −1 − q Z D + · · · ,

where the listed terms are linearly independent from · · · in each component.

166


Formula (67) gives an expression for the R-matrix R (λ) of the weak Hopf algebra HJT in terms of R JT (λ), and it is easy to see that the image of R (λ) contains the matrix units Eµµ , µ, µ ∈ L, and elements Ki , i = 1, . . . , m, generating T. Showing that it also contains generators Ei (resp., Fi ) amounts to proving that the matrices Aνη (λ) (resp., Bνη (λ)), where (102) Aνη (λ)ij = Pη ⊗ Pν q aij Kj ⊗ Ki−1 bij (λ) + δij q −1 − q Z , ˜ (103) Bνη (λ)ij = Pη ⊗ Pν bij (λ), and Pη , Pν are minimal idempotents in kT, are invertible for all η, ν, λ. Using the formula for bij (λ), one can show that these matrices are equivalent (up to permuting and multiplying the rows and columns by nonzero constants) to the matrix  1 1 ... 1 F ˜ 1 . . . 1   . . . ..  , . . . . . . . ˜ ˜ F F ... 1 

(104)

˜ = F−n q (λ+αj ,s) (Pη ⊗ Pν )(Ks ⊗ Ks )Z is a multiple of Pη ⊗ Pν independent where F i ˜ n−1 , so it is invertible from i and j . The determinant of this matrix is equal to (1 − F) n(T )@ " = 1. when Fi Thus, Theorem 5.3.2 extends to the case when T is an automorphism of the Dynkin diagram of g. theorem 5.4.1 For every generalized Belavin-Drinfeld triple ( , , T ), the R-matrix R (λ) defines ∗ op an isomorphism between weak Hopf algebras HJT and DJT ∼ = HJT . Acknowledgments. We thank Ping Xu and Philippe Roche for useful discussions. Nikshych thanks Massachusetts Institute of Technology for the warm hospitality during his visit.

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Etingof Department of Mathematics, Massachusetts Institute of Technology, Building 2, Room 2-165, Cambridge, Massachusetts 02139-4307, USA; [email protected] Nikshych Department of Mathematics, University of California at Los Angeles, 405 Hilgard Avenue, Los Angeles, California 90095-1555, USA; [email protected]

ON THE abc CONJECTURE, II C. L. STEWART and KUNRUI YU

Abstract Let x, y, and z be coprime positive integers with x + y = z. In this paper we give upper bounds for z in terms of the greatest square-free factor of xyz. 1. Introduction Let x, y, and z be positive integers, and define G = G(x, y, z) by p. G = G(x, y, z) = p|xyz p a prime

Thus G is the greatest square-free factor of xyz. In 1985, D. Masser [6] proposed a refinement of a conjecture that had been recently formulated by J. Oesterlé. Masser conjectured that for each positive real number ε there is a positive number c(ε), which depends on ε only, such that, for all positive integers x, y, and z with x+y =z

and

(x, y, z) = 1,

(1)

we have z < c(ε)G1+ε .

(2)

The conjecture is now known as the abc conjecture. It captures in a succinct way the idea that the additive and the multiplicative structure of the integers should be independent, and, accordingly, it has profound consequences (cf. [1], [3], [4], [5], [11], [13]). In 1986, C. Stewart and R. Tijdeman [11] obtained an upper bound for z as a function of G. They proved that there exists an effectively computable positive constant c1 such that, for all positive integers x, y, and z satisfying (1), z < exp c1 G15 . (3) The proof depends on a p-adic estimate for linear forms in the logarithms of algebraic DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 1, Received 1 September 1999. Revision received 22 August 2000. 2000 Mathematics Subject Classification. Primary 11J86; Secondary 11D75, 11J61. Stewart’s work supported in part by grant number A3528 from the Natural Sciences and Engineering Research Council of Canada. Yu’s work supported by Hong Kong Research Grants Council Competitive Earmarked Research Grants grant number HKUST 633/95p.

169

170

STEWART AND YU

numbers due to A. van der Poorten [8]. In 1991 Stewart and K. Yu [12] strengthened (3). They proved, by combining a p-adic estimate for linear forms in the logarithms of algebraic numbers due to Yu [15] with an earlier Archimedean estimate due to M. Waldschmidt [14], that there exists an effectively computable positive constant c2 such that, for all positive integers x, y, and z, with z > 2, satisfying (1), z < exp G2/3+c2 / log log G . (4) Our purpose in this paper is to present two further improvements on (4). theorem 1 There exists an effectively computable positive number c such that, for all positive integers x, y, and z with x + y = z and (x, y, z) = 1, (5) z < exp c G1/3 (log G)3 . The key new ingredient in our proof of Theorem 1 is an estimate of Yu [17] for p-adic linear forms in the logarithms of algebraic numbers which has a better dependence on the number of terms in the linear form than previous p-adic estimates; for the Archimedean case, see the earlier work of E. Matveev [7]. We employ this estimate in order to control the p-adic order of x, y, and z at the small primes p dividing x, y, and z. Next we combine the contributions from the small primes in order to reduce the number of terms in our linear forms. We conclude with a further application of estimates for linear forms in the logarithms of algebraic numbers in a fashion similar to [12]. Here we appeal to a p-adic estimate due to Yu [16] and its earlier Archimedean counterpart due to A. Baker and G. Wüstholz [2]. An examination of our proof reveals that the impediment to a further refinement of Theorem 1 is not the dependence on the number of terms in the estimates for linear forms in logarithms but instead is the dependence on the parameter p in the p-adic estimates. This fact is highlighted by our next result, which shows that if the greatest prime factor of one of x, y, and z is small relative to G, then the estimate for z from Theorem 1 can be improved. In particular, let px , py , and pz denote the greatest prime factors of x, y, and z, respectively, with the convention that the greatest prime factor of 1 is 1. Put p = min px , py , pz . Denote the ith iterate of the logarithmic function by logi , so that log1 t = log t and logi t = log(logi−1 t) for i = 2, 3, . . . . theorem 2 There exists an effectively computable positive number c such that, for all positive integers x, y, and z with x + y = z, (x, y, z) = 1, and z > 2,

ON THE abc CONJECTURE, II

171

∗ z < exp p Gc log3 G / log2 G ,

(6)

where G∗ = max(G, 16). Thus, for each ε > 0 there exists a number c3 (ε), which is effectively computable in terms of ε, such that for all positive integers x, y, and z with x + y = z and (x, y, z) = 1, z < exp c3 (ε)p Gε . Observe that

1/3 ≤ G1/3 , p ≤ px py pz

and so we immediately obtain z < exp c3 (ε)G1/3+ε , a slightly weaker version of Theorem 1. On the other hand, if p is appreciably smaller than G1/3 , (6) gives a sharper upper bound than (5). For any integer n with n > 1, let P (n) denote the greatest prime factor of n. As an illustration of the above remark, we deduce from Theorem 2 that there exists an effectively computable positive number c4 such that if x and y are coprime positive integers with x < y and y ≥ 16, then P = P (xy(x + y)) > c4

log2 y log3 y , log∗4 y

(7)

where log∗4 y = max{log4 y, 1}, a result that improves upon the lower bound of c5 log2 y obtained by T. Shorey, van der Poorten, Tijdeman, and A. Schinzel [10]. Suppose y ≥ 16, and suppose (1) holds. Then by (6) there exists an effectively computable positive number c6 such that log log y < log P + c6

log G log3 G∗ , log2 G

(8)

where G denotes the greatest square-free factor of xy(x + y). Plainly, p ≤ exp log p , G= p|xy(x+y)

p≤P

and so, by the prime number theorem, we have log G < c7 P ,

(9)

where c7 is an effectively computable positive number (cf. J. Rosser and L. Schoenfeld [9, Theorem 9]). Estimate (7) follows directly from (8) and (9).

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2. Preliminary lemmas For any algebraic number α, let h0 (α) denote its absolute logarithmic Weil height, so that   d log max 1, α (j )  , h0 (α) = d −1 log |ad | + j =1

where the minimal polynomial for α over Z is ad x d + · · · + a1 x + a0 = ad x − α (1) · · · x − α (d) . Let p be a prime number, and put q=

2

if p > 2,

3

if p = 2

and

  if p ≡ 3 (mod 4),  −1 √ α0 = i = −1 if p ≡ 1 (mod 4),   e2π i/3 if p = 2.

Put K = Q(α0 ). Let p be a prime ideal of OK , the ring of algebraic integers in K. Suppose that p lies above p with ramification index ep and residue class degree fp . Note that 1 if p > 2, ep = 1 and fp = (10) 2 if p = 2 (see [15, appendix] for the case p ≡ 1 mod 4 and the case p = 2). For nonzero α in K, we write ordp α for the exponent to which p divides the fractional ideal generated by α in K. Let α1 , . . . , αn be nonzero elements of K, and put hj = max h0 (αj ), log p , for j = 1, . . . , n. Let b1 , . . . , bn be rational integers with absolute values at most B (≥ 3). Next put = α1b1 · · · αnbn − 1. lemma 1 1/q 1/q Suppose that K α0 , . . . , αn : K = q n+1 , ordp αj = 0 for j = 1, . . . , n, and = 0. Then there exists an effectively computable positive number c8 such that ordp
exp −(c10 n)2n log B j =1

Proof This is a consequence of [2, Theorem]. lemma 4 Let α1 , . . . , αn be prime numbers with α1 < α2 < · · · < αn . Let q = 2 and α0 ∈ {−1, i} or q = 3 and α0 = e2π i/3 , and put K = Q(α0 ). Then 1/q 1/q 1/q : K = q n+1 K α0 , α1 , . . . , αn except when q = 2, α0 = i, and α1 = 2, and in this case 1/2 1/2 1/2 : K = 2n+1 . K α0 , (1 + i)1/2 , α2 , . . . , αn Proof This follows from [12, Lemma 3] except when q = 2 and α0 = −1. In this case, the proof of [12, Lemma 3] again applies.

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lemma 5 Let 2 = p1 , p2 , . . . be the sequence of prime numbers in increasing order. There is an effectively computable positive constant c11 such that, for every positive integer r, we have r pj r + 3 r+3 > . log pj c11 j =1

Proof This is [12, Lemma 4]. 3. Proofs of Theorems 1 and 2 Note that Theorem 1 holds for z = 2 trivially. Henceforth, let x, y, and z be positive integers with x + y = z, (x, y, z) = 1, and z > 2. We may suppose, without loss of generality, that x ≤ y. Since z > 2, we see that x < y < z and G ≥ 6. Note that max{ordp x, ordp y, ordp z} ≤ Put ˜ = max G

G , 16 px py pz

log z . log 2

(11)

and r = ω(xyz), the number of distinct prime factors of xyz. Let c12 , c13 , . . . denote effectively computable positive constants. By Lemma 5, r ˜ > r , G c12 and so r < c13

˜ log G . ˜ log2 G

(12)

Put m = r − 2 if x = 1 (whence px = 1) and m = r − 3 otherwise. Notice that, by the arithmetic-geometric mean inequality,  m ˜ m log G 1  log p ≤  log p  ≤ , (13) m m p|xyz p|xyz p∈{px ,py ,pz }

p∈{px ,py ,pz }

provided that m is positive. From (12) and (13), we deduce that ˜ log3 G ˜ log G , log p < exp c14 ˜ log2 G p|xyz p∈{px ,py ,pz }

(14)


175

with the usual convention that the empty product is 1. It also follows from (12) that ˜ ˜ log3 G log G 2r (log r) < exp c15 . (15) ˜ log2 G We now estimate ordp (xyz) for each prime p that divides xyz and satisfies p < e(log r) . 2

(16)

First suppose that p | z. Since (x, y, z) = 1 and x + y = z, we have (x, y) = (x, z) = (y, z) = 1. Thus, for each prime p that divides z, 4 z x x ordp z = ordp = ordp − 1 ≤ ordp −1 . (17) −y −y y Let α1 < · · · < αn be the primes that divide either x or y except in the case when p ≡ 1 mod 4 and α1 = 2. In that case, we take α1 = 1 + i in place of α1 = 2. Note that 24 = (1 + i)8 . Write 4 x = α1b1 · · · αnbn , y with b1 , . . . , bn rational integers. We choose q, α0 , K = Q(α0 ), as in §2. Let p be a prime ideal of OK lying above p. Since p | z and (x, z) = (y, z) = 1, we have ordp αj = 0, for j = 1, . . . , n. Let B denote the maximum of the absolute values of the bj ’s. Then, by (11), log z . (18) log B ≤ log 8 log 2 Put 4 x − 1 = α1b1 · · · αnbn − 1, = y and note that ordp = ordp . (19) By (16), (18), and Lemmas 1 and 4, n ordp < pc16 (log r)2n log log z

log l.

(20)

l|xy l a prime

It follows from (12), (14)–(17), (19), and (20) that ˜ ˜ log3 G log G ordp z < exp c17 log(2px ) log py log log z. ˜ log2 G In a similar fashion, we deduce, for p satisfying (16), that

(21)

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and that

˜ ˜ log3 G log G ordp y < exp c18 log(2px ) log pz log log z ˜ log2 G

(22)

˜ log3 G ˜ log G log py log pz log log z. ordp x < exp c19 ˜ log2 G

(23)

We now define R, S, and T by l ordl x , S= R= l|x,l=px 2

l<e(log r)

l ordl y ,

T =

l|y,l=py 2

l|z,l=pz

l ordl z , 2

l<e(log r)

l<e(log r)

where l runs through primes. Observe that   R h0 < r(log r)2 max  max ordl x, max ordl y ; l|x l|y −S 2

l<e(log r)

2

l<e(log r)

hence, by (12), (22), and (23), ˜ log3 G ˜ log G R h0 log max(px , py ) log pz log log z. < exp c20 ˜ −S log2 G Similarly, we find that ˜ ˜ log3 G T log G log max(px , pz ) log py log log z < exp c21 h0 ˜ R log2 G and that ˜ ˜ log3 G T log G log max(py , pz ) log(2px ) log log z. < exp c22 h0 ˜ S log2 G

(24)

(25)

(26)

We are now in a position to estimate ordp (xyz) for each prime p that divides xyz. In particular, we no longer require condition (16). We first estimate ordp z for p | z. As in (17), we have x −1 . (27) ordp z = ordp −y Put α1 = R/(−S), and let x = α1 α2b2 · · · αnbn , −y where α2 , . . . , αn are distinct prime numbers and where b2 , . . . , bn are nonzero rational integers. Since α 2 · · · αn | G and

αj ≥ e(log r) , 2

for j = 2, . . . , n with αj ∈ {px , py }, we deduce that


177

n−3≤ Next observe that

˜ log G . (log r)2

˜ log G n2n < exp c23 ˜ log2 G

(28)

(29)

˜ 1/2 , then the result is immediate on noting that n is at since if r is at most (log G) ˜ 1/2 , then (29) follows from (28). most r, while if r exceeds (log G) Let B = max(|b2 |, . . . , |bn |, 3), and note that (18) follows from (11) as before. Put x − 1 = α1 α2b2 · · · αnbn − 1, = −y and observe that (19) holds. Next put ˜ log3 G ˜ p log G Wp = exp c24 log max(l, p) · (log log z)2 . (30) ˜ (log p)3 log2 G l∈{px ,py ,pz }

We now apply Lemma 2, taking into account (12), (14), (24), (27), and (29), to conclude that (31) (ordp z) log p < Wp log max(px , py ). Similarly, if p | y, then, by considering ordp z/x − 1 and applying Lemma 2, we find that (32) (ordp y) log p < Wp log max(px , pz ); while if p | x, then, by considering ordp z/y − 1 and applying Lemma 2, we obtain

Certainly,

(ordp x) log p < Wp log max(py , pz ).

(33)

(ordp z) log p ≤ r max(ordp z) log p . log z =

(34)

p|z

p|z

Put L = log max(px , py ) · log max(px , pz ) · log max(py , pz ). By (12), (30), (31), and (34), we find that ˜ pz ˜ log3 G log z log G < exp c L. 25 ˜ (log log z)2 (log pz )2 log2 G

(35)

Since y > z/2 and z ≥ 3, 1 log z. (36) 4 Plainly, (34) holds with z replaced by y, and so from (12), (30), (32), and (36), we deduce that log y > log z − log 2 >

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˜ py ˜ log3 G log z log G < exp c26 L. ˜ (log log z)2 (log py )2 log2 G

(37)

Next, either x ≥ y 1/2 , in which case log x ≥

1 1 log y > log z, 2 8

or x < y 1/2 , in which case 1/2 x+y x 1 2 1 log = log 1 + < log 1 + 1/2 < 1/2 < . y y z y y

(38)

(39)

In the former case, we may appeal to (34) with z replaced by x, and so from (12), (30), (33), and (38), ˜ log3 G ˜ px log G log z < exp c L. (40) 27 2 ˜ (log log z)2 (log(2p log2 G x )) In the latter case, put α1 = T /S, and write z = α1 α2b2 · · · αnbn , y where α2 , . . . , αn are distinct prime numbers and where b2 , . . . , bn are nonzero rational integers. Then z x+y = log = log α1 + b2 log α2 + · · · + bn log αn . 0 < log y y Note that we again have (29). Thus, on applying Lemma 3 and appealing to (11), (12), (14), (26), and (29), we obtain ˜ ˜ log3 G log G x+y log max(py , pz ) > − exp c28 log log ˜ y (41) log2 G · log(2px ) log py log pz (log log z)2 . On comparing (39) and (41), we see that ˜ log3 G ˜ log z log G < exp c29 log max(py , pz ) log(2px ) log py log pz . ˜ (log log z)2 log2 G Therefore, in both cases x ≥ y 1/2 and x < y 1/2 , (40) holds. Suppose that {px , py , pz } = {p , p , p }, and suppose that p < p < p .


179

It follows from (35), (37), and (40) that ˜ ˜ log3 G p log G log z < exp c log p (log p )2 . 30 2 ˜ (log log z) (log(2p ))2 log2 G

(42)

Just as for (14), we have p|xyz

log G log3 G∗ , log p < exp c31 log2 G

whence, by (42), p log G log3 G∗ log z < exp c . 32 log2 G (log log z)2 (log(2p ))2

(43)

Theorem 2 follows directly from (43).

To prove Theorem 1, we remark that from (35), (37), and (40), 3 log z (log log z)2 ˜ log3 G ˜ px py pz log G < exp c33 × (log p )3 (log p )6 . 2 ˜ (log(2p ) log p log p ) log2 G x y z

Note that we may assume that

p > G1/4

since otherwise Theorem 1 follows from (43). Thus we have 3 log z ˜ x py pz (log G)3 , < c34 Gp (log log z)2 and so log z < c35 G1/3 (log G)3 , as required. Acknowledgment. C. L. Stewart would like to thank the Universität Basel for its hospitality during the period when this paper was written.

References [1]

A. BAKER, “Logarithmic forms and the abc-conjecture” in Number Theory (Eger,

Hungary, 1996), de Gruyter, Berlin, 1998, 37–44. MR 99e:11101

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[2]

¨ A. BAKER and G. WUSTHOLZ , Logarithmic forms and group varieties, J. Reine

[3]

N. D. ELKIES, ABC implies Mordell, Internat. Math. Res. Notices 1991, 99–109.

[4]

S. LANG, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc.

[5]

M. LANGEVIN, Cas d’égalité pour le théoréme de Mason et applications de la

Angew. Math. 442 (1993), 19–62. MR 94i:11050 MR 93d:11064 (N.S.) 23 (1990), 37–75. MR 90k:11032

[6]

[7]

[8]

[9] [10]

[11] [12] [13] [14] [15] [16] [17]

conjecture (abc), C. R. Acad. Sci. Paris Sér. I. Math. 317 (1993), 441–444. MR 94j:11027 D. W. MASSER, Conjecture in “Open Problems” section in Proceedings of the Symposium on Analytic Number Theory, ed. W. W. L. Chen, Imperial College, London, 1985, 25. E. M. MATVEEV, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers, Izv. Math. 62, no. 4 (1998), 723–772. MR 2000g:11071 A. J. VAN DER POORTEN, “Linear forms in logarithms in the p-adic case” in Transcendence Theory: Advances and Applications (Cambridge, 1976), ed. A. Baker and D. W. Masser, Academic Press, London, 1977, 29–57. MR 58 #16544 J. B. ROSSER and L. SCHOENFELD, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 25 #1139 T. N. SHOREY, A. J. VAN DER POORTEN, R. TIJDEMAN and A. SCHINZEL, “Applications of the Gelfond-Baker method to Diophantine equations” in Transcendence Theory: Advances and Applications (Cambridge, 1976), ed. A. Baker and D. W. Masser, Academic Press, London, 1977, 59–77. MR 57#12383 C. L. STEWART and R. TIJDEMAN, On the Oesterlé-Masser conjecture, Monatsh. Math. 102 (1986), 251–257. MR 87k:11077 C. L. STEWART and K. YU, On the abc conjecture, Math. Ann. 291 (1991), 225–230. MR 92k:11037 P. VOJTA, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239, Springer, Berlin, 1987. MR 91k:11049 M. WALDSCHMIDT, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257–283. MR 82h:10049 K. YU, Linear forms in p-adic logarithms, II, Compositio Math. 74 (1990), 15–113. MR 91h:11065a , p-adic logarithmic forms and group varieties, I, J. Reine Angew. Math. 502 (1998), 29–92. MR 99g:11092 , p-adic logarithmic forms and group varieties, II, Acta Arith. 89 (1999), 337–378. MR 2000e:11097


181

Stewart Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; [email protected] Yu Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; [email protected]

REALIZATION OF LEVEL ONE REPRESENTATIONS g ) AT A ROOT OF UNITY OF Uq ( VYJAYANTHI CHARI and NAIHUAN JING

Abstract Using vertex operators, we construct explicitly Lusztig’s Z[q, q −1 ]-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the character is given by the Weyl-Kac character formula. 0. Introduction In [L1] and [L3] G. Lusztig proved that a quantum Kac-Moody algebra U defined over Q(q) admits an A = Z[q, q −1 ]-lattice UA and that any irreducible highest weight integrable representation V of U admits a corresponding A -lattice, say, VA . This allows us to specialize q to a nonzero complex number ζ , and we let Uζ , Wζ denote the corresponding objects. If ζ is not a root of unity, Lusztig proved that Wζ is irreducible and its character is the same as that of the corresponding classical representation. On the other hand, when ζ is a primitive lth root of unity, the situation is more interesting, even for finite-dimensional Kac-Moody algebras. In that case, Wζ is not always irreducible: a sufficient condition for irreducibility [APW] is that the highest weight of V should be “small” in the sense that ( , α) < l for all positive roots α. The corresponding question for infinite-dimensional Kac-Moody algebras at roots of unity is open, and in this paper we answer it in the case of level one representations of quantum affine algebras of ADE type. Note that the condition ( , α) < l never holds in this case; nevertheless, we find that Wζ is irreducible, provided that l is coprime to the Coxeter number of the underlying finite-dimensional Lie algebra. The level one representations of an affine Lie algebra of ADE type can be explicitly constructed in the tensor product of a symmetric algebra and a twisted group algebra (see [FK], [S]). Essentially, these representations are built from the canonical representation of an infinite-dimensional Heisenberg algebra. Later, in [FJ], this construction was extended to the case of the basic representations of the quantum affine DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 1, Received 30 September 1999. Revision received 6 June 2000. 2000 Mathematics Subject Classification. Primary 17B37; Secondary 17B67. Jing’s work partially supported by National Science Foundation grant number DMS-9970493.

183

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CHARI AND JING

algebras of ADE type. Again, the representations are built from the representation of a suitable quantum Heisenberg algebra. In this paper we identify the natural lattice VA of the level one representation explicitly as the tensor product of the lattice of Schur functions with the obvious A -lattice in the twisted group algebra (see also [J2]). We also describe the action of the divided powers of the Chevalley (and Drinfeld) generators on an A -basis of VA , and this allows us to realize the level one irreducible representations Wζ explicitly and to prove that they are irreducible. Our methods also apply to the study of highest weight representations of affine Lie algebras in characteristic p, and the results obtained are similar. The details will appear elsewhere. 1. The algebras U and UA Throughout this paper g denotes a simply-laced, finite-dimensional complex simple Lie algebra, and (aij )i,j ∈I , I = {1, . . . , n}, denotes its Cartan matrix. Let (aij )i,j ∈Iˆ , Iˆ = I ∪ {0}, be the extended Cartan matrix of g, and let g be the corresponding affine

Lie algebra. Let R (resp., R + ) denote a set of roots (resp., positive roots) of g, and let αi (i ∈ I ) be a set of simple roots. Let Q be the root lattice of g, let P be the weight lattice, and let ωi ∈ P (i ∈ I ) be the fundamental weights of g. For ω ∈ P , η ∈ Q, define an integer |ω| · |η| by extending bilinearly the assignment |ωi | · |αj | = δij . Notice that |αi | · |αj | = aij . Let θ be the highest root of g. Let q be an indeterminate, let Q(q) be the field of rational functions in q with rational coefficients, and let A = Z[q, q −1 ] be the ring of Laurent polynomials with integer coefficients. For r, m ∈ N, m ≥ r, define q m − q −m [m]! m [m] = . , [m]! = [m][m − 1] · · · [2][1], = −1 r [r]![m − r]! q −q Then mr ∈ A for all m ≥ r ≥ 0. proposition 1.1 There is a Hopf algebra U over Q(q) which is generated as an algebra by elements Eαi , Fαi , Ki±1 (i ∈ Iˆ), and D ±1 with the following defining relations: DD

−1

Ki Ki−1 = Ki−1 Ki = 1, =D

−1

D = 1,

DEαi D

Ki Kj = Kj Ki , −1

Ki Eαj Ki−1 = q aij Eαj ,

=q

δi0

Eαi ,

DFαi D −1 = q −δi0 Fαi ,

Ki Fαj Ki−1 = q −aij Fαj ,

Ki − Ki−1 Eαi , Fαj = δij , q − q −1

Ki D = DKi ,

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LEVEL ONE REPRESENTATIONS AT ROOTS OF UNITY 1−aij

(−1)

r

r=0 1−aij

(−1)

r=0

r

1−aij −r 1 − aij r =0 Eαi Eαj Eαi r

if i = j,

1−aij −r 1 − aij r F α i Fα j F α i =0 r

if i = j.

The comultiplication of U is given on generators by ! Eαi = Eαi ⊗ 1 + Ki ⊗ Eαi , ! Fαi = Fαi ⊗ Ki−1 + 1 ⊗ Fαi , !(D) = D ⊗ D, ! Ki = Ki ⊗ Ki , for i ∈ Iˆ. Let U+ (resp., U− , U0 ) be the Q(q)-subalgebras of U generated by the Eαi (resp., Fαi , Ki±1 , and D ±1 ) for i ∈ Iˆ. The following result is well known (see [L3], for instance). lemma 1.1 U∼ = U− ⊗ U0 ⊗ U+ as Q(q)-vector spaces. For i ∈ Iˆ, r ≥ 0, set Eα(r) = i

Eαr i

[r]!

.

(r)

The elements Fαi are defined similarly. Let UA denote the A -subalgebra of U (r) (r) ± are defined in generated by Eαi , Fαi , Ki±1 (i ∈ Iˆ), and D ±1 . The subalgebras UA the obvious way. For i ∈ Iˆ, r ≥ 1, m ∈ Z, define elements r Ki q m−s+1 − Ki−1 q −m+s−1 Ki , m , = r q s − q −s s=1 r Dq m−s+1 − D −1 q −m+s−1 D, m . = r q s − q −s s=1

K ,m ±1 0 be the A -subalgebra of U ±1 i , and D,m , Let UA A generated by Ki , D , r r i ∈ Iˆ, r ≥ 1, and m ∈ Z. The following is well known (see [L3]). lemma 1.2 − 0 + UA UA . We have UA ∼ = UA We also need another realization of U, due to [Dr], [B], and [J1].

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theorem 1 ± There is an isomorphism of Q(q)-algebras from U to the algebra with generators xi,r (i ∈ I , r ∈ Z), Ki±1 (i ∈ I ), hi,r (i ∈ I , r ∈ Z\{0}), and C ±1 , and the following defining relations: C ±1

are central,

Ki Ki−1 = Ki−1 Ki = 1,

Ki Kj = Kj Ki ,

Ki hj,r = hj,r Ki ,

DD −1 = D −1 D = 1, Dhj,r D

−1

r

= q hj,r ,

CC −1 = C −1 C = 1,

± ± Ki xj,r Ki−1 = q ±aij xj,r ,

DKi = Ki D, ±1 −1 ± Dxj,r D = q r xj,r , r −r C −C

1 , hi,r , hj,s = δr,−s [raij ] r q − q −1 1 ± ± hi,±r , xj,s , r > 0, = ± [raij ] xj,s±r r 1 ± ± hi,∓r , xj,s , r > 0, = ± C r [raij ] xj,s±r r

± ± ± ± ± ± ± ± xi,r+1 xj,s − q ±aij xj,s xi,r+1 = q ±aij xi,r xj,s+1 − xj,s+1 xi,r , + − C −s ψi,r+s − C −r ψi,r+s + − , xj,s , = δi,j xi,r q − q −1 m ± ± k m (−1) · · · xi,r x± x± · · · xi,r = 0, x± π(k) j,s i,rπ(k+1) π(m) k i,rπ(1)

if i = j,

π ∈)m k=0

for all sequences of integers r1 , . . . , rm , where m = 1 − aij , )m is the symmetric ± are determined by equating powers of u in the formal group on m letters, and the ψi,r power series

∞ ∞ ± ψi,±r u±r = Ki±1 exp ± q − q −1 hi,±s u±s . r=0

s=1

Following [CP, Section 3], we define elements Pk,i and P˜k,i via the generating functions

∞

∞ hi,±k h˜ i,±k ± k uk = exp − uk , Pi± (u) = Pi,k u = exp − (1.1) [k] k k≥0 k=1 k=1

∞

∞ hi,±k h˜ i,±k ± ± k k k ˜ ˜ Pi (u) = (1.2) Pi,k u = exp u = exp u , [k] k k≥0

k=1

k=1

where h˜ i,k = khi,k /[k]. Notice that these formulas are exactly those that relate the elementary symmetric functions (resp., complete symmetric functions) to the power

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LEVEL ONE REPRESENTATIONS AT ROOTS OF UNITY

sum symmetric functions [M]. For a vertex operator approach to this and to Schur functions, see [J2]. The following result was proved in [CP, Section 5]. lemma 1.3 For all i ∈ I , k ∈ Z, k ≥ 0, we have ± , Pi,k

± P˜i,k ∈ UA .

˜ A (resp., U ˜ ± ) be the A -subalgebra generated by (x ± )(r) , r, n ∈ Z, r ≥ 0, Let U i,n A 0 . The following result is proved in [BCP, i ∈ I (resp., n ∈ Z, ±n ≥ 0), and by UA Section 2]. proposition 1.2 We have

Ã, UA = U

± ˜±. UA ⊂U A

Finally, let U(0) (resp., UA (0)) be the Q(q)-subalgebra of U (resp., the A -subalgebra ± , i ∈ I , k ∈ Z, k > 0), of UA ) generated by the elements hi,n , i ∈ I , n ∈ Z (resp., Pi,k ± ±1 ± and C . The subalgebras U (0) and UA (0) are defined in the obvious way. proposition 1.3 (i) The algebra U(0) is defined by the relations 1 C n − C −n hi,n , hj,m = δm,−n [naij ] , n q − q −1

C ±1 hi,n = hi,n C ±1 , for all i, j ∈ I and m, n ∈ Z. In particular, U± (0) is commutative. (ii) For all i ∈ I , k > 0, we have ± =− Pi,k

k 1 ˜ ± hi,m Pi,k−m , k m=0

k 1 ˜ ± ± hi,m P˜i,k−m P˜i,k = . k m=0

± ∈ UA (0), and as Q(q)-spaces we have In particular, h˜ i,k , P˜i,k

U(0) ∼ = Q(q) ⊗A UA (0),

± U± (0) ∼ (0). = Q(q) ⊗A UA

± ± ± (iii) Monomials in Pi,n (resp., P˜i,n ), i ∈ I , n > 0, form a basis for UA (0).

Proof Part (i) is a consequence of the Poincaré-Birkhoff-Witt (PBW) theorem for U proved

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± in [B]. Parts (ii) and (iii) follow from the definition of the elements P˜i,k (see [BCP] for details).

2. The level one representations of U and UA We begin this section by recalling the natural irreducible representation of U(0), and we construct a natural UA (0)-lattice in this representation. We then recall the definition of the highest weight representations Vq ( ) of U and of the lattice VA ( ) of UA (see [L3]). Finally, we recall the explicit construction of the level one representations given in [FJ], and we state and prove the main theorem of the paper. Consider the left ideal I in U(0) generated by C ±1 − q ±1 and U+ (0). Then U(0)/I is a left U(0)-module through left multiplication. It is easy to see that as Q(q)-vector spaces we have U− (0) ∼ = U(0)/I . Thus U− (0) acquires the structure of a left U(0)-module, and we let π : U(0) → End(U− (0)) be this representation. Then elements of U− (0) act by left multiplication, and it is easy to see that, for n > 0, i ∈ I , π(hi,n ) is the derivation of U− (0) obtained by extending the assignment [naij ][n] . π hi,n hj,−m = δn,m n proposition 2.1 (i) The mapping π is an irreducible representation of U(0). (ii) For i, j ∈ I , we have + − (v) = fi,j (u, v)P − (v), (u) · P π P i j j + − π Pi (u) · Pj (v) = fi,j (u, v)Pj− (v), − (v) = P − (v), fi,j (u, v)π Pi+ (u) · P j j where the power series fi,j is defined by fi,j (u, v) = 1

if aij = 0,

= (1 − uv) if aij = −1, −1 = (1 − quv)−1 1 − q −1 uv

if aij = 2.

− − (0) ⊂ UA (0). (iii) We also have π(UA (0))UA

Proof Part (i) is well known. For part (ii), notice that the relations in Proposition 1.3 imply that

189


+ − (u) · P − (v) π P j

∞

∞ π(h˜ i,k ) h˜ j,−k k k = exp u exp v k k k=1 k=1

∞

∞

∞ [kaij ] h˜ j,−k π(h˜ i,k ) = exp uk v k exp v k exp uk · 1 k[k] k k k=1

k=1

− (v). = fi,j (u, v)P j

k=1

The second equality above follows by using the Campbell-Hausdorff formula. The calculation of fi,j (u, v) is now straightforward. The other equations are proved similarly. Part (iii) follows immediately from part (ii). ˆ

By a weight, we mean a pair (µ, n) ∈ Z|I | × Z. If n = 0, we denote the pair (µ, 0) as µ. A representation W of U is said to be of type 1 if

Wµ,n , W = (µ,n)

q µi w, D

· w = q n w}. If Wµ,n = 0, then Wµ,n is where Wµ,n = {w ∈ W | Ki · w = called the weight space of W with weight (µ, n). Throughout this paper we consider only type-1 representations. Writing θ = i∈I di αi , we define the level of (µ, n) to be i∈I di µi + µ0 . For i ∈ Iˆ, let i be the Iˆ-tuple with 1 in the ith place and zero elsewhere. Given a weight = i ni i , ni ≥ 0, let Vq ( ) be the irreducible highest weight Umodule with highest weight , and let v be the highest weight vector. Thus, Vq ( ) is generated by v with relations Eαi · v = 0,

Ki · v = q ni v ,

D · v = v ,

Fαnii +1 · v = 0,

for i ∈ Iˆ. Clearly, Vq ( ) is of type 1. We say that Vq ( ) has level one if has level one. Set VA ( ) = UA · v . − · v . The following result is now an immediate By Lemma 1.2 we see that VA = UA consequence of Proposition 1.2.

lemma 2.1 We have

˜ A · v = U ˜ − · v . VA ( ) = U A

The following result is due to Lusztig [L3].

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proposition 2.2 VA ( ) is a UA -submodule of Vq ( ) such that Vq ( ) ∼ = VA ( ) ⊗A Q(q). Further, VA ( ) = and

µ,n

VA ( ) ∩ Vq ( )µ,n ,

dimA VA ( ) ∩ Vq ( )µ,n = dimQ(q) Vq ( )µ,n .

We turn now to the realization of the level one representations of U. In fact, we restrict ourselves to constructing the basic representation of gˆ , that is, the representation corresponding to 0 . The construction of the other level one representations is identical except that one adjoins v i to the twisted group algebra (see [FJ]). Fix a bilinear map 2 : Q × Q → {±1} such that for all i ∈ I , α, β, γ ∈ Q, we have 2(α, 0) = 2(0, α) = 1, 2(α, β)2(α + β, γ ) = 2(α, β + γ )2(β, γ ), 2(α, β)2(β, α) = (−1)|α|·|β| . Let Q(q)[Q] be the twisted group algebra over Q(q) of the weight lattice of g. Thus, Q(q)[Q] is the algebra generated by elements eη , η ∈ Q, subject to the relation eη · eη = 2 η, η eη+η . Set

Vq = U− (0) ⊗ Q(q)[Q].

Let zi∂ : Vq → Vq [z, z−1 ] be the Q(q)-linear map defined by extending z∂i v ⊗ eη = v ⊗ eη z|η|·|αi | , v ∈ U− (0), η ∈ Q. ± on Vq by means of the following generating series: Define operators Xi,n − + −1 −1 α ∂ + −n−1 iz i = (z) π P q z Xi,n z , e Xi+ (z) = π P i i n∈Z

Xi− (z)

+ −1 −α −∂ − −n−1 z = π Pi− (qz) π P Xi,n z . e iz i = i n∈Z

The following result was proved in [FJ]. theorem 2 ± ± → Xi,n , hi,n → π(hi,n ) ⊗ 1 defines a representation of U on The assignment xi,n Vq . In fact, as U-modules we have


191

Vq ( 0 ) ∼ = Vq . Further, for all i ∈ I , u ∈ U− (0), η ∈ Q, we have Ki u ⊗ eη = q |η|·|αi | u ⊗ eη , C u ⊗ eη = u ⊗ eη . The highest weight vector in Vq ( 0 ) maps to 1 ⊗ 1 under this isomorphism. Let VA be the image of VA ( 0 ) under this isomorphism. Clearly, VA is a UA submodule of Vq ( 0 ), and Vq ( 0 ) ∼ = Q(q) ⊗A VA . Set − L = UA (0) ⊗ A [Q], where A [Q] is the A -span in Q(q)[Q] of the elements eη . It follows from Proposition 1.3 that Vq ∼ = Q(q) ⊗A L . We now state our main result. theorem 3 The lattice L is preserved by UA and L ∼ = VA

as UA -modules. Remark. The case g = sl2 was studied in [J2]. In that paper the author worked over A = Z[q 1/2 , q −1/2 ], proved that the corresponding lattice L was preserved by UA , and gave the action of the divided powers of the Drinfeld generators on the Schur functions. The rest of the section is devoted to proving Theorem 3. We begin with the following two lemmas, which are easily deduced from the definition of Xi± (z) and Proposition 2.1. lemma 2.2 Let i ∈ I , η ∈ Q and m = |η| · |αi |. Then + xi,−m−1 1 ⊗ eη = 2 αi , η ⊗ eαi +η , − 1 ⊗ eη = 2(−αi , η) ⊗ e−αi +η . xi,m−1 lemma 2.3 Let r, l ∈ Z, r, l ≥ 0, and let i, j1 , j2 , . . . , jl ∈ I . We have

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− (w1 )P − (w2 ) · · · P − (wl ) ⊗ eη Xi+ (z1 )Xi+ (z2 ) · · · Xi+ (zr ) P j1 j2 jl =2 ·

r k=1

r−k+|η|·|αi |

zk

−1 fi,i (qzk )−1 , zs

1≤k<s≤r

fi,js (qzk )−1 , ws

1≤k≤r,1≤s≤l

− (z2 ) · · · P − (zr )P − (w1 )P − (w2 ) · · · P − (wl ) ⊗ erαi +η − (z1 )P ×P i i i j1 j2 jl |η|·|αi | −2 = 2 · z 1 z2 · · · zr zk − q zs (zk − zs )

×

1≤k<s≤r

fi,js (qzk )−1 , ws

1≤k≤r,1≤s≤l − (z2 ) · · · P − (zr )P − (w1 )P − (w2 ) · · · P − (wl ) ⊗ erαi +η , − (z1 )P ×P i i i j1 j2 jl

where 2 = 2(rαi , η)

r−1

k=1 2(αi , kαi ).

Let Sr be the symmetric group on r letters, and, for σ ∈ Sr , let l(σ ) be the length of σ . lemma 2.4 We have

(−1)l(σ )

zσ (k) − q −2 zσ (s) = q −r(r−1)/2 [r]! zk − z s .

k<s

σ ∈Sr

k<s

Proof Observe that the left-hand side of the equation is an antisymmetric polynomial in z1 , z2 , . . . , zr and hence is divisible by the right-hand side. Hence, by comparing degrees, we can write (−1)l(σ ) zσ (k) − q −2 zσ (s) = C(q) zk − zs . σ ∈Sr

k<s

k<s

But it is easy to see that the coefficient of z1r−1 z2r−2 · · · zr−1 on the left-hand side is q −2l(σ ) = q −r(r−1)/2 [r]!, σ ∈Sr

thus proving the proposition. lemma 2.5 (i) Let δ = (δ1 , δ2 , . . . , δr ) ∈ Zr be the r-tuple (r − 1, r − 2, . . . , 1, 0). We have   µρ(1) µρ(2) 2 µ z1 z2 · · · zr ρ(r) , zj − z k =  aµ j 0, we know by Lemma 3.1 that the matrix [A]ζ k is invertible. Let bij (k) denote the inverse of this matrix. For i ∈ I , k ∈ Z, k > 0, set bij (k)h˜ j,k . hi,k = j ∈I

Clearly, hi,k satisfies

hi,k , h˜ j,m = δk,−m δi,j .

The second formula in (i) is now clear from Proposition 1.3. To prove (ii), assume that W is a submodule of Uζ− (0), and let 0 = w ∈ W . By Proposition 1.3 we can choose i ∈ I , k ∈ Z, k > 0, such that w=

n r=0

− r wr , P˜i,k

− − , j = i, 1 ≤ l ≤ k, and P˜i,l , 1 ≤ l < k. where wr is a polynomial in the elements P˜j,l i,k Applying h to w repeatedly, we find that wn ∈ W . Repeating the argument, we find that 1 ∈ W , thus proving the proposition.

We now turn to the representations of Uζ . Given =

i∈Iˆ ni i ,

ni ≥ 0, set

Wζ ( ) = VA ( ) ⊗A Cζ . It follows from Proposition 2.2 that Wζ ( ) is a representation of Uζ . Again, for v ∈ VA ( ), we let v ∈ Wζ ( ) be the element v ⊗ 1. Clearly, Uζ+ · v = 0 and

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Wζ ( ) = Uζ · v . Set Wζ ( )µ,n = (VA ( ) ∩ Vq ( )µ,n ) ⊗A Cζ . Then one knows from [L3] and [L2] that

Wζ ( )µ,n , dimC Wζ ( ) = dimQ(q) Vq ( ), Wζ ( ) = µ,n

and w ∈ Wζ ( )µ,n if and only if µi

Ki · v = ζ v,

D · v = ζ n v,

Ki , 0 · v = µi v, l D, 0 · v = n v, l

where µi = µi + lµi , 0 ≤ µi < l, and n and n are defined similarly. Turning now to the level one basic representation, we see from Theorem 3 that Wζ ( 0 ) ∼ = Uζ− (0) ⊗ Cζ [Q]. The main result of this section is the following theorem. theorem 4 Wζ ( 0 ) is irreducible. Proof Set W = Wζ ( 0 ), and let 0 = W be a submodule of W . Then W contains a nonzero vector w ∈ Wµ,n such that Uζ+ · w = 0. It is clear from Theorem 2 that w must be of the form wµ ⊗ eµ for some wµ ∈ Uζ− (0) with Uζ+ (0) · wµ = 0. By Proposition 3.1 we see that this forces wµ = 1 and hence that 1 ⊗ eµ ∈ W . Proposition 2.2 now shows that 1⊗eν for all ν ∈ Q and hence finally that W = W .

References [APW]

H. H. ANDERSEN, P. POLO, and K. WEN, Representations of quantum algebras, Invent.

[B]

J. BECK, Braid group action and quantum affine algebras, Comm. Math. Phys. 165

[BCP]

J. BECK, V. CHARI, and A. PRESSLEY, An algebraic characterization of the affine

Math. 104 (1991), 1–59. MR 92e:17011 (1994), 555–568. MR 95i:17011


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canonical basis, Duke Math. J. 99 (1999), 455–487. MR 2000g:17013 [BFJ]

J. BECK, I. B. FRENKEL, and N. JING, Canonical basis and Macdonald polynomials,

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R. BORCHERDS, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat.

[Bo]

´ ements de mathématique, fasc. 34: Groupes et algèbres de Lie, N. BOURBAKI, El´

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V. CHARI and A. PRESSLEY, Quantum affine algebras at roots of unity, Represent.

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I. DAMIANI, A basis of type Poincaré-Birkhoff-Witt for the quantum algebra of sl(2),

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V. G. DRINFELD, A new realization of Yangians and of quantum affine algebras, Soviet

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I. B. FRENKEL and N. JING, Vertex representations of quantum affine algebras, Proc.

[FK]

I. B. FRENKEL and V. G. KAC, Basic representations of affine Lie algebras and dual

[J1]

N. JING, “On Drinfeld realization of quantum affine algebras” in The Monster and Lie

Adv. Math. 140 (1998), 95–127. MR 2000k:33041 Acad. Sci. U.S.A. 83 (1986), 3068–3071. MR 87m:17033 chapitres 4–6, Actualités Sci. Indust. 1337, Hermann, Paris, 1968. MR 39:1590 Theory 1 (1997), 280–328, http://www.ams.org/ert MR 98e:17018 J. Algebra 161 (1993), 291–310. MR 94k:17021 Math. Dokl. 36, no. 2 (1988), 212–216. MR 88j:17020 Nat. Acad. Sci. U.S.A. 85 (1988), 9373–9377. MR 90e:17028 resonance models, Invent. Math. 62 (1980), 23–66. MR 84f:17004

[J2] [L1] [L2] [L3] [M] [S]

Algebras (Columbus, Ohio, 1996), Ohio State Univ. Math. Res. Inst. Publ. 7, de Gruyter, Berlin, 1998, 195–206. MR 99j:17021 , Symmetric polynomials and Uq ( sl2 ), Represent. Theory 4 (2000) 46–63, http://www.ams.org/ert MR CMP 1740180 G. LUSZTIG, Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237–249. MR 89k:17029 , Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89–113. MR 91j:17018 , Introduction to Quantum Groups, Progr. Math. 110, Birkhäuser, Boston, 1993. MR 94m:17016 I. G. MACDONALD, Symmetric Functions and Hall Polynomials, 2d ed., Oxford Math. Monogr., Clarendon, New York, 1995. MR 96h:05207 G. SEGAL, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981) 301–342. MR 82k:22004

Chari Department of Mathematics, University of California, Riverside, California 92521–0135, USA; [email protected] Jing Department of Mathematics, North Carolina State University, Raleigh, North California 27695–8205, USA; [email protected]

ON THE MONODROMY OF COMPLEX POLYNOMIALS ´ NEMETHI ´ ALEXANDRU DIMCA and ANDRAS

Abstract Consider a polynomial function f : Cn → C with generic fiber F . Let Bf be the bifurcation set of f; hence f induces a smooth locally trivial fibration over C \ Bf . Then, for any integer q ≥ 0 and any coefficient ring R, there is an associated monodromy representation ρ(f )q : π1 C \ Bf , pt −→ Aut H˜ q (F, R) in (reduced) homology. Going around a circle in C large enough to contain all of the bifurcation set gives rise to the monodromy operators at infinity, which we denote by M∞ (f )q . We show that these monodromy operators at infinity and a certain natural direct sum decomposition of the homology of F in terms of vanishing cycles determine the monodromy representation. The role played by this decomposition is crucial since there are examples of polynomials C2 → C having distinct complex monodromy representations but whose monodromy operators at infinity have the same Jordan normal form. 1. Introduction Consider a polynomial function f : Cn → C with generic fiber F . Let Bf be the bifurcation set of f ; hence f induces a smooth locally trivial fibration over C \ Bf (see Section 2.1). Then, for any integer q ≥ 0 and any coefficient ring R, there is an associated monodromy representation ρ(f )q : π1 C \ Bf , pt −→ Aut H˜ q (F, R) in (reduced) homology as well as a monodromy representation in (reduced) cohomology ρ(f )q : π1 C \ Bf , pt −→ Aut H˜ q (F, R) . DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 2, Received 10 December 1999. Revision received 21 August 2000. 2000 Mathematics Subject Classification. Primary 32S40; Secondary 32S20, 32S30.

199

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´ DIMCA AND NEMETHI

Going around a circle in C large enough to contain all of the bifurcation set gives rise to the monodromy operators at infinity, which we denote by M∞ (f )q and M∞ (f )q , respectively. Our main result is the following theorem. theorem A The following statements are equivalent: (i) M∞ (f )q = Id; (ii) the homology representation ρ(f )q is trivial. If we assume that R is a field or that R = Z and the integral (co)homology of F is torsion free, then (i) and (ii) are equivalent to the following: (iii) M∞ (f )q = Id; (iv) the cohomology representation ρ(f )q is trivial. This result was first obtained in the case n = 2 by A. Dimca [D2] using specific properties of this low-dimensional situation plus some Hodge theory. The same case n = 2 was also proved by E. Artal-Bartolo and P. Cassou-Noguès [AC] using the splice diagrams. Here we prove it in full generality; actually we obtain a much finer result, namely, that the monodromy at infinity and a certain natural direct sum decomposition of the homology of F in terms of vanishing cycles determine the monodromy representation (see Theorem 3.1). The role played by this decomposition is crucial since there are examples of polynomials C2 → C having distinct complex monodromy representations but whose monodromy operators at infinity have the same Jordan normal form (see the polynomials fB and fC in [D2]). Theorem 3.1 is very unexpected since in many questions it is much easier to deal with the monodromy at infinity than with the whole of the monodromy representation. For instance, to compute the monodromy at infinity under the Thom-Sebastiani construction is not so difficult (see [N1], [N2]), but to treat the corresponding monodromy representations is much harder (see [DN]). On the other hand, a lot of information on the monodromy at infinity is known, at least in some special cases (see, e.g., [GN]), and this can hopefully be used in conjunction with Theorem 3.1 to get information on the monodromy representation itself. As a by-product of Theorem A we give conditions under which the generic fiber of the polynomial f is acyclic in a certain degree q, generalizing in this way (see [D2, Corollary 2]). Of course, in the case n > 2, the Abhyankar-Moh-Suzuki Theorem is missing, so we cannot deduce from the acyclicity of the fibers that the polynomial f is equivalent to a linear form. (See M. Zaidenberg [Z] for examples of such polynomials.)

ON THE MONODROMY OF COMPLEX POLYNOMIALS

201

corollary B Assume that R = Z, and assume that q is a fixed nonnegative integer. Then the following are equivalent: 2n−q−1 (Fb , R) = 0 for all b ∈ Bf ; (i) M∞ (f )q = Id and H˜ c ˜ (ii) Hq (F, R) = 0 and Ker(M∞ (f )q−1 − Id) = 0. Here we use the notation Fb = f −1 (b) for any b ∈ C. For an m-dimensional affine j variety Y , the reduced cohomology with compact supports H˜ c (Y, R) is defined to j be just Hc (Y, R) when j = 2m and H˜ c2m (Y, R) = Coker(p ∗ : Hc2m (Cm , R) → Hc2m (Y, R)), where p : Y → Cm is any finite morphism. To simplify the notation, from now on we omit the coefficient ring R when it plays no special role. To give more geometric meaning to the cohomology with compact supports, note that if Sb is the singular part of the fiber Fb and d = dim(Sb ), then the natural morphism j j Hc Fb \ Sb −→ Hc (Fb ) is surjective for j = 2d + 1 and an isomorphism for j > 2d + 1. In particular, for R a field, (i) dim Hc2n−2 (Fb ) is equal to the number of irreducible components of Fb as in [NN2]; (ii) dim Hc2n−3 (Fb ) = dim H1 (Fb \ Sb ) if d < n − 2.

2. A Picard formula for some monodromy operators 2.1. Let f : Cn → C be a polynomial function. It is well known that there is a (minimal) finite bifurcation set Bf in C such that f is a C ∞ -locally trivial fibration over C \ Bf . If c0 ∈ C is not in Bf , then F = f −1 (c0 ) is called the generic fiber of f . The fundamental group π1 (C \ Bf , c0 ) is a free group with t = |Bf | generators. Note that in this paper we use the following convention: the product loop a · b is obtained by going first along the loop b and then along the loop a. Set Bf = {b1 , . . . , bt }. Then a convenient way to construct a set of free generators γi for this group is the following. First fix some paths pi : [0, 1] → C, i = 1, . . . , t, such that we have (i) pi (0) = c0 , pi (1) = bi ; (ii) (image pi ) ∩ (image pj ) = {c0 } for any i = j ; (iii) the paths p1 , . . . , pt leave c0 in counterclockwise order.


202

Definition 2.2 A family of paths (pi )i=1,...,t with properties (i), (ii), and (iii) is called a star with base point c0 and end points b1 , . . . , bt . Once we fix a star as above, we have automatically a base (γi )i=1,...,t of π1 (C \ Bf , c0 ). Indeed, consider small discs Di centered at bi such that (image pi ) ∩ Di is a topological radius in Di . Set (image pi ) ∩ ∂Di = ci = pi (ti ). Then p˜ i : [0, ti ] → C, p˜ i (t) = pi (t), connects c0 to ci . Let ∂Di be the loop with both end points ci and going once around bi counterclockwise. Then γi = (p˜ i )−1 ◦ ∂Di ◦ p˜ i for i = 1, . . . , t are loops based at c0 , and their homotopy classes (still denoted by γi ) generate the group π1 (C \ Bf , c0 ). We emphasize that the ordered set of homotopy classes γ1 , . . . , γt is canonically associated with the star p1 , . . . , pt and that it is independent of any other choices. 2.3. The sum decomposition of H˜ q (F ) associated with a star Consider as above a polynomial f : Cn → C with a bifurcation set Bf , a point c0 not in Bf , and a star with base point c0 and end points in Bf . This provides a direct sum decomposition of H˜ q (F ) (see [B], [DN], [NN2], [ST1], [ST2] for various degrees of generality). Indeed, first we have the obvious isomorphism ∂ −1 : H˜ q (F ) → Hq+1 (Cn , F ), next we can replace the second group by Hq+1 (f −1 (∪i pi ([0, 1]) ∪ ∪i Di ), F ) using a deformation retract, and finally by excision we may replace the last group by ⊕i Hq+1 (f −1 (Di ), Fci ). Next we can define a subgroup of vanishing cycles Vq,i in H˜ q (F ) for i = 1, . . . , t by pushing the subgroup ∂Hq+1 (f −1 (Di ), Fci ) in H˜ q (Fci ) to H˜ q (F ) along the path pi . This sum decomposition, (∗q )

H˜ q (F ) = ⊕ti=1 Vq,i ,

depends essentially on the choice of the star. If R → K is a ring extension with K = R or with K a field, then by tensor product by K over R in (∗q ) we get the corresponding direct sum decomposition of H˜ q (F, K) which is denoted sometimes by (∗q (K)). Some notation: let mj = ρ(f )q (γj ) = H˜ q (Tj ) ∈ Aut(H˜ q (F )) and m∗j = ρ(f )q (γj ) = H˜ q (Tj )−1 ∈ Aut(H˜ q (F )) be the monodromy operators in homol-

ogy and in cohomology associated to a star as above. Here Tj : F → F is the homeomorphism obtained by parallel transport along the path γj . It is clear that these operators completely determine the corresponding monodromy representation and that mk · . . . · m1 = M∞ (f )q and m∗k · . . . · m∗1 = M∞ (f )q . When we need to keep track of the degree in which these operators act, we q write mq,j instead of mj and mj instead of m∗j .


203

The following result provides key information on these monodromy operators in homology (exactly as in the case of isolated hypersurface singularities and/or the classical theory of Lefschetz pencils) and can be found in the global setting in various recent preprints (see [DN, Lemma 2.4], [NN2, Theorem 1.4], [NN1, Theorem 3.1], [ST2, Proposition 2.2]). The proof of it being very simple, we reproduce it below. lemma 2.4 For any cycle [c] ∈ H˜ q (F ):

mj ([c]) − [c] ∈ Vq,j .

Proof The geometric monodromy along the path γj is defined as follows: take a q-cycle c, and move it along the path γj to get a cycle mgeom (c) in F . This means that there is a (q + 1)-cycle d such that ∂d = mgeom (c) − c and f (d) ⊂ γj ([0, 1]). But then mj ([c]) − [c] = ∂[d] ∈ Vq,j . 2.5. Lemma 2.4 shows that in the sum decomposition ⊕ti=1 Vq,i , the monodromy mj = ρf (γj ) (1 ≤ j ≤ t) has a block decomposition of the following form:   1   ..   .     mj 1 · · · mjj mj,j +1 · · · mj t  mj =  (∗∗q ) ,   1    ..  .   1 where all the entries, excepting the diagonal and the j th line, are trivial. Here mj i : Vq,i → Vq,j is given by mj i (ai ) = mj (ai ) − ai (for i = j ), and mjj : Vq,j → Vq,j is mjj (aj ) = mj (aj ). 2.6. Remarks (i) Lemma 2.4 is a weak global affine Picard-Lefschetz formula (corresponding to the result of Picard), and it says that mj ([c]) − [c] can be localized in f −1 (Dj ). (ii) We have similar decompositions (∗q ) and (∗∗q ) in the more general case of a regular function f : X → C where X is an algebraic variety with Hq+1 (X) = H˜ q (X) = 0. Hence all our results below hold in this more general setting. (iii) Note that in some cases the decomposition (∗q ) is completely determined by the individual monodromy operators mj or m∗j , and this may be a first step in


204

the difficult task of fitting together these operators in order to obtain the monodromy representation (see, e.g., [DN], where this idea is applied to the Briançon polynomial). 3. The monodromy at infinity determines the monodromy representation In this section we prove the following result, which implies Theorem A. theorem 3.1 Let R = Z, and fix a nonnegative integer q and an extension Z → K. Then (i) the monodromy operator at infinity M∞ (f )q and the decomposition (∗q ) completely determine the monodromy representation ρ(f )q ; (ii) if v ∈ H˜ q (F, K) has components (v1 , . . . , vt ) with respect to the decomposition (∗q (K)), then M∞ (f )q (v) = av for some a ∈ K if and only if mk · . . . · m1 (v) = av1 , . . . , avk , vk+1 , . . . , vt for all k = 1, 2, . . . , t. In particular, Ker M∞ (f )q − 1 = H˜ q (F )inv , the invariant homology under the monodromy representation. The precise meaning of Theorem 3.1(i) is that if we know the operator M∞ (f )q as a matrix of blocks of type (∗∗q ), then this gives the matrices corresponding to all the operators mk , k = 1, . . . , t. Before giving the proof, we would like to emphasize some points. (a) Theorem 3.1(i) holds for the cohomology monodromy representation as well when H∗ (F, R) is torsion free. Then M∞ (f )q is the inverse transpose of M∞ (f )q in the following sense. The decomposition (∗q ) induces a similar decomposition for H˜ q (F ) (see [DN, (8.2)]), which we denote by (∗q ). If the matrix A of M∞ (f )q is known with respect to this decomposition (∗q ), then the matrix of M∞ (f )q with respect to the decomposition (∗q ) is just the inverse transpose matrix (At )−1 . (b) Theorem 3.1(ii) is definitely false for cohomology; that is, for R a field, there are examples when dim Ker M∞ (f )q − 1 > dim H˜ q (F )inv (see, e.g., the Briançon polynomial in [D2]). More precisely, we have the following situation when R is a field. For a single monodromy operator we have an obvious equality


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dim Ker(mk − 1) = dim Ker m∗k − 1 , and various formulas for this dimension have been obtained in [ACD] and, in full generality, in [NN2]. Moreover, the vector spaces (Ker(m∗k − 1))k are in general position in H˜ q (F ) (see [ACD], [NN2]). This implies the following inequality (which is strict in general): dim H˜ q (F )inv = dim Ker M∞ (f )q −1 = dim Ker M∞ (f )q −1 ≥ dim H˜ q (F )inv . This discussion shows that homology and cohomology monodromy representations each have their advantages, and hence it is useful to study and to use them both. 3.2. Proof of Theorem 3.1 The shape of the matrices in (∗∗q ) implies the following obvious and equivalent facts. For any k = 1, . . . , t we have that (·) the operator mk is determined by the composition pk · mk where pk : H˜ q (F ) → Vq,k is the projection onto the kth factor in (∗q ); (··) mk (v1 , . . . , vt ) = (v1 , . . . , vk−1 , vk , vk+1 , . . . , vt ); that is, the operator mk acts only on the kth component of any vector v. The formula (∗ ∗ ∗)

m · mk · m = M∞ (f )q ,

where m = mk−1 · . . . · m1 and m = mt · . . . · mk+1 , implies that pk · mk = pk · M∞ (f )q · m−1 . Indeed, the operator m leaves the first k components of a vector v unchanged. But this shows that if m1 , . . . , mk−1 are already determined by M∞ (f )q , then the same holds for mk ; that is, Theorem 3.1(i) follows by induction on k. To get Theorem 3.1(ii), we use again the decomposition (∗ ∗ ∗) and the property of m to get that M∞ (f )q (v) = av implies mk · m (v) = av1 , . . . , avk , vk+1 , . . . , vt . Note for this that mk · m leaves unchanged the last (t − k) components of any vector. 3.3. Proof of Corollary B This proof uses Theorem A and the following exact sequence from [NN2, Theorem 1.4]: 2n−q−1 (E) : 0 −→ Im mq,j − 1 −→ Vq,j −→ H˜ c Fbj −→ Ker mq−1,j − 1 −→ 0. 2n−q−1 (Fbj ) is replaced by the group In fact, in [NN1, Theorem 1.4], the group H˜ c


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H 2n−q−1 (Y, ∂Y ) where Y (resp., ∂Y ) is the intersection of the special fiber Fbj with a very large closed ball B in Cn (resp., with the boundary of this ball). The two groups are clearly isomorphic, and we prefer to work with the cohomology with compact supports. To end the proof note that condition (i) (resp., (ii)) in Corollary B is equivalent to the vanishing of the first and the third (resp., the second and the fourth) terms in the exact sequence (E).

4. Some examples Example 4.1 (Nontrivial monodromy on torsion in homology) Here we give an example of a polynomial whose generic fiber F has torsion in integral homology, and the action of the monodromy on this torsion is nontrivial. Hence, here we take R = Z. Let Q be a three-cuspidal quartic curve in the projective plane P2 . In an appropriate coordinate system on P2 , the curve Q is the zero set of the polynomial f (x, y, z) = x 2 y 2 + y 2 z2 + x 2 z2 − 2xyz(x + y + z). It is well known that the fundamental group π1 (P2 \ Q) is a finite nonabelian group of order 12 (see, e.g., [D1, p. 131]). Now regarding f as a polynomial function C3 → C, we have that (i) t = 1 and b1 = 0; (ii) the only monodromy homeomorphism h : F → F can be represented by h(x, y, z) = (ix, iy, iz), i2

where = −1, and hence h4 = 1; (iii) if h denotes the cyclic group of order 4 spanned by h, then the quotient space F /h is just P2 \ Q. This implies that H1 (F ) = π1 (F ) = Z/3Z (see also [D1, Corollary 4.14, p. 133]). In the exact sequence (E) we have in this case: V1,1 = H1 (F ) = Z/3Z, H˜ c4 (F0 ) = 0 since F0 is irreducible, and hence Im(m1,1 − 1) = Z/3Z. In other words, the monodromy operator in homology satisfies m1,1 = M∞ (f )1 = 1. On the other hand, H 1 (F ) = 0, and so M∞ (f )1 = 1. This example explains our restrictions on torsion in Theorem A. The fact that H1 (F ) is torsion was already remarked by D. Siersma in [S, Example 7.4]. The methods developed in his paper can also be used to show that in this case m1,1 = M∞ (f )1 = 1.


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Example 4.2 (Trivial monodromy versus trivial intersection form) It was shown by W. Neumann and P. Norbury [NN1] that the total space of the fibration f : f −1 (Sr1 ) → Sr1 for r 0 which gives the monodromy at infinity for the polynomial f can be embedded in a natural way as an open subset of a large sphere S 2n−1 in Cn . This gives a Seifert form L : H˜ n−1 (F ) × H˜ n−1 (F ) → Z defined in the usual way, as well as the usual relation L 1 − M∞ (f )n−1 = S, where S is the intersection form on H˜ n−1 (F ). This equality gives the following. 4.3. M∞ (f )n−1 = 1 implies S = 0; that is, S(x, y) = 0 for any x, y ∈ H˜ n−1 (F ). The converse implication is false in general since the Seifert form L can be degenerate. For instance, the polynomial f = x 2 y 2 + y : C2 → C has M∞ (f )1 = 1 and S = 0 since F is homeomorphic to C \ {0, 1} (see G. Bailly-Maitre [BM] for details). This converse implication is nevertheless true for the M-tame polynomials introduced by A. Némethi and A. Zaharia in [NZ1] as a generalization of the tame polynomials introduced by S. Broughton in [B]. In fact, it was shown in [NZ2] that for an M-tame polynomial the fibration f : f −1 (Sr1 ) → Sr1 for r 0 which gives the monodromy at infinity for the polynomial f is equivalent to the Milnor fibration at infinity φ : Sr2n−1 \f −1 (0) → S 1 , φ(x) = f (x)/|f (x)|. This implies in the standard way that in this case the Seifert form is nondegenerate. There is an analogy between Example 4.2 and the relations between Ker(M(f )∗ − 1) and Ker S in the case of a local nonisolated hypersurface singularity as discussed in detail by D. Siersma in [S, Section 7]. Here M(f )∗ stands for the corresponding local monodromy operators.

References [AC]

` , Polynôme d’Alexander a` l’infini d’un E. ARTAL-BARTOLO and P. CASSOU-NOGUES

[ACD]

polynôme a` deux variables, preprint, 1998, http://www.math.u-bordeaux.fr/ Math Pures/ ` and A. DIMCA, “Sur la topologie des E. ARTAL-BARTOLO, P. CASSOU-NOGUES, polynômes complexes” in Singularities (Oberwolfach, 1996), Progr. Math. 162, Birkhäuser, Basel, 1998, 317–343. MR 99k:32069


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[BM]

G. BAILLY-MAITRE, Sur le système local de Gauss-Manin d’un polynôme de deux

[B]

S. A. BROUGHTON, Milnor numbers and the topology of polynomial hypersurfaces,

[D1]

A. DIMCA, Singularities and Topology of Hypersurfaces, Universitext, Springer,

variables, Bull. Soc. Math. France 128 (2000), 87–101. MR 1765827 Invent. Math. 92 (1988), 217–241. MR 89h: 32025

[D2] [DN] [GN]

[N1] [N2] [N3] [NZ1]

[NZ2] [NN1] [NN2] [S] [ST1] [ST2] [Z]

New York, 1992. MR 94b:32058 , Monodromy at infinity for polynomials in two variables, J. Algebraic Geom. 7 (1998), 771–779. MR 99h:32040 ´ A. DIMCA and A. NEMETHI , Thom Sebastiani construction and monodromy of polynomials, preprint, 1999, http://www.math.u-bordeaux.fr/Math Pures/ ´ ´ R. GARCÍA LOPEZ and A. NEMETHI , On the monodromy at infinity of a polynomial map, I, Compositio Math. 100 (1996), 205–231 MR 97g:32047; II, Compositio Math. 115 (1999), 1–20. MR 2000a:32062 ´ A. NEMETHI , Generalized local and global Sebastiani-Thom type theorems, Compositio Math. 80 (1991), 1–14. MR 92m:32061 , Global Sebastiani-Thom theorem for polynomial maps, J. Math. Soc. Japan 43 (1991), 213–218. MR 92d:32054 , On the Seifert form at infinity associated with polynomial maps, J. Math. Soc. Japan 51 (1999), 63–70. MR 2000a:32068 ´ A. NEMETHI and A. ZAHARIA, On the bifurcation set of a polynomial function and Newton boundary, Publ. Res. Inst. Math. Sci. 26 (1990), 681–689. MR 92c:32046 , Milnor fibration at infinity, Indag. Math. (N.S.) 3 (1992), 323–335. MR 93i:32051 W. D. NEUMANN and P. NORBURY, Unfolding polynomial maps at infinity, Math. Ann. 318 (2000), 149–180. MR 1785580 , Vanishing cycles and monodromy of complex polynomials, Duke Math. J. 101 (2000), 487–497. MR 1740685 D. SIERSMA, Variation mappings on singularities with a 1-dimensional critical locus, Topology 30 (1991), 445–469. MR 92e:32022 ˘ , Singularities at infinity and their vanishing cycles, Duke D. SIERSMA and M. TIBAR Math. J. 80 (1995), 771–783. MR 96m:32040 , Vanishing cycles and singularities of meromorphic functions, arXiv:math. AG/9905108 M. ZAIDENBERG, Lectures on exotic algebraic structures on affine spaces, Schriftenreihe des Graduiertenkollegs Geometrie und Mathematische Physik, no. 24, Ruhr-Universität, Bochum, 1997.

Dimca Laboratoire de Mathématiques Pures de Bordeaux, Université Bordeaux I, 33405 Talence Cedex, France; [email protected]


Némethi Department of Mathematics, Ohio State University, Columbus, Ohio 43210, USA; [email protected]

209

GEODESICS, PERIODS, AND EQUATIONS OF REAL HYPERELLIPTIC CURVES PETER BUSER and ROBERT SILHOL

Abstract In this paper we start a new approach to the uniformization problem of Riemann surfaces and algebraic curves by means of computational procedures. The following question is studied: Given a compact Riemann surface S described as the quotient of the Poincaré upper half-plane divided by the action of a Fuchsian group, find explicitly the polynomial describing S as an algebraic curve (in some normal form). The explicit computation given in this paper is based on the numerical computation of conformal capacities of hyperbolic domains. These capacities yield the period matrices of S in terms of the Fenchel-Nielsen coordinates, and from there one gets to the polynomial via theta-characteristics. The paper also contains a list of worked-out examples and a list of examples—new in the literature—where the polynomial for the curve, as a function of the corresponding Fuchsian group, is given in closed form. 0. Introduction The uniformization theorem of Koebe and Poincaré states that any Riemann surface has a universal covering conformally equivalent to either the Riemann sphere P1 , the complex plane C, or the Poincaré upper half-plane H. One of the consequences is that any smooth complex algebraic curve C of genus g > 1 is conformally equivalent to H/G, where G ⊂ PSL2 (R) is a Fuchsian group. Conversely, any compact Riemann surface is isomorphic to an algebraic curve. Hence, any curve of genus g > 1 may be described in two ways, either by an equation or by a Fuchsian group. Going explicitly from one description to the other is, in either direction, a difficult problem. This is the classical uniformization problem. In this paper we study the direction from the Fuchsian groups to the curves. We also provide a large number of new examples, all in genus 2, where the correspondence is given in exact form. Since H has a natural hyperbolic metric and this induces one on the corresponding curve C, one can reformulate the problem by asking how one relates explicitly DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 2, Received 1 August 2000. Revision received 18 October 2000. 2000 Mathematics Subject Classification. Primary 30F10; Secondary 14H15, 32G15. Authors’ work supported in part by European Community Human Capital and Mobility Programme plan number CHRX-CT93-0408 and by Swiss National Science Foundation contract numbers 21-50847.97, 21-57251.99.

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equations for C with the hyperbolic metric on C. This is the approach we have taken, and we present here a method based on the computation of period matrices in terms of the hyperbolic metric. If H/G has specific symmetries such that the associated algebraic curve is hyperelliptic and real, then the period matrix may be expressed in terms of conformal capacities of certain geodesic polygons. This reduces the computations to quite simple matters, and for this reason we have restricted this paper to real hyperelliptic curves, although the method is somewhat more general. For the conformal capacities we use approximation by harmonic polynomials, and combining this with Theta characteristics we obtain an equation of the curve in terms of the Fenchel-Nielsen coordinates of H/G. While experimenting with this method, we noticed a large number of examples that suggested that the correspondence between the Fenchel-Nielsen coordinates and the equation was expressible in exact form. For many of these we have proved that this is indeed the case by using a sort of uniformization “in families,” that is, by exhibiting one-parameter families of curves which are algebraic with respect to the natural projective structure of the moduli space and which at the same time are defined by algebraic relations between the Fenchel-Nielsen coordinates. A list of exact correspondences and also some “possibly exact” correspondences, for which we have no proof, can be found in the final section. 1. Preliminaries As a general reference we use [FK]; for more specific results on real curves we use [GH] or [SS], and for results on hyperbolic surfaces we use [Bu]. Let C be a smooth complex algebraic curve. Then C is naturally endowed with the structure of a Riemann surface. Conversely, any compact Riemann surface is conformally equivalent to an algebraic curve. By a classical theorem of Weil, C can be defined by real polynomial equations if and only if C admits an antiholomorphic involution σ . Moreover, the polynomials can be chosen such that σ is the involution induced by complex conjugation. Hence a real curve is a couple (C, σ ), where C is a complex curve and σ is an antiholomorphic involution on C. We also say that σ is a real structure on C. If the antiholomorphic involution is clear, for example, if C is a curve already defined by polynomials with real coefficients and σ is complex conjugation, we simply say that C is a real curve. Note, however, that such a real curve may have additional real structures. If C is of genus g, then the fixed-point set of σ , C(R) can have at most g + 1 connected components, and in this case we say that C is an M-curve. If C is an M-curve, then C C(R) has two connected components, each homeomorphic to a sphere with g + 1 disks removed.

EQUATIONS OF REAL HYPERELLIPTIC CURVES

213

Let (C, σ ) be a real curve of genus g 2, and assume C is hyperelliptic with C(R) = ∅. Then we can define (C, σ ) by an equation of the form y 2 = P (x), with P a real polynomial of degree either 2g + 2 or 2g + 1, with distinct roots and such that σ is induced by complex conjugation. For such curves we note the following fairly trivial facts. (i) (C, σ ) is an M-curve if and only if all the roots of P are real. (ii) C has two real structures. If the first is (C, σ ), then the second is (C, −σ ), where −σ is σ composed with the hyperelliptic involution. If an equation for the first is y 2 = P (x), then an equation for the second is y 2 = −P (x). We say that the real components of (C, −σ ) are the pure imaginary components of (C, σ ). For M-curves the intersection points of the real and pure imaginary components are precisely the Weierstrass points of C. By the uniformization theorem any algebraic curve of genus g > 1 carries a unique hyperbolic metric compatible with the underlying conformal structure. The terms geodesic and isometry should always be understood with respect to this metric. The next lemma is the starting point of this paper. lemma 1.1 Let (C, σ ) be a real hyperelliptic M-curve of genus g > 1. Then the union of the real and pure imaginary components of (C, σ ) separates C into four isometric geodesic right-angled (2g + 2)-gons. Conversely, let P be a hyperbolic geodesic right-angled (2g + 2)-gon. Glue P and a mirror image of P so as to obtain a sphere S with g + 1 disks removed. Let S be a mirror image of S . Glue S and S in a way that preserves the mirror symmetries. Then the surface thus obtained is a real hyperelliptic M-curve. Proof Since σ is antiholomorphic, it is an orientation-reversing isometry. This implies that the real components of (C, σ ) are simple closed geodesics. The same is true, of course, for the pure imaginary components. To see that they intersect at right angles, note that −σ commutes with σ , so that σ fixes the pointwise invariant geodesics of −σ and vice versa. As σ and −σ are reflections along these sets, the sets intersect each other orthogonally. The rest of the first half of the lemma follows from the above considerations. Call S the Riemann surface obtained in the second half. By construction S has 2 orientation-reversing symmetries, σ and σ . Since by construction σ is an orientationreversing isometry, it defines an antiholomorphic involution. Also by construction, the real curve (S, σ ) has g + 1 real components and hence is an M-curve. On the other hand, τ = σ ◦ σ defines a holomorphic involution. The fixed points of τ are the

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vertices of P , and since there are 2g + 2 of these, τ is an hyperelliptic involution. Remark 1.2 The preceding discussion does not extend immediately to genus-1 curves since an elliptic curve does not carry a natural hyperbolic metric. On the other hand, an elliptic curve with one point removed does, and we can do the following. Let C be an elliptic curve defined by an equation of the form y 2 = x(x−1)(x−a), = C {p} has a natural hyperbolic a ∈ R, and let p be the point at infinity. Then C into four metric. The union of the real and pure imaginary components of C separate C isometric quadrangles with three right angles and one zero angle. (The corresponding vertex is a parabolic point.) Conversely, starting with four copies of a hyperbolic quadrangle with three right angles and one zero angle, we can glue these so as to obtain an “ideal” pair of pants with two geodesic boundary components, β1 and β2 , of equal length and a third of length zero. Gluing β1 and β2 with zero twist, we obtain an elliptic curve with one point p removed. For the same reasons as in the proof of Theorem 1.1, this curve has a real structure with two real components. Hence it has an equation of the form y 2 = x(x − 1)(x − a), a ∈ R. Moreover, since by construction point p is at the intersection of a real and a pure imaginary component, we can choose a so that point p is the point at infinity. 2. The standard period matrices of real hyperelliptic M-curves Let (C, σ ) be a real hyperelliptic M-curve of genus g. Then by the considerations in Section 1, (C, σ ) can be defined by an equation of the form y = P (x) = 2

2g+1

x − xi ,

with all xi ∈ R and x1 < · · · < x2g+1 .

(2.1)

i=1

We associate with it a standard form of the corresponding period matrices as follows. Since P is nonzero in the upper half-plane H and the latter is simply connected, √ we can choose on H a determination of the square root P (x). Obviously we can extend this determination to R and even to strips below the ]xi , xi+1 [’s. We take the one that is positive on [x1 , x2 ]. It is then negative on [x3 , x4 ], positive on [x5 , x6 ], . . . , and so on; it is also pure imaginary with positive imaginary part on ] − ∞, x1 ], pure imaginary with negative imaginary part on [x2 , x3 ], . . . , and so on. Let π : C → P1 be the projection (x, y) → x. Let βi = π −1 ([x2i−1 , x2i ]) for 1 i g, βg+1 = π −1 ([x2g+1 , ∞[), γ1 = π −1 (] − ∞, x1 ]), and γi+1 = √ π −1 ([x2i , x2i+1 ]) for 1 i g. For x ∈ H, the map x → (x, P (x)) is a conformal inverse of π.

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We choose on these cycles (which correspond to the real and pure imaginary √ components) the orientation defined by the map x → (x, P (x)) and x increasing. At the points (xi , 0), y is a local coordinate on C, and from this it is easy to compute the intersection numbers of the βi ’s and γj ’s. We have (γi · βi ) = 1 and (γi+1 · βi ) = −1, for 1 i g, and (γ1 · βg+1 ) = −1, all other intersection numbers being zero. Hence if αi = − ik=1 γk , then {α1 , . . . , αg , β1 , . . . , βg } defines a symplectic basis of H1 (C, Z), that is, one for which the intersection matrix is 0 −1g , 1g 0 where 1g is the g × g identity matrix. lemma 2.2 Let (C, σ ) be a real hyperelliptic M-curve defined by an equation of the form in (2.1). √ Let P (x) be the above determination of the square root on H, and set i−1 x2k+1 j −1 x1 x j −1 dx x2i x j −1 dx

x dx , B= . A= − − √ √ √ P (x) P (x) P (x) −∞ x2i−1 k=1 x2k i,j

i,j

Then Z = A · B −1 is a period matrix for C with e(Z) = 0. We say that a period matrix obtained in this way is a standard period matrix of (C, σ ) or, more precisely, the standard period matrix associated to equation (2.1). Proof Let ωj = x j −1 dx/y. Then it is well known that the ωj ’s form a basis of the space 1C of holomorphic 1-forms on C. Since {α1 , . . . , αg , β1 , . . . , βg }, αi ’s and βi ’s as above, is a symplectic basis of H1 (C, Z), this means that if we replace A and B by ( αi ωj )i,j and ( βi ωj )i,j , Z is a period matrix of C. But clearly

αi

ωj = −2

βi

ωj = 2

x1

−∞ x2i

x2i−1

i−1 x2k+1 j −1

x j −1 dx x dx −2 √ √ P (x) P (x) k=1 x2k

and

x j −1 dx . √ P (x)

Finally, we note that B is real while A is pure imaginary, and this ends the proof. lemma 2.3 Let (C, σ ) be as in Lemma 2.2, and let the βi ’s be as above. Let

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y = 2

2g+1 i=1

x − xi ,

, with xi ∈ R and xi < xi+1

be another equation defining (C, σ ), and let β1 , . . . , βg+1 be constructed in the same way as the βi ’s, but from the xi ’s. } is either a cyclic permutation of the ordered Then the ordered set {β1 , . . . , βg+1 set {β1 , . . . , βg+1 } or a cyclic permutation of {−βg+1 , . . . , −β1 }.

Proof To simplify notation, call (C , σ ) the curve defined by the second equation. The hypothesis is now that (C, σ ) and (C , σ ) are isomorphic. Let ψ : (C, σ ) → (C , σ ) be an isomorphism. Then, since the hyperelliptic involution is in the centre of the automorphism group of an hyperelliptic curve, ψ also induces an isomorphism between (C, −σ ) and (C , −σ ). This implies that ψ sends real components to real components and pure imaginary components to pure imaginary components. Since an isomorphism respects the intersection form, this means that if ψ transforms βi into βj (resp., −βj ), then it must transform γi into γj or −γj +1 (resp., −γj or γj +1 ), where the γi ’s are again constructed in the same way as the γi ’s and γg+2 = γ1 . Hence the cyclic order is either respected or reversed. To see that in the first case the orientations are preserved while they are reversed in the latter, recall how the cycles are oriented and recall the fact that ψ is induced by a projective transformation of P1 , taking the xi ’s to the xi ’s. (Hence the only possibilities are, in fact, βi goes to βj and γj goes to γj or βi goes to −βj and γj goes to −γj +1 .) corollary 2.4 Let (C, σ ) be as in Lemma 2.2, and let Z and Z be two standard period matrices of (C, σ ). Then Z = MZ tM, where M is in Gg , the subgroup of GLg (Z) generated by the matrices     0 0 0 ··· 0 1 0 0 · · · 0 −1   .. .. 1 0 · · · 0 −1 . .   1 0     0 1 0 −1    . . and N2 =  N1 = −  .  . ..  .. 0 0 1  ..  . 0 1 0 · · · 0 0 . .  0 0 1 −1 1 0 0 ··· 0 0 Proof Let ψ, the βi ’s, and the γi ’s be as in the proof of Lemma 2.3. Now recall that by construction we have βg+1 = −β1 − β2 − · · · − βg and γg+1 = −γ1 − γ2 − · · · − γg . This, together with Lemma 2.3, implies that the matrix

217


of ψ∗ (the induced mapping), in the bases {αi , βi } and {αi , βi }, is of the form t −1 N 0

0 N

for a matrix N in Gg . Applying this, with M = N −1 , to the construction of Z, we have Corollary 2.4.

Remarks 2.5 (i) It is easily seen that the group Gg is isomorphic to the dihedral group Dg+1 . (ii) Let Z = (zij )i,j . Then the diagonal elements of N2 Z tN2 are zgg ,

z11 − 2z1g + zgg ,

z22 − 2z2g + zgg ,

...,

zg−1,g−1 − 2zg−1,g + zgg .

(Recall that Z is symmetric.) Those of N22 Z tN22 are zg−1,g−1 − 2zg−1,g + zgg ,

zg−1,g−1 ,

z11 − 2z1,g−1 + zg−1,g−1 ,

...,

and so forth. This means that we can recover the coefficients of Z from the diagonal elements of the matrices N2k Z tN2k . 3. Period matrices of real hyperelliptic M-curves in terms of capacities Let (C, σ ) again be a real hyperelliptic M-curve defined by an equation of the form in (2.1), and let notation be as in Section 2. Let B and Z be as in Lemma 2.2, and write tB −1 = (cij )i,j and Z = (zij )i,j . Let z cij x j −1 dx fi : z −→ . √ P (x) x1 √ By the choice of the determination of P (x), this defines holomorphic functions fi on the simply connected domain formed by H and the vertical strips below the ]xi , xi+1 [’s. By the choice of the cij ’s we have fi x2i −fi x2i−1 = 1, fi x2j −fi x2j −1 = 0, for 1 i, j g and i = j. (3.1) Recalling the definition of the αi ’s and the fact that γ1 = −γ2 − · · · − γg+1 , we also have g

fi x2j +1 − fi x2j = zii . (3.2) j =i

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Define ui = e(fi ) and vi = m(fi ). These are harmonic functions on H. To √ obtain the boundary behaviour of the ui ’s, note that since P (x) is pure imaginary on the intervals [x2i , x2i+1 ] and ] − ∞, x1 ], we have by (3.1), ui (x) = 0 on ] − ∞, x1 ] and on x2j , x2j +1 , for j < i, (3.3) ui (x) = 1 on x2j , x2j +1 , for i j. √ Writing x = ξ + −1ζ with ξ and ζ real, we have from the Cauchy-Riemann equations ∂fi (x) ∂ui (x) = −m . (3.4) ∂ζ ∂ξ This holds, in particular, for x ∈ ]x2k−1,2k [, k = 1, . . . , g and x ∈ ]x2g+1 , ∞[. Since the imaginary part of fi is constant on these intervals, we get ∂ui (x) =0 ∂ζ

for x ∈ ]x2k−1 , x2k [ and ]x2g+1 , ∞[.

(3.5)

Now recall that the capacity of a harmonic function h on a domain M is c = 2 M ∇h dM and that by Green’s theorem we can rewrite this as h · ν[h] dµ, c= ∂M

where ν[h] is the derivative of h with respect to the outward pointing unit normal vector field. In our situation, where M = H, the upper half-plane, and h = ui , the corresponding capacity is ∂ui (t) dt. (3.6) ∇ui 2 dξ dζ = − ui (t) · ci = ∂ζ H R Applying this and recalling (3.2), (3.3), (3.4), and (3.5), we find the following proposition. proposition 3.7 (x − xi ), 1 i 2g + 1, xi < xi+1 . Let Let (C, σ ) be defined by y 2 = √ Z = −1(yij )ij be the standard period matrix associated to this equation, as in Lemma 2.2. Let ci be the capacity introduced in (3.6) for ui , harmonic, satisfying conditions in (3.3) and (3.5). Then, for 1 i g, ci = yii . corollary 3.8 Let (C, σ ) be as in Proposition 3.7, and let P be the hyperbolic (2g + 2)-gon


219

associated to (C, σ ) as in Lemma 1.1. Then a standard period matrix of (C, σ ) can be computed in terms of capacities of harmonic functions with mixed boundary conditions on P . Proof There is a conformal map sending P to H ∪ {∞} in such a way that its vertices go 1 , 2 , . . . , γ1 , β γg+1 , to x1 , x2 , . . . , x2g+1 , ∞. We label the sides of P successively β γ1 to [x2 , x3 ], and so on. such that β1 goes to the interval [x1 , x2 ], We observe that yii is also the capacity ci of the harmonic function hi on P which is uniquely defined by the boundary conditions ν[hi ] = 0

1 , . . . , β g+1 , on β

hi = 0

on γ1 , . . . , γi ,

hi = 1

on γi+1 , . . . , γg+1 .

i−κ , i [κ] = β γi [κ] = γi−κ Now let κ ∈ {1, . . . , g}, and set for i = 1, . . . , g +1, β (subscripts modulo g + 1). This is just a cyclic renumbering of the sides. It induces a renumbering βi [κ] = βi−κ , γi [κ] = γi−κ , i = 1, . . . , g + 1, and provides the symplectic basis {α1 [κ], . . . , αg [κ], β1 [κ], . . . , βg [κ]} with corresponding standard period matrix Z[κ]. A simple check using the relation βg+1 [κ] = −β1 [κ]−· · ·−βg [κ] shows that βj = nij [κ]βi [κ] with (nij )i,j = N2κ . Hence Z[κ] = N2κ Z tN2κ . √ The diagonal elements of Z[κ] = −1(yij [κ])i,j are the capacities ci [κ] of the harmonic functions hi [κ] on P which are defined as the hi ’s but with respect to 1 [κ], γ1 [κ], . . . ; that is, the hi [κ]’s are the harmonic functions with the numbering β respect to the boundary conditions ν[hi [κ]] = 0

1 , . . . , β g+1 , on β

hi [κ] = 0

on γ1−κ , . . . , γi−κ ,

hi [κ] = 1

on γi+1−κ , . . . , γg+1−κ

(3.9)

(subscripts modulo g + 1). Using Remark 2.5, we can easily compute the coefficients yij ’s from these data. Explicitly, we have for 1 i < j g, 2yij = yii + yjj − yj −i,j −i [g + 1 − i]. Remark 3.10 All of the results of this section generalize to the case g = 1 in even simpler form since for g = 1 we only have to compute one capacity of the quadrangle introduced in Remark 1.2 to obtain the period of the corresponding real elliptic curve.

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4. Computing the equation for the curve associated to a hyperbolic (2g + 2)-gon We detail in this section the method for computing an equation of the curve associated to a (2g+2)-gon as in Lemma 1.1 and Remark 1.2. The numerical part in our approach reduces to the computation of the conformal capacities of plane domains. After testing various finite element methods based on Euclidean and hyperbolic triangulations, we found that the approximation of a harmonic function by harmonic polynomials—that is, functions of the form r k (ak cos(kx) + bk sin(kx))—is by far the fastest and the most accurate method. Moreover, it is simple to implement and uses only standard subroutines. For the achieved accuracy we refer to Section 8. The practical computation of an approximating harmonic polynomial on a polygon domain with respect to given boundary conditions is as follows. First we realize the polygon as a domain in the unit disk and distribute a finite number of points along the boundary, more or less equidistantly (with respect to the Euclidean metric). Imposing that the polynomial satisfy the boundary conditions in these points gives us a family of linear equations in the ak ’s and bk ’s. We overdetermine the system and solve it in the sense of least squares using a standard SVD (singular values decomposition) subroutine. Using the Green-Riemann formula, we get the capacity of the function directly out of the ak ’s and bk ’s. From the capacities we can, using the method described in Section 3, recover a standard period matrix Z of the curve. Since the curve is hyperelliptic, we can use Theta characteristics to obtain an equation of the curve (see, e.g., [FK, pp. 348–350]). For genus 2 the domain is a hyperbolic hexagon H in the unit disk with sides l3 , l2 , l1 , l3 , l2 , in that order (see Figure 1). labeled l1 ,

l2 l1

l3 l1

l3

l2

Figure 1

We assume that the sides {li } (resp., { li }) are mapped by the inclusion H 7→ C into the real and pure imaginary components, respectively. The three capacities c1 , c2 ,


221

and c3 are computed corresponding, respectively, to u1 , which is zero on l1 , 1 on l2 and l3 ; u2 , which is 1 on l2 , zero on l1 and on l3 ; and u3 , which is 1 on l3 , zero on l1 and on l2 . From Section 3 we conclude that the matrix   1 c c + c − c 1 1 2 3   2 Z = i 1  c1 + c2 − c3 c2 2 is a period matrix of C. To compute an equation for C, let

2α (Z) = exp πi t(n + α)Z(n + α) + 2πi t(n + α)β ϑ 2β n∈Zg

for 2α and 2β in Zg , and set

2 ϑ 00 00 (Z) · ϑ 00 01 (Z) x1 = , ϑ 01 00 (Z) · ϑ 01 01 (Z) 2 ϑ 00 01 (Z) · ϑ 00 01 (Z) , x2 = ϑ 01 01 (Z) · ϑ 01 01 (Z) 2 ϑ 00 00 (Z) · ϑ 00 01 (Z) . x3 = ϑ 01 00 (Z) · ϑ 01 01 (Z)

(4.1)

Then y 2 = x(x − 1)(x − x1 )(x − x2 )(x − x3 ) is an equation for C (see, e.g., [FK, pp. 348–350]). The generalization of this method for g > 2 is straightforward. Actually, this method also works in the hyperbolic genus-1 case as considered in Remark 1.2 and is in fact even simpler. Let Q be a hyperbolic quadrangle with three right angles and with one zero angle, that is, with one vertex at infinity. We denote by l and l the two sides of Q with finite length. Let C be the real curve associated with Q such that the inclusion map Q 7→ C sends l and l upon a real and a pure imaginary component, respectively. l, 1 We let τ be the capacity of the harmonic function u on Q which is zero on on the opposite side of l, and which has zero normal derivatives on the remaining two sides. By the considerations of Section 3, there is a conformal map ψ1 sending Q onto the rectangle (0, 1, 1 + iτ, iτ ) and sending vertices to vertices. Let 8 ∞ exp(πτ ) 1 + exp − (2n − 1)πτ . (4.2) x1 = 8 16 1 + exp(−2nπτ ) n=1

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l l

Figure 2

Then it is well known (see, e.g., Z. Nehari [Ne]) that there exists a conformal map ψ2 sending the rectangle (0, 1, 1 + iτ, iτ ) onto the upper half-plane and such that ψ2 (0) = 0, ψ2 (1) = 1, ψ2 (1 + iτ ) = x1 , and ψ2 (iτ ) = ∞. But this means that y 2 = x(x − 1)(x − x1 ) is an equation for C, and that if h is the projection (x, y) → x, then the composition h

Q 7→ C −→ P1

is equal to ψ2 ◦ ψ1 and maps Q conformally onto the upper half-plane H. For the numerical computation of τ , we place Q as in Figure 2. Let, as before, ak and bk be the coefficients of the harmonic polynomial. Then all bk ’s and also all a2k ’s are zero and the capacity becomes

τ= (−1)k a2k+1 |p|2k+1 , where p is the nonzero end point of side l. In this case the computation is particularly fast and efficient. With a variant of this method we can also handle “half-twists.” We describe this here in the case g = 1; for g = 2 we refer to the end of Section 7. We first need some remarks on real elliptic curves with one real component. Let C1 be the elliptic curve defined by the lattice 91 generated by 1 and 1/2+iµ, µ ∈ R, µ > 0. Then C1 is a real curve with one real component, the image in C1 of the horizontal lines R + niµ, n ∈ Z. C1 has also one pure imaginary component, the image of the vertical lines n/2 + i R, n ∈ Z. We note also that we can normalize equations of such curves in the form y 2 = (x 2 + 1)(x 2 − α), α ∈ R, α > 0. Now let C2 be the elliptic curve defined by the lattice 92 generated by 1/2 and iµ. The inclusion 91 ⊂ 92 yields a double covering of C2 by C1 . Such a double covering

223


is easy to describe in terms of equations. If the equation of C1 is as above, then the equation of C2 is y 2 = x(x + 1)(x − α), the covering map being (x, y) → (x 2 , xy).

h

h

h b

b

a

a

Figure 3

Of course, the procedure can be reversed, and this is what we are going to do. Thus, consider again the quadrangles and the “ideal” pair of pants described in Remark 1.2, but assume this time that we glue the two boundary components β1 and β2 with a half-twist. We again obtain a torus with one point removed. It also has an obvious orientation-reversing symmetry with a connected fixed-point set and hence corresponds to a real genus-1 curve with one real component. Call C this curve. Looking at two copies of the quadrangle, the situation is as in Figure 3. Note that in the corresponding Riemann surface the points a, a , b, b , . . . are identified and so are the points h, h , h , . . .. For the orientation-reversing symmetry we can take the one induced by reflection along the (h, b) geodesic, which is the same as the one induced by taking reflection along the (a, h ) geodesic. In this case the real part consists of the images of these two geodesic arcs. The pure imaginary components consist of the images of the geodesic arcs (a, h) and (b, h ). But now we are exactly in the situation described above. (In terms of fundamental parallelograms the situation is as in Figure 4.) h

b

h

b

a

h

a

h

Figure 4

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In particular, if τ is the capacity of a harmonic function on the shaded area a, h, b, h , with boundary conditions zero on (a, h ) and 1 on (b, h) and zero normal derivative on the two remaining sides, then the period of the curve C obtained with a half-twist is 1/2+i(τ/2). Using the Legendre map we can easily find from this period an equation for C. We can also use the above remarks to simplify the computations. Namely, let x1 be associated to τ as in (4.1), and let α = x1 − 1. Then an equation of C is y 2 = (x 2 + 1)(x 2 − α). 5. A D5 action on a subspace of the real genus-2 moduli space In Section 6 we construct various examples of real genus-2 curves with three real components for which the uniformization is given exactly. The construction uses group actions and curve families which have an explicit algebraic description in terms of the coefficients of equations of algebraic curves. At the same time these actions and curve families have an explicit description that is algebraic in terms of the Fenchel-Nielsen coordinates. In this section we describe the group actions. We use the following notation. The group of all orientation-preserving automorphisms of a real curve C = (C, σ ) preserving the real structure σ is denoted by Aut + σ (C). The real moduli space of the real genus-2 curves with three components is (2,3,0) (2,3,0) . As usual we write C ∈ MR to say that the isomorphism denoted by MR (2,3,0) (2,3,0) . For any finite group G we let MR (G) denote the class of C belongs to MR (2,3,0) + with G ⊂ Autσ (C). subspace of all C ∈ MR Let P be the moduli space of pairs of pants. Taking for any P ∈ P the Schottky double, we get a Riemann surface SP with a real structure. (2,3,0) , and we identify P This sets up a natural isomorphism between P and MR (2,3,0) with MR . An element P in P consists of two copies of a hyperbolic geodesic hexagon. l3 , l2 , The lengths of the sides of the hexagon listed in cyclic order are denoted by l1 , l2 . We also denote by hi the common orthogonal between li and li . l1 , l3 , The hexagon is determined up to isometry by l1 , l2 , l3 . The lengths of the remaining quantities are given by the following formulas, where we abbreviate ui = cosh(li ), i = 1, 2, 3: 2 ui + ui−1 ui+1 2 (5.1) cosh li = 2 ui−1 − 1 u2i+1 − 1 (subscripts modulo 3), cosh2 (hi ) =

u21 + u22 + u23 + 2u1 u2 u3 − 1 u2i − 1

.

(5.2)

P is obtained by pasting the hexagons together along the sides li and has boundary

225


geodesics of lengths 2l1 , 2l2 ,2l3 . We use the unordered triple {l1 , l2 , l3 } as a set of coordinates for P ∈ P . Since the pasting that gives SP has zero twist, {l1 , l2 , l3 } also serves as a set of coordinates for SP . (2,3,0) are written either in the form Equations for the elements in MR y 2 = x(x − 1)(x − x1 )(x − x2 )(x − x3 )

(5.3)

as earlier, with x1 , x2 , x3 real, or in the form y 2 = (x − d)(x + 1)(x − c)(x − a)(x − 1)(x − b), d < −1 < c < a < 1 < b. (5.4) Now let P2 be the subspace of P formed by the pairs of pants with two boundary components of equal length. From the point of view of hyperbolic geometry, P2 may be characterized as the set of elements in P with coordinates {l1 , l2 , l2 }. To describe P2 in terms of coefficients of equations, we note that any P ∈ P2 has an orientation-preserving involution with one fixed point. It induces an involution of SP with exactly two fixed points. Since there is also the hyperelliptic involution, we have (2,3,0)

Z/2 × Z/2 ⊂ Aut + σ (SP ). Conversely, any C ∈ MR

with Z/2 × Z/2 ⊂ Aut+ σ (C) (2,3,0)

is obtained in this way. Hence, under the above identification of P with MR , (2,3,0) (Z/2 × Z/2) and we identify P2 the subspace P2 ⊂ P is identified with MR (2,3,0) with MR (Z/2 × Z/2). In (5.4) we may choose the constants such that the fixed points of the nonhyperelliptic involution are (0, ±y). This allows us to normalize equations of elements in P2 in the form y 2 = x 2 − a x 2 − 1 x 2 − b , with 0 < a < 1 < b. (5.5) Using Lemma 2.3, it is straightforward to show that two curves C, C with equations in this form are real isomorphic if and only if a = a and b = b . Each element in P2 is therefore represented by a unique equation, so that we have here a one-to-one correspondence from P2 to the set of all (a, b) ∈ R2 , with 0 < a < 1 < b. l1 , l2 , l3 }. From Our first action is ϕ : P → P defined by ϕ : {l1 , l2 , l3 } → { the point of view of equations, this corresponds to replacing the curve defined by y 2 = P (x) by the one defined by y 2 = −P (x). If the equation of the curve is written in the form of (5.4), then this action can also be described by (a, b, c, d) → (1/b, 1/a, 1/d, 1/c) or by (a, b, c, d) → (−1/c, −1/d, −1/a, −1/b) (which in cer(2,3,0)

tain cases is more useful). This defines an algebraic action on MR (2,3,0) MR (Z/2

. Of course, ϕ

restricts to an action on P2 = × Z/2). In terms of the coordinates (a, b) based on (5.5), ϕ : P2 → P2 is described by 1 1 ϕ : (a, b) −→ , , 0 < a < 1 < b. (5.6) b a

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To describe the action of ϕ in terms of the Fenchel-Nielsen coordinates we recall that for P ∈ P2 , l2 = l3 and all twist parameters are zero. Hence the action can be described by l1 , l2 , l1 , l2 > 0, (5.7) (l1 , l2 ) −→ where l1 and l2 are given by (5.1). Remarks 5.8 (i) From the point of view of complex isomorphism classes, the action of ϕ is of course trivial. The action on real isomorphism classes, however, is nontrivial, and we see that even from the complex point of view it is quite important (see Remark 5.18). (ii) The action of ϕ can also be described in terms of period matrices. If we take the standard period matrix as in Section 3, then the action is Z = iY → (i/ det(Y ))Y . The fixed space is simply characterized by the condition det(Y ) = 1. To describe the next action we begin with the genus-3 curves given by an equation 1 1 (5.9) y 2 = x 2 − x12 x 2 − x22 x 2 − 2 x 2 − 2 , x1 x2 where 0 < x1 < x2 < 1. We denote this family by C2 . Any curve C ∈ C2 has the two fixed-point free involutions 1 −y j2 : (x, y) −→ , . j1 : (x, y) −→ (−x, −y), x x4 Each ji fixes two components of the real part of C and interchanges the other two. The mapping ψ1 : (x, y) → (x 2 , xy) is the natural projection onto the quotient of C by j1 and thus is a twofold covering from C to the genus-2 curve κ, the claim is proved. Call ω the order of G. Of course, i > 2κ − (2κ − ω) ⇒ Gi ≡ 0. Obviously this implies ∀I = (i1 , . . . , i8 ) ∈ Z8+ , i8 > 2κ − n =⇒ GI (θ) = 0 if we take n = 2κ − ω(≥ 2). By Lemma 6.12, ascending induction on n ≥ 2κ − ω proves that ∀I ∈ Z8+ , GI (θ) = 0. Combining this with (91) allows us to use induction on 8 and to conclude that G ≡ 0. The proof of Lemma 6.8 is complete. We are now in a position to complete the proof of Theorem 4.2. Suppose µ ≥ 4. In this case, A satisfies conditions (93) and (94) according to (91) and Lemma 6.7. We can therefore apply Lemma 6.8 with G = A, to conclude that A ≡ 0, contradicting the definition of the order µ of P . Acknowledgments. I am indebted to several colleagues, first of all to Nalini Joshi (University of Adelaide, South Australia), who proposed a joint effort (unsuccessful so far) on the extension of meromorphic solutions of the KdV equation and helped with the first numerical computations that gave some credence to the truth of the theorem. Others who gave me a helping hand are Andrew Pickering (then at the

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University of Adelaide) and Mark Ablowitz and his students at the University of Colorado in Boulder. Needless to say, any errors in the paper are entirely my fault.

References [FPMC] G. FALQUI, M. PEDRONI, F. MAGRI, and P. CASATI, Soliton Equations, Bi-Hamiltonian Manifolds and Integrability, 21◦ Coloq. Bras. Mat., Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1997. MR 2000g:37114 [GD] I. M. GELFAND and L. A. DIKI˘I, Fractional powers of operators and Hamiltonian systems (in Russian), Funkcional. Anal. i Prilozhen 10 (1976) 13–29; English translation in Functional Anal. Appl. 10 (1976), 259–273. MR 55:6484 [KMGZ] M. D. KRUSKAL, R. M. MIURA, C. S. GARDNER, and N. J. ZABUSKY, Korteweg-de Vries equation and generalizations, V: Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys. 11 (1970), 952–960. MR 42:6410 [L1] P. D. LAX, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 38:3620 [L2] , “A Hamiltonian approach to the KdV and other equations” in Group Theoretical Methods in Physics (Montreal, 1976), ed. R. T. Sharp and B. Kolman, Academic Press, New York, 1977, 39–57. MR 58:31222 [L3] , “A Hamiltonian approach to the KdV and other equations” in Nonlinear Evolution Equations (Madison, Wisc., 1977), ed. M. G. Crandall, Publ. Math. Res. Center Univ. Wisconsin 40, Academic Press, New York, 1978, 207–224. MR 80d:35129 [M] F. MAGRI, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162. MR 80a:35112 [Mi] R. M. MIURA, Korteweg-de Vries equation and generalizations, I: A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), 1202–1204. MR 40:6042a [MGK] R. M. MIURA, C. S. GARDNER, and M. D. KRUSKAL, Korteweg-de Vries equation and generalization, II: Existence of conservation laws and constants of motion, J. Math. Physics 9 (1968), 1204–1209. MR 40:6042b [MC] M. MUSETTE and R. CONTE, The two-singular manifold method, I: Modified Korteweg-de Vries and sine-Gordon equations, J. Phys. A 27 (1994), 3895–3913. MR 95c:35221 [O] P. J. OLVER, Applications of Lie Groups to Differential Equations, Grad. Texts in Math. 107, Springer, New York, 1986. MR 88f:58161 [W] J. WEISS, The Painlevé property for partial differential equations, II: Bäcklund transformations, Lax pairs, and the Schwarzian derivative, J. Math. Phys. 24 (1983), 1405–1413. MR 85c:35086

Department of Mathematics, Rutgers University, Hill Center, 110 Freilinghuisen Road, Piscataway, New Jersey, 08854-8019 USA; [email protected]

ALL TIME C ∞ -REGULARITY OF THE INTERFACE IN DEGENERATE DIFFUSION: A GEOMETRIC APPROACH P. DASKALOPOULOS, R. HAMILTON, and K. LEE

Abstract We study the connection between the geometry and all time regularity of the interface in degenerated diffusion. Our model considers the porous medium equation ut = um , m > 1, with initial data u0 nonnegative, integrable, and compactly supported. is smooth up to the interface and in We show that if the initial pressure f0 = um−1 0 addition it is root-concave and also satisfies the nondegeneracy condition |Df0 | = 0 at ∂suppf0 , then the pressure f = um−1 remains C ∞ -smooth up to the interface and root-concave, for all time 0 < t < ∞. In particular, the free boundary is C ∞ -smooth for all time. 1. Introduction We study in this work the connection between the geometry and the regularity of the interface in degenerated diffusion. We consider as our model the porous medium equation ut = um , m > 1 with initial data nonnegative integrable and compactly supported. It is well known that this equation describes the evolution in time of various diffusion processes, in particular, the spread of antisocial biological species and the flow of a gas through a porous medium. In the last case u represents the density, while f = m um−1 represents the pressure of the gas and satisfies the equation 1 |Df |2 . m−1 When u = 0, then f = 0 and both of the above equations become degenerate. This degeneracy results in the interesting phenomenon of the finite speed of propagation: if the initial data u0 is compactly supported in Rn , the solution u(·, t) remains compactly supported for all time t. In [8] P. Daskalopoulos and R. Hamilton showed that, under certain assumptions on the initial data, the free boundary = ∂ supp u ft = f f +

DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 2, Received 6 January 2000. Revision received 8 September 2000. 2000 Mathematics Subject Classification. Primary 35; Secondary 53.

295

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is a smooth surface when 0 < t < T for some T > 0. It is well known [1] that, in general, the free boundary does not remain smooth for all time: advancing free boundaries may hit each other, creating singularities. In this paper we address the question: under what geometric assumptions on the initial data does the free boundary remain smooth for all time? Let us consider the initial value problem for the pressure f , namely, the problem ft = f f + r|Df |2 , (x, t) ∈ Rn × [0, ∞), (1.1) x ∈ Rn , f (x, 0) = f 0 , with r = 1/(m − 1), where f 0 is nonnegative and supported on the compact set Ω = x ∈ Rn : f 0 (x) > 0 . It is well known that for any integrable initial data f 0 ≥ 0 the initial value problem (1.1) admits a unique weak solution f on Rn × (0, ∞). Moreover, it follows by the results in [4] that the pressure f is a Hölder continuous function. If the initial interface ∂Ω is convex, then it does not necessarily remain convex since its shape at time t > 0 depends on the speed of the free boundary, namely, the gradient Df of the pressure near the interface. However, we show in this work that if the pressure f is initially a concave function, which in particular, implies that its interface is convex, then the support of f (·, t) remains convex for all time 0 ≤ t < ∞. In particular, under certain regularity initial assumptions the free boundary is a smooth surface for all t > 0. One may ask: is the matrix inequality Dij2 f ≤ 0 preserved under the flow? In other words, if the initial pressure f 0 is weakly concave, does f (·, t) remain weakly concave for all time? Surprisingly, this is not the case. Instead we show in Section 2 that the matrix inequality Dij2 f ≤ 0 is preserved under the flow. If the initial pressure f 0 is root-concave on its support, then f (·, t) remains root-concave on its support for all time. Hence, the interface = ∂ x ∈ Rn : f (x, t) > 0 is convex for all t > 0. Using the geometry of the level sets of f , we show that the pressure f is C ∞ -smooth up to the interface for all t > 0. In particular, the interface is smooth. The above discussion is summarized in the following result, which is shown in Section 5.

ALL TIME C ∞ -REGULARITY IN DEGENERATE DIFFUSION

297

theorem 1.1 Assume that the function f 0 is smooth up to the boundary of Ω and that in addition it is root-concave in Ω and satisfies the nondegeneracy condition 2 (1.2) f 0 + Df 0 ≥ c > 0 for some c > 0. Then the solution f of the initial value problem (1.1) is a smooth function smooth up to the interface , and f (·, t) is root-concave for all 0 ≤ t < ∞. In particular, the free boundary is a smooth surface. Remark. If the initial pressure f 0 is concave on its support, then it is also rootconcave. Hence, by Theorem 1.1 the solution to (1.1) is smooth up to the interface for ¯ in Theorem 1.1 can be weakened to asall time. The assumption that f 0 ∈ C ∞ (Ω) 0 1 ¯ as shown in the next result, which is proven in Section 6. sume only that f ∈ C (Ω) theorem 1.2 ¯ is root-concave in Ω and that it satisfies the nondegeneracy Assume that f 0 ∈ C 1 (Ω) condition (1.2) and the lower bound on the Laplacian, f 0 ≥ −K

in Rn ,

(1.3)

in the distributional sense for some constant K > 0. Then the solution f of the initial value problem (1.1) is a function smooth up to the interface , and f (·, t) is root-concave for all 0 < t < ∞. In particular, the free boundary is a smooth surface. 2. The root-concavity estimate Let A be a compact subset of Rn × [0, T ], T > 0, with smooth lateral boundary, and let f be a smooth solution of equation ft = f f + r|Df |2

on A

for some 0 < r < ∞, with f = 0, Df = 0, on the following theorem. theorem 2.1 √ If f is weakly concave at t = 0, it remains so for all t. Proof We must show that the matrix inequality Dij2 f ≤ 0 is preserved in time. To simplify the notation we denote Dxk f by fk . Then we can write the evolution

298


of f as ft = ffkk + rfk2

(2.1)

where the summation convention is used. Since f i fj 1 2 , Dij f = √ fij − 2f 2 f it is enough to show that the matrix inequality Aij = fij −

f i fj ≤0 2f

is preserved. We show that Aij ≤ δIij for all δ > 0. Of course, this implies that Aij ≤ 0. To do this we choose a positive function ψ = ψ(t) with (2.2) ψt > c ψ + ψ 2 for a suitable constant c which depends on bounds for f , fi , and fij and which is sufficiently small at t = 0. Consider the quadratic f i fj − ψIij V i V i Z = fij − 2f in the vector V . We show that the inequality Z 0 when Z = 0 at a point x0 ∈ A and at vector V0 with |V0 | = 1, while Z < 0 for all t < t0 at all x and in all directions V . Interior estimate Assume that x0 belongs to the interior of the set A. We extend V0 to be a smooth vector field V in a neighborhood of (x0 , t0 ) in space time, so that Vji =

1 fk V k Iij 2f

(2.3)

at the point (x0 , t0 ). (Note that there may be an obstruction to (2.3) holding this in the full neighborhood, but we only need it to hold at the point (x0 , t0 ).) Also (to simplify the notation) we choose the extension so that j

Vt = 0 at (x0 , t0 ). Now

and

j

Vkk = 0

f i fj − ψIij V i V j Z = fij − 2f


299

is a function of x and t only. Differentiating equation (2.1), we compute fit = ffikk + fi fkk + 2rfk fik and fij t = ffij kk + fi fj kk + fj fikk + 2rfk fij k + fij fkk + 2rfik fj k . Thus

fij V i V j t = ffij kk + 2fi fj kk + 2rfk fij k + fij fkk + 2rfik fj k V i V j

at (x0 , t0 ). Also, using (2.3), we compute

fj k f i i j fij V V k = fij k + V iV j f and

i

f fij V V

j

= ffij kk

kk

at (x0 , t0 ). Hence

fij V i V j

t

fkk fi fj f j k fi fk + V iV j + 2fkkj fi + fj k fik − 2f 2f

= f fij V i V j kk + 2rfk fij V i V j k

f j k fi fk i j f i fj i j V V − 2r V V + fkk fij − 2f f

f i fk i j Vi Vj + 2rfik fj k V V − fj k fik − 2f

(2.4)

at (x0 , t0 ). On the other hand,

rfi fj fk2 fi f j 4rfj fk fik f i fj i j V V fkk + − = 2fj fikk + V iV j , 2 f f f f t

while

and

Thus

fi fj i j V V f

fi fj i j V V 2f

kk

=

fi fj i j V V f

k

=

2fik fj i j V V f

fik fj fk fkk fi fj 2fikk fj 2fik fj k V iV j . − + + f f f2 f2

fi fj i j fi fj i j V V V V =f + 2rfk 2f 2f kk k

2 rf f f f j fk i j k Vj Vi − − fik fj k − V iV j . 2f 2f 2

t

(2.5)

300


Since the vector V0 must be a null eigenvector for the matrix fij − at (x0 , t0 ), we have

fi f j − ψIij 2f

f i fj Aij V ≡ fij − V i = ψIij V i 2f

i

at (x0 , t0 ). If ψ = 0, all terms involving Aij V i in (2.4) and (2.5) drop out; if not, at least they are bounded by cψ|V |2 . for some constant c. Therefore from (2.4) and (2.5) we compute Zt ≤ f Zkk + 2rfk Zk + [cψ − ψt ]|V |2 + R with fj k fi fk i j rfI fj fk2 i j R = 2rfik fj k V i V j − 2r V V V V + f 2f 2

fj f k f i fk rfi fk i j V V − fj k − V j V i, = 2rfj k fik − 2f f 2f which again can be estimated as R ≤ cψ|V |2 . We conclude that at (x0 , t0 ), Zt ≤ f Zkk + 2rfk Zk + (2cψ − ψt )|V |2 ,

(2.6)

and by choosing ψ satisfying (2.2), we can make Zt < f Zkk + 2rfk Zk at (x0 , t0 ). Regardless of the way we extended V0 to V we still have Z = 0 at (x0 , t0 ) and Z ≤ 0 for all t ≤ t0 at all x in a neighborhood of x0 . Thus Zt ≥ 0, Zk = 0, and Z ≤ 0 at (x0 , t0 ). Now if we choose ψt > c ψ + ψ 2 , then since |V0 | = 1, we get that 0 < 0 in (2.6), which is a contradiction. Hence x0 , the point where Z < 0 fails for the first time t = t0 , does not belong to the interior of A.


301

Estimate at the boundary We show next that the quadratic fi fj − ψIij V i V i Z = fij − 2f remains strictly negative at the free boundary for all directions V i . Let (x0 , t0 ) be a free boundary point. If V i is not a tangent direction, then fi fj V i V j > 0 at the point (x0 , t0 ) (since Df = 0 at the free boundary), and therefore the quadratic Z is negative in a neighborhood of the point (x0 , t0 ) where f = 0. Hence we only need to consider the case where V i is a tangent direction of ∂{f (·, t0 ) > 0} at x0 , so that fi V i = 0 at x0 at time t0 . Let us denote by Z(x, t0 ) the value of the quadratic Z at a point x ∈ {f (·, t0 ) > 0} at time t0 , and let us assume that limx→x0 Z(x, t0 ) < 0 fails at a point x0 on the boundary in a tangent direction V0 = 0 for the first time t = t0 . We can assume without loss of generality that |V0 | = 1. Let us notice that since f is smooth up to the interface and vanishes only of a first order, the function fi V i f extends to be smooth at x0 . Hence f i fj V i V j =0 f at x0 at time t0 and therefore the quadratic Z is equal to the well-defined function Q = fij − ψIij V i V j at (x0 , t0 ). We show the following lemma. lemma 2.2 At x0 at time t0 in the direction V0 we have fk fij k V i V j ≤ 0. Proof Choose a path x(s) parameterized by s with x = x0 at s = 0 and dx k = fk , ds so that the path lies in the interior region f > 0 for small s > 0. Then choose a vector field V (s) along the path with V = V0 and

302


1 dV k = fi fij V j fk 2 ds |fl | at s = 0. Along this path the functions f and fi V i are both smooth and both zero at s = 0. Then by l’Hôpital’s rule d fi V i /ds fi V i = lim . lim s→0 f s→0 df/ds Now

while

dx k d f = fk = |fk |2 ds ds dx j dV i d fi V i = fij V i + fi = 2fj fij V i , ds ds ds

which gives

2fi fij V j fi V i = . s→0 f |fk |2 lim

Consider the function

f i fj − ψIij V i V j Q = fij − 2f

along the path x(s) in the direction V (s). Therefore, for s > 0, i k f i fj fi f j dV j dQ i j dx = fij − − ψIij V V − ψIij + 2 fij − V ds 2f 2f ds ds k i f i f j fk fik fj f i fj dV j i j V . = fk fij k − V + − ψI V + 2 f − ij ij ds f 2f 2f 2 In fact, the function Q extends to be smooth at s = 0 because fi V i /f does (since f only vanishes at first order). Therefore we can evaluate dQ/ds at s = 0 by taking the limit. Note that dQ/ds ≤ 0 at s = 0 since Q = 0 at s = 0, while Q ≤ 0 for s ≥ 0. Rewrite fj V j fj V j r fi V i dQ = rfk fij k V i V j − r fk fik V i + |fk |2 ds f 2 f f j i i j fj V dV j dV dV + 2fij V − fi − 2ψV i . ds ds f ds Now we can take the limit as s → 0. Using our chosen value of dV i /ds and our limit of fi V i /f , we get dQ = rfk fij k V i V j ds at s = 0 (after several cancellations!). This proves the lemma since dQ/ds ≤ 0.


303

Now we study the time evolution. Pick a path x(t) for t ≤ t0 with x(t) always in the boundary and with x = x0 at t = t0 . We also pick a path V (t) for t ≤ t0 , with V (t) always tangent to the boundary at the point x(t) at time t for t ≤ T0 and with V = V0 at t = t0 . The variation dx k /dt is constrained by the equation f = 0 on the boundary, so d f =0 dt along the path x(t), which makes ft + fk

dx k = 0. dt

From the equation ft = ffkk + rfk2 = rfk2 on the boundary where f = 0. Thus we need to have k dx fk + rfk = 0, dt and this is the only constraint on dx k /dt. Therefore we choose dx k = −rfk , dt wherein the variation dV k /dt is constrained by the equation fk V k = 0, so d fk V k = 0. dt The vector V (t) along the path x(t) makes fk

dV k dx j k + fkt V k + fj k V = 0. dt dt

From the equation ft = ffkk + rfk2 we get fti = ffikk + fi fkk + 2rfk fik , and on the boundary where f = 0 in the direction V i , where fi V i = 0, we get fti V i = 2rfk fik V i . Using dx k /dt = −rfk , we see that dV k /dt is constrained by

304


fk

dV k i + rfik V = 0, dt

and this is the only constraint. Therefore we choose dV k = −rfik V i . dt Now consider the function Q = fij − ψIij V i V j along the path x(t) in the direction V (t). Since fi V i = 0 along this path, this agrees with our previous quadratic, and we have Q ≤ 0 for t ≤ t0 while Q = 0 at t = t0 . Therefore dQ/dt ≥ 0 at t = t0 . We compute dQ/dt along the path dV i j dx k i j dQ = fij t − ψ Iij V i V j + fij k V V + 2 fij − ψIij V . dt dt dt From the equation fij t = ffij kk + fi fj kk + fj fikk + fij fkk + 2rfk fij k + 2rfik fj k , since f = 0 and fi V i = 0 on our path, we get fij t V i V j = 2rfk fij k V i V j + fkk fij V i V j + 2r fik V i fj k V j . Now use fij V i V j = ψ|V |2 at t = t0 and dx k /dt = −rfk and dV k /dt = −rfik V i and fk fij k V i V j ≤ 0 to compute dQ ≤ − ψ + fkk ψ + 2rψ 2 |V |2 dt (after some cancellations!). If fkk ≤ c and ψ > cψ + 2rψ 2 , we must have dQ/dt < 0, contradicting dQ/dt ≥ 0. The contradiction shows that the quadratic ffj − ψIij V i V j fij − 2f √ must stay strictly less than zero. This proves our claim that f is concave since ψ can be as small as we like. 3. Gradient estimates Assuming throughout this section that f is a solution of the initial value problem


  ∂f = f f + r|Df |2 ∂t f (x, 0) = f 0

305

on Rn × [0, T ], in Rn ,

which is smooth up to the interface for 0 ≤ t ≤ T , we establish certain a priori gradient estimates which constitute, together with Theorem 2.1, the main step in the proofs of Theorems 1.1 and 1.2. We begin with a simple modification of the well-known Aronson-Bénilan inequality. lemma 3.1 If f satisfies f ≥ −K in the distributional sense at t = 0, then f ≥ −

Kσ Kt + σ

for all t > 0, with σ = (m − 1 + 2/n)−1 . In particular, f ≥ −K, for all t > 0. Proof One can observe that the proof of the Aronson-Bénilan inequality f ≥ −

σ t

in [2] can be slightly modified to show that f ≥ −

σ t +τ

for t > 0 provided that f ≥ −σ/τ at t = 0. Setting K = σ/τ , the desired inequality follows. We prove next that if f (·, t) is root-concave, then the upper bound of the gradient of f is preserved under the flow. theorem 3.2 If f (·, t) is root-concave for all t in 0 ≤ t ≤ T and satisfies the gradient bound |Df | ≤ C at t = 0, then |Df | ≤ C for all t in 0 ≤ t ≤ T . We show first the next simple lemma.

306


lemma 3.3 Assume that f (·, t) is root-concave for all t in 0 ≤ t ≤ T . Then for any point P = (x0 , t0 ) at the free boundary of f and any unit vector V tangent to the boundary of the support Ω(t0 ) of f (·, t0 ) at P , we have Dij2 f V i V j ≤ 0

at P .

Proof The result follows by a simple approximation argument. Consider a sequence of points Pk = (xk , t0 ) converging to the point P and such that the sequence ck = f (Pk ) decreases to zero. Let Vk = (Vki ) be a sequence of unit vectors tangent to the level set f (x, t0 ) = ck of f at Pk and such that Vk → V as k → ∞. Since Dij2 f ≤ 0 at Pk , we have Dij2

i j f Vk Vk ≤ 0,

which implies that j

f (Pk )Dij2 f (Pk )Vki Vk ≤

1 j fi (Pk )fj (Pk )Vki Vk , 2

where fi = Dxi f . Since each vector Vk = (Vki ) is tangent to the level set f (xk , t0 ) = ck at Pk , we have j fi (Pk )fj (Pk )Vki Vk = 0 showing the inequality

j

f (Pk )Dij2 f (Pk )Vki Vk ≤ 0.

Because f (Pk ) > 0, we must have j

Dij2 f (Pk )Vki Vk ≤ 0, which implies (3.7). We are now in position to prove Theorem 3.2. Proof of Theorem 3.2 For every ) > 0, we use the maximum principle on fi2 − )t, 2 where fi = Dxi f for i = 1, 2, . . . , n and the summation convention is used. Then we send ) to zero to get the desired estimate. Since X=

ft = ffkk + rfk2 ,


307

we compute

Xt = fi fit = fi ffkk + rfk2 i = fi2 fkk + ffi fikk + 2rfk fi fki − ).

(3.1)

On the other hand, Xk = fi fik and Xkk = fi fikk + fik2 , so that f Xkk = ffikk + ffik2 . Hence, using (3.1), we conclude that Xt = f Xkk − ffik2 + fi2 fkk + 2rfk Xk − ).

(3.2)

Interior estimate At an interior maximum point P of X we must have f Xkk ≤ 0, while Xk = 0 for each k. Hence from (3.2) we deduce that ∂X(P ) ≤ fi2 fkk − ). ∂t Therefore it is enough to show that f = fkk ≤ 0 at an interior maximum point P of X. We can assume by rotating the coordinates that fn > 0

fi = 0,

and

i = 1, . . . , n − 1,

(3.3)

at the point P . Moreover, Xn = fn fnn = 0

at P ,

which implies that fnn = 0

at P .

It remains to show that fkk ≤ 0,

∀k = 1, . . . , n − 1.

But this follows directly from the root-concavity inequality ffij V i V j ≤

1 fi fj V i V j 2

by taking V = (V i ) with V i = δ ik and using (3.3). Boundary estimate Assume now that X attains its maximum at a free boundary point P = (x0 , t0 ), and also assume that

308


fn > 0

fi = 0,

and

i = 1, . . . , n − 1,

(3.4)

at the point P . Since f (P ) = 0, we conclude by (3.2) and (3.4) that Xt ≤ fn2 fkk + 2rfn Xn − ) at P . Also, since fn > 0 at the maximum point P of X, we must have Xn ≤ 0 concluding that We show that fkk

at P ,

∂X ≤ fn2 fkk − ) at P . ∂t ≤ 0. Indeed, by (3.4) we have

(3.5)

Xn = fi fin = fn fnn at P . Since Xn ≤ 0 at P , we conclude that fnn ≤ 0 at P . It remains to show that fii ≤ 0,

i = 1, . . . , n − 1.

But this follows immediately from Lemma 3.3 since each of the unit vectors Vk = (δik ), k = 1, . . . , n − 1, are tangent of the boundary of the set Ω(t0 ) = {f (·, t0 ) > 0} at P . lemma 3.4 Assume that at t = 0 the function f satisfies the nondegeneracy condition αft + f ≥ c > 0

on f > 0

(3.6)

and the lower bound on the Laplacian f ≥ −K for some positive constants c, α, and K. Then, for all t in 0 ≤ t ≤ T , f satisfies (α + t)ft + f ≥ ce−Kt

on f > 0.

Proof Set F = (t + α)ft + f. We prove, using the maximum principle, that F ≥ ce−Kt

on {f > 0}

provided that F ≥ c on {f > 0} at time t = 0. By a direct computation we find that F evolves by Ft = f F + 2rDf · DF + f · F


309

on {f > 0}. Since f ≥ −K at t = 0, by Lemma 3.1, f ≥ −K for t > 0. Hence for F ≥ 0 we obtain Ft ≥ f F + 2rDf · DF − KF, and therefore, for ) > 0, F˜ = F eKt + )t satisfies the differential inequality F˜t ≥ f F˜ + 2rDf · D F˜ + ),

(3.7)

with r = 1/(m − 1) > 0. It is clear by (3.7) that the minimum of F˜ cannot be attained in the interior of the set {f > 0}. Let P be a free boundary point where F˜ is minimum. By rotating the coordinates we can assume that at the point P , Dn f > 0, while Di f = 0 for all i = 1, . . . , n − 1. Then, since F˜ achieves local minimum at P , we must have Dn F˜ ≥ 0 at P . Therefore Df · D F˜ ≥ 0 at P , and hence by (3.7) we obtain F˜t ≥ ) at P , which is impossible since F˜t ≤ 0 at P . We conclude that F˜ ≥ c on {f > 0} for t > 0 provided that F˜ ≥ c on {f > 0} at time t = 0, showing the desired result. corollary 3.5 Assume that at time t = 0 the initial pressure f is root-concave and satisfies the nondegeneracy condition f + |Df |2 ≥ c > 0

on f > 0

and the lower bound on the Laplacian f ≥ −K for some positive constants c and K. Then, for t > 0, f satisfies the nondegeneracy estimate 1 1 r+ |Df |2 ≥ ce−Kt on f > 0, f+ t+ K +r 2 with r = 1/(m − 1). Proof Notice first that by Lemma 3.1 the lower bound on the Laplacian f ≥ −K holds for all t > 0. We apply Lemma 3.4 to F = (t + α)ft + f , with α = 1/(K + r). At t = 0 we have F = αft + f = αf f + αr|Df |2 + f, and therefore, since f ≥ −K and α = 1/(K + r), we have

310


r f + |Df |2 . K +r The nondegeneracy estimate on f implies that at t = 0, cr F ≥ K +r on {f > 0}. Hence, by Lemma 3.4 we have cr −Kt F ≥ e K +r for t > 0 on {f > 0}, which implies that cr −Kt (t + α)f f + r(t + α)|Df |2 + f ≥ e K +r on {f > 0}. On the other hand, we have F ≥ (1 − αK)f + αr|Df |2 =

(3.8)

1 |Df |2 (3.9) 2 since f (·, t) is root-concave for t > 0, as shown in Theorem 2.1. Combining (3.8) and (3.9), we finally obtain 1 cr −Kt f + (t + α) r + |Df |2 ≥ e 2 K +r f f ≤

on {f > 0}, as desired. An immediate consequence of Theorem 3.2 and Corollary 3.5 is the following, for our purposes, important result. theorem 3.6 Assume that at time t = 0 the function f is root-concave and satisfies the upper gradient bound |Df | ≤ C, the nondegeneracy estimate f + |Df |2 ≥ c > 0

on {f > 0},

and the lower bound on the Laplacian f ≥ −K for some positive constants C, c, and K. Then, given a free boundary point P = P (x0 , t0 ), 0 ≤ t0 ≤ T , there exist positive constants d0 and c0 depending only on C, c, K, and T such that for all x in Ω(t0 ) = {f (·, t0 ) > 0} with d(x, x0 ) < d0 we have Df (x, t0 ) ≥ c0 > 0.


311

Let us denote by Bρ (x0 ) the ball centered at x0 of radius ρ. Based on Theorem 3.6, we now prove the main result in this section. theorem 3.7 Assume that at time t = 0 the function f is root-concave and satisfies the upper gradient bound |Df | ≤ C, the nondegeneracy estimate f + |Df |2 ≥ c > 0

on {f > 0},

and the lower bound on the Laplacian f ≥ −K for some positive constants C, c, and K. Then there exist positive constants ρ and c0 depending only on C, c, K, T , and the shape of the initial support such that, given a free boundary point P = P (x0 , t0 ), ρ ≤ t0 ≤ T , there exists a unit direction ν = νx0 depending only on x0 such that Df (x, t) · ν ≥ c0 > 0 for all x ∈ Ω(t) ∩ Bρ (x0 ), t0 − ρ ≤ t ≤ t0 . Since by assumption f is smooth up to the interface at t = 0, the initial support Ω = x ∈ Rn : f (x, 0) > 0 is a domain with smooth boundary. Moreover, since f is root concave, the domain Ω is strictly convex. Let us also assume without loss of generality that B1 (0) ⊂ Ω. For each x = (r, θ ) ∈ Rn , the half-line {(λr, θ); λ > 0} intersects the boundary of ¯ Let us denote by νθ¯ the convex domain Ω at a unique point, which we denote by θ. the exterior unit normal to Ω at the point θ¯ ∈ ∂Ω. We define the vector field νx by νx = νθ¯ . Let us denote by ∠(v, w) the angle between two vectors v and w. The proof of Theorem 3.7 is an immediate consequence of Theorem 3.6 and the next lemma. lemma 3.8 There exist positive numbers ρ and η depending only on the initial data f (·, 0) such that if P = (x0 , t0 ) is a free boundary point of f with 2ρ ≤ t0 ≤ T , then cos ∠ nx (t), νx0 ≥ η for all x ∈ Bρ (x0 ) ∩ Ω(t), t0 − ρ ≤ t ≤ t0 , where nx (t) denotes the outer unit normal

312


vector to the level set Ω(t, x) = y ∈ Rn : f (y, t) ≥ f (x, t) . Let us denote as before by Ω the initial support Ω = {x ∈ Rn : f (x, 0) > 0}. Since the solution f is smooth up to the interface on 0 ≤ t ≤ T and f (·, 0) satisfies the nondegeneracy condition f + |Df |2 ≥ c > 0, given a small number ρ > 0, there exists a number δ > 0 such that Ω(ρ) = x ∈ Rn : f (x, ρ) > 0 ⊃ (1 + δ)Ω = (1 + δ)x : x ∈ Ω . The proof of Lemma 3.8 is based on the following geometric result along the lines of [3, Lemma 2.2]. proposition 3.9 Assume that f (·, t) is root-concave on its support for all 0 ≤ t ≤ T , and let x0 , x1 be two distinct points in Rn \ (1 + δ)Ω, where Ω denotes the initial support, such that |x0 |, |x1 | ≤ R. Then there exists a constant α0 = α0 (R, δ) with 0 < α0 < 1 such that if (3.10) cos ∠ x1 − x0 , νx0 ≥ α0 , then f (x1 , t) ≤ f (x0 , t),

t ≥ 0.

Proof Since the domain Ω is convex, one can easily observe that there exists a constant c0 ∈ (0, 1) depending on R and the shape of the domain Ω such that if cos ∠ x1 − x0 , νx0 ≥ α0 , then Ω lies on one side of the hyperplane bisecting the line segment x0 x1 vertically. Let us assume for simplicity that the vertical bisector is the line xn = λ and that Ω ⊂ {x : xn < λ}. Define the function f˜ on {x : xn < λ}, t > 0, by f˜(x, t) = f x , 2λ − xn , t , where we use the notation x = (x , xn ). We show by the comparison principle that at time t > 0 we have f˜ ≤ f on {x : xn < λ}. Indeed, both f˜ and f are solutions of equation ft = f f + r|Df |2 . Moreover, they coincide at xn = a, while at t = 0 and for x ∈ {x : xn < λ} we have


313

f˜(x, 0) = f x , 2λ − xn , 0 = 0 ≤ f (x, 0) since (x , 2λ − xn ) lies outside the support Ω of f . Since both f and f˜ are compactly supported, from the standard comparison principle we deduce that for t > 0, f˜(x, t) ≤ f (x, t) on {x : xn < λ}, implying the desired inequality f (x1 , t) = f˜(x0 , t) ≤ f (x0 , t). We now present the proof of Lemma 3.8. Proof of Lemma 3.8 Fix a point P = (x0 , t0 ) on the free boundary of f , and denote by Ω(t) the set Ω(t) = x ∈ Rn : f (x, t) > 0 . Let x ∈ Bρ (x0 ) ∩ Ω(t), t0 − ρ ≤ t ≤ t0 , where ρ is a small number to be determined later. Define the cones Cx1 = y ∈ Bρ (x0 ) : ∠(y − x, −νx ) ≤ δ0 and

Cx2 = y ∈ Bρ (x0 ) : ∠(y − x, νx ) ≤ δ0

where δ0 ∈ (0, 1) is a constant sufficiently small, so that by Proposition 3.9 we have (3.11) Cx1 ⊂ y : f (y, t) ≥ f (x, t) and

Cx2 ⊂ y : f (y, t) ≤ f (x, t)

(3.12)

for all ρ ≤ t0 − ρ ≤ t ≤ t0 . Since the vector field νx is smooth, we can choose ρ > 0 depending only on the initial data such that δ0 (3.13) ∠ νx , νx0 < 2 for all x ∈ Bρ (x0 ). Thus combining (3.11)–(3.13), we obtain that

δ0 C˜ x1 ≡ y ∈ Bρ (x0 ) : ∠ y − x, −νx0 ≤ ⊂ y : f (y, t) ≥ f (x, t) 2 and

δ0 2 ˜ ⊂ y : f (y, t) ≤ f (x, t) Cx ≡ y ∈ Bρ (x0 ) : ∠ y − x, νx0 ≤ 2

for all t0 − ρ ≤ t ≤ t0 . This in particular implies that

314


π δ0 ∠ nx (t), νx0 ≥ − , 2 2

∀x ∈ Bρ (x0 ) ∩ Ω(t),

t0 − ρ ≤ t ≤ t0 ,

where nx (t) denotes the outer unit normal vector to the level set Ω(t, x) = y ∈ Rn : f (y, t) ≥ f (x, t) . Hence there exists a positive number η such that cos ∠ nx (t), νx0 ≥ η for all x ∈ Bρ (x0 ) ∩ Ω(t), t0 − ρ ≤ t ≤ t0 , showing the desired result. 4. Local coordinate change and preliminary results Let us assume in this section that f is a solution of the initial value problem   ∂f = f f + r|Df |2 , (x, t) ∈ Rn × [0, T ], ∂t f (x, 0) = f 0 , x ∈ Rn ,

(4.1)

with r = 1/(m − 1), where f 0 is a nonnegative and compactly supported function which satisfies the hypotheses of Theorem 1.1. As in Section 3, we denote by Ω(t), 0 ≤ t ≤ T , the set Ω(t) = x ∈ Rn : f (x, t) > 0 with Ω = Ω(0). Also for 0 < τ < T , let us denote by Ωτ the set Ωτ = (x, t) ∈ Rn × (0, τ ) : f (x, t) > 0 = ∪ Ω(t) × {t} 0 0 for which Dn f (P ) ≥ c > 0, where

∀P ∈ Aδ (P0 ),

(4.2)


Aδ (P0 ) = (x, t) : x ∈ Ω(t) ∩ Bδ (x0 )

315

t0 − δ ≤ t ≤ t0 .

Hence we can apply the implicit function theorem to solve the equation z = f x , xn , t , x , xn , t ∈ Aδ (P0 ) with respect to xn , yielding to a function xn = h x , z, t . To simplify the notation, let us introduce the new coordinates yi = xi ,

i = 1, . . . , n − 1,

yn = z,

t = t,

(4.3)

where time is still denoted by t. Denote by R0 the point R0 = (y0 , t0 ) = y0 , 0, t . Then we can choose ρ > 0 sufficiently small, so that the function h(y, t) is well defined in the parabolic cube (4.4) Qρ (R0 ) = |y − y0 | ≤ ρ, 0 ≤ yn ≤ ρ, t0 − ρ 2 ≤ t ≤ t0 . One can show (see [8], [9]) that the function h(y, t) satisfies the equation n−1 2 2 1 + n−1 1 + n−1 2hi i=1 hi i=1 hi ht = yn Rn−1 h − − r hin + h , (4.5) nn hn hn h2n i=1

where for i = 1, . . . , n and j = 1, . . . , n we use the notation hi = Dyi h, hij = Dy2i yj h, and Rn−1 = n−1 i=1 hii . Equation (4.5) can also be expressed in divergence form [9] as n−1 2 (m−2)/(m−1) 1/(m−1) 1 + i=1 hi ht = yn Rn−1 h − yn . (4.6) Dyn yn hn The linearization of equation (4.5) at a point h is n−1 n 1 + n−1 h2i 2h i i=1 ˜ht = yn Rn−1 h˜ − ˜hin + ˜hnn + bi h˜ i hn h2n i=1

with bi = − and

2rhi 2yn hi 2yn + hnn − hin , 2 hn hn hn

(4.7)

i=1

i = 1, . . . , n − 1,

(4.8)

316


bn = r

1+

n−1 i=1 h2n

h2i

n−1 2 2yn 1 + n−1 2yn i=1 hi − hnn + 2 hi hin . h3n hn

(4.9)

i=1

The linearization of the divergence form equation (4.6) at a point h is n−1 (m−2)/(m−1) 1/(m−1) Dyn yn Ai h˜ i , h˜ t = yn Rn−1 h˜ + yn

(4.10)

i=1

with Ai = −

2hi , hn

and An =

i = 1, . . . , n − 1, 1+

n−1 i=1

h2i

. hn It has been shown in [9] that equation (4.10) has the form ut = yn−σ Di yn1+σ a ij Dj u + σ a nj Dj u, where σ > −1 and

(a ij )

(4.11)

satisfies the conditions

2 a ij ξi ξj ≥ min 1, h−2 n |ξ | for all ξ, η ∈ Rn . Hence a ij ξi ξj ≥ λ|ξ |2

2 ij a ξi ηj ≤ 2 1 + |Dh| |ξ ||η| 2 hn

and

and

ij a ξi ηj ≤ λ−1 |ξ ||η|

(4.12)

for some positive constant c provided that hn = Dn h ≥ c > 0

and

|Dh| ≤ c−1 .

It is easy to observe that these bounds are indeed satisfied by h on the cube Qρ (R0 ) = |y − y0 | ≤ ρ, 0 ≤ yn ≤ ρ, t0 − ρ 2 ≤ t ≤ t0 since f ∈ C 1 (Aδ ) and (4.2) holds. We use in Section 5 the following result by H. Koch [9]. theorem 4.1 (Hölder regularity) Assume that u is a solution of equation (4.11) on the cube Qρ = Qρ (R0 ) with σ > −1 and coefficients which satisfy conditions (4.12). Then there exists a number γ > 0 depending only on n, σ , and λ such that u ∈ C γ (Qδ ), with δ = ρ/2, and |u|dµσ , uC γ (Qδ ) ≤ C(n, σ, λ)ρ −γ |Qρ |−1 σ Qρ

where dµσ denotes the measure dµσ = ynσ dy dt and |Qρ (R0 ))|σ =

Qρ (R0 ) dµσ .

We need the following generalization of Theorem 4.1, also proven by Koch in [9].


317

theorem 4.2 (Hölder estimate) Let u be a solution of the equation ut = yn−σ Di yn1+σ a ij Dj u + yn−σ Di ynσ f i

(4.13)

in the cube Qρ = Qρ (R0 ) with σ > −1. Assume that the coefficients a ij satisfy conditions (4.12) and f i ∈ C β (Qρ (R0 )) for some β > 0. Then there exists a number 0 < γ < β, depending only on n, σ , λ, β, for which u ∈ C γ (Qδ ), with δ = ρ/2, and u

C γ (Q

δ)

≤ C(n, σ, λ)ρ

−γ

|Qρ |−1 σ

where dµσ = ynσ dy dt and |Qρ (R0 ))|σ =

Qρ

|u| dµσ +

n

f i C γ (Qρ ) ,

i=1

Qρ (R0 ) dµσ .

It has been shown in [8] that the linearization of the nondivergence form in equation (4.7) is of the form ut = yn a ij uij + bi ui , where the matix (a ij ) satisfies conditions (4.12). In addition, if we assume that yn uij are continuous in Qρ (R0 ), then the coefficients bi given by (4.8) and (4.9) are bounded and in addition, r (4.14) bn ≥ 2 ≥ λ > 0, 2hn provided that ρ and λ are is sufficiently small. Daskalopoulos and Hamilton in [8] showed a Schauder-type estimate for solutions of equation (4.15) ut = yn a ij uij + bi ui + g, where the coefficients (a ij ) and bi satisfy conditions (4.12) and (4.14). Since the equation is degenerate, the Hölder norms need to be scaled according to a singular metric. More precisely, let us consider the half-space H = {yn > 0} and define on H the Riemannian metric n 1 2 dyi . ds 2 = yn i=1

(x , xn )

The distance between two points x = s(x, y) which is equivalent to the function s(x, y) = √

and y = (y , yn ) in H is a function

|x − y | + |xn − yn | . √ xn + yn + |x − y |

For the parabolic problem we use the parabolic distance

318


s((x, t), (y, s)) = s(x, y) +

|t − s|.

As in [8] we denote by Csα the Banach space of all Hölder continuous functions with respect to the distance s, where the Hölder norm is also defined with respect to s. Suppose next that the set A is the closure of its interior, that the function f on A has continuous derivatives ft , Di f , Dij2 f , i, j = 1, . . . , n in the interior of A , that ft , Di f

yn Dij2 f,

and

i, j = 1, . . . , n,

extend continuously to that the boundary, and that the extensions are Hölder continuous on A of class Csα (A ). We define Cs2+α (A ) to be the Banach space of all such functions with norm f C 2+α (A ) = f Csα (A ) + s

n i=1

Di f Csα (A ) +

n yn D 2 f i,j =1

ij

Csα (A )

.

Define the box of side ρ around a point R0 = (y0 , t0 ) to be Br (R0 ) = |yi − y0 i | ≤ ρ, yn ≥ 0, t0 − ρ ≤ t ≤ t0 . theorem 4.3 (Second Schauder estimate) For any α in 0 < α < 1 and ρ > 0, there exists a constant C depending on n, λ, α, and ρ such that for δ = ρ/2 we have uC 2+α (Bδ ) ≤ C f Cs◦ (Bρ ) + gCsα (Bρ ) s

for all solutions u ∈

Cs2+α (Bρ )

of equation (4.15).

The above theorem is proven in [8] in the case of dimension n = 2. The proof of the theorem in dimensions n ≥ 3 is very similar, with the obvious changes. Before we finish this section, we state for the convenience of the reader the short time C ∞ -regularity result proven in [8]. This is used together with Theorems 4.1 and 4.2 in the proof of Theorem 1.1. Let Ω be a domain in Rn . Imitating the case where Ω is the half-space H , we define the distance function s in Ω to coincide with the standard Euclidean distance at the interior of Ω and, around any point P ∈ ∂Ω, to be the pullback of the distance s on the half-space H via a map ϕ : H → Ω that straightens the boundary of Ω near P . The parabolic distance s¯ is defined by s¯ (P1 , t1 ), (P2 , t2 ) = s(P1 , P2 ) + |t1 − t2 |. Similarly to the half-space case we can define the Banach space Csα (Ω), Cs2+α (Ω), as well as the spaces Csα (A ), Cs2+α (A ), for a subset A of Ω × [0, ∞). The following two results are proven in [8].


319

theorem 4.4 Assume that the initial data f 0 ∈ Cs2+α (Ω) and satisfies the nondegeneracy condition 2 (4.16) f 0 + Df 0 ≥ c > 0 for some α > 0 and c > 0. Then there exists a number T > 0 for which the solution f of the initial value problem (4.1) belongs to the space Cs2+α (Ωτ ) for all τ < T . theorem 4.5 Assume that for some T > 0 and some number α in 0 < α < 1, f ∈ Cs2+α (ΩT ) is a solution of the free boundary problem (4.1) satisfying the nondegeneracy condition |Df (x, t)| + f (x, t) ≥ c > 0,

(x, t) ∈ ΩT .

Then f is smooth up to the interface on 0 < t < T , and, in particular, the free boundary T is smooth. Combining the previous two theorems we obtain the following theorem. theorem 4.6 Assume that the initial data f 0 ∈ Cs2+α (Ω) and satisfies the nondegeneracy condition (4.16) for some α > 0 and c > 0. Then there exists a number T > 0 for which the solution f of the initial value problem (4.1) is smooth up to the interface on 0 < t < T , and, in particular, the free boundary T is smooth. 5. All time C ∞ -regularity This section is devoted to the proof of Theorem 1.1. Using the notation of Section 4, we first show the following result. theorem 5.1 Assume that the initial data f 0 is smooth in the closure of its support Ω and that in addition f 0 is root-concave and satisfies the nondegeneracy condition 2 (5.1) f 0 + Df 0 ≥ c > 0 for some c > 0. Then there exists a number β > 0 such that the solution f of the 2+β initial value problem (1.1) belongs to the class Cs (ΩT ) for all 0 < T < ∞. Proof By Theorem 4.5 there exists a maximal time T > 0 for which f is smooth up to the interface on 0 ≤ t < T . Assuming that T < ∞, we show that at time t = T the function f satisfies the nondegeneracy condition

320


f (x, T ) + |Df (x, T )|2 ≥ c(T ) > 0 2+β

and also that f (·, T ) ∈ Cs

(5.2)

(Ω(T )) with f (·, T )C 2+β (Ω(T )) ≤ C s

(5.3)

for some β > 0. Therefore, by Theorem 4.6 there exists a number T > 0 for which 2+β f is in Cs (Ωτ ) for all τ < T + T . Theorem 4.5 then implies that f is smooth up to the interface on 0 < t < T + T , contradicting the fact that T is maximal. It remains to prove conditions (5.2) and (5.3). Condition (5.2) is implied by Corollary 3.5. Observe first that since f is smooth up to the interface for τ < T , we can choose τ0 small enough so that f (·, τ0 ) + |Df (·, τ0 )|2 ≥

c > 0, 2

x ∈ Ω(τ0 ).

The Aronson-Bénilan inequality f ≥ −σ/t implies that at t = τ0 we have f ≥ −K = −

σ . τ0

Hence we can apply Corollary 3.5 to conclude that 1 c 1 r+ |Df |2 ≥ e−K(t−τ0 ) f+ t+ K +r 2 2 on {f > 0} for τ0 ≤ t ≤ T , proving (5.2). We next prove condition (5.3). Let P0 = (x0 , t0 ) be a free boundary point with t0 < T . By Theorem 3.7 there exist positive numbers δ, c0 depending only on n, m, T , and the initial data f 0 , and unit direction ν = νx0 such that Df · ν ≥ c0 > 0, where

∀(x, t) ∈ Aδ (P0 )

(5.4)

Aδ (P0 ) = (x, t) : x ∈ Ω(t) ∩ Bδ (x0 ), t0 − δ ≤ t ≤ t0 .

We can assume without loss of generality that ν = en , so that Dn f ≥ c0 > 0,

∀(x, t) ∈ Aδ (P0 ).

Hence we can perform the local coordinate change (4.3) on Aδ (P0 ) to obtain a function xn = h(y, t) defined on the parabolic cube Qη (R0 ) = |y − y0 | ≤ η, 0 ≤ yn ≤ η, t0 − η2 ≤ t ≤ t0 , with R0 = (y0 , t0 ) = (x0 , 0, t0 ) and η > 0. Notice that since f is continuous on Rn × [t0 − δ, t0 ], we can choose η sufficiently small depending on δ and the modulus


321

of continuity of f such that (x, t) ∈ Aδ (P0 ) if (y, t) ∈ Qη (R0 ). Denoting for any η > 0 the cube Qη (R0 ) by Qη , we show the following lemma. lemma 5.2 There exist numbers γ > 0 and C > 0 depending only on n, m, T , and the initial data f 0 such that the gradient Dh satisfies DhC γ (Qρ ) ≤ C

(5.5)

with 2ρ = η. The significance of this lemma is that the norm DhC γ (Qρ ) remains uniformly bounded as t0 ↑ T . Let us continue with the proof of Theorem 5.1 and leave the proof of the lemma for the end. Observe first that (5.5) implies that DhC β (Q s

ρ)

≤C

for some β < γ . Hence the Schauder estimate of Theorem 4.3 applied to equation 2+β (4.5) implies that h ∈ Cs (Qρ/2 ), with hC 2+β (Q s

ρ/2 )

≤ C.

Since C remains uniformly bounded as t ↑ T , we can go back to the original coordinates to finally conclude that f C 2+β (Ω(T )) ≤ C, s

finishing the proof of Theorem 5.1. Before we prove Lemma 5.2, we show the next simple lemma. lemma 5.3 Under the hypotheses of Lemma 5.2, there exists a constant c > 0 depending only on n, m, T , and the initial data f 0 such that |Dh| ≤ c−1 and Dyn h ≥ c > 0 in Qη = Qη (R0 ). Proof One can easily compute that

322


Dy i h = −

Dxi f , Dx n f

Dyn h =

1 Dxn f

.

Since |Df | ≤ C and Dxn f ≥ c0 > 0 in Aδ (P0 ), the lemma follows. We now prove Lemma 5.2. Proof of Lemma 5.2. We show that the derivatives Dyi h belong to the Hölder class C γ (Qρ ), with ρ = δ/2, by differentiating equation (4.6) with respect to yi and applying Theorem 4.1. Let us first show the conclusion of the lemma for the derivatives u = Dyi h = hi , i = 1, . . . , n − 1. Differentiating (4.6) with respect to yi , we find that u = Dyi h satisfies the equation ut = yn Rn−1 u + yn−σ Dyn yn1+σ Ai ui , with σ = (2 − m)/(m − 1), and 2 n−1 i=1 hi Ai = − , i = 1, . . . , n − 1, (5.6) hn and 2 1 + n−1 i=1 hi An = , (5.7) hn where to simplify the notation we denote hi = Dyi h. It follows from (4.12) and Lemma 5.3 that the coefficients Ai , i = 1, . . . , n, satisfy the hypotheses of Theorem 4.1. Hence −γ −1 uC γ (Qρ ) ≤ C(n, σ, λ)ρ |Qδ |σ |u|ynσ dy dt. Qδ

Since σ > −1, the last estimate in combination with Lemma 5.3 implies that Di hC γ (Qρ ) ≤ C,

i = 1, . . . , n − 1,

(5.8)

with C depending only on n, σ , T , and f 0 . It remains to prove the same estimate for the derivative u = Dyn u. Differentiating (4.6) with respect to yn , we find that u = Dyn h = hn satisfies the equation ut = yn Rn−1 u + yn−(1+σ ) Dyn yn2+σ Ai ui + Rn−1 h, (5.9) where the Ai , i = 1, . . . , n, are given by (5.6) and (5.7). To verify this let us denote by


Q=−

1+

n−1 i=1

hn

323

h2i

and by F = Dyn Q = Ai ui where u = Dyn h = hn and the summation convention is used. Indeed, under this notation (4.6) can be simply written as ht = yn Rn−1 h + yn−σ Dyn yn1+σ Q . Differentiating the last equation with respect to yn , we obtain ut = yn Rn−1 u + Dyn yn−σ Dyn yn1+σ Q + Rn−1 h, which results, after some calculations, to the equation ut = yn Rn−1 u + (2 + σ )F + yn Dyn F + Rn−1 h. This can be rewritten as ut = yn Rn−1 u + yn−(1+σ ) Dyn yn2+σ F + Rn−1 h where F = Dyn Q = Ai ui , yielding to (5.9). We observe next that equation (5.9) is of the form of equation (4.13), with f i = Di h, i = 1, . . . , n − 1, f n = 0, and coefficients (a ij ) which satisfy condition (4.12) because of the bounds of Lemma 5.3. Hence by Theorem 4.2 and (5.8) we obtain Dn hC γ (Qρ ) ≤ C, with C depending only on n, σ , T , and f 0 , finishing the proof of the lemma. 6. Less regular initial data In this section we show the proof of Theorem 1.2. Let f be a weak solution ofthe initial value problem (1.1) with continuous initial data f 0 . We assume that φ = f 0 is weakly concave on its support Ω, namely, that it satisfies the inequality φ(x) + φ(y) x+y −φ ≤ 0, ∀x, y ∈ Ω. (6.1) 2 2 Denoting by d(x) the distance function from the boundary of Ω, we first show the following result. theorem 6.1 If f 0 is continuous and strictly positive on the compact domain Ω with f 0 = 0 at

324

∂Ω, and if in addition condition


f 0 is weakly concave in Ω and satisfies the nondegeneracy f 0 (x) ≥ cd(x)

for some constant c > 0, then the weak solution f of the initial value problem (1.1) √ has f (·, t) weakly concave for all 0 < t < ∞. Proof Let us approximate the initial data f 0 by a sequence of functions fk0 , so that each fk0 is supported on a compact domain Ωk , fk0 ∈ C ∞ (Ω¯ k ), and satisfies the estimates Dij2 fk0 ≤ 0 on Ωk (6.2) and

fk0 (x) + Dfk0 ≥ c˜

on Ωk

(6.3)

for some constant c˜ > 0. It is easy to observe that such an approximation is possible since f 0 is weakly concave on Ω and satisfies (6.1). Moreover, {f0k } can be chosen so that fk0 → f 0 uniformly on Rn . Let fk be the unique weak solution of the initial value problem (1.1) with initial data fk0 . Since fk0 satisfies conditions (6.2) and (6.3), it follows from Theorem 1.1 that the solution fk is smooth up to the interface for 0 ≤ t < ∞ and, moreover, that fk (·, t) is root-concave for all t > 0. In particular, √ each fk (·, t) satisfies inequality (6.1), namely, √ √ fk (x, t) + fk (y, t) x+y − fk ,t ≤ 0 (6.4) 2 2 for all x, y in its support. Since the sequence of solutions fk is uniformly bounded, it is equicontinuous on compact subsets of Rn × (0, ∞) by the result in [10]. Therefore one can show by standard arguments that f k converges uniformly on compact subsets of Rn × (0, ∞) to the solution f of (1.1). By taking the limit k → ∞ in (6.4), we √ obtain that f (·, t) is weakly concave, finishing the proof of the theorem. We assume next that f 0 belongs to the weighted Hölder space Cs2+α (Ω), as defined in Section 4. We show the following theorem. theorem 6.2 Assume that the function f 0 ∈ Cs2+α (Ω) is root-concave in Ω and satisfies the nondegeneracy condition 2 f 0 + Df 0 ≥ c > 0 for some c > 0. Then the solution f of the initial value problem (1.1) is a smooth function smooth up to the interface , and f (·, t) is root-concave for all 0 < t < ∞. In particular, the free boundary is a smooth surface on t > 0.


325

Proof The proof of this theorem follows quite immediately by combining the short time regularity results in [8, Theorems 4.6 and 4.7] with Theorems 1.1 and 6.1. By [8, Theorems 4.6 and 4.7], there exists a number T > 0 for which f is C ∞ -smooth up to the interface on 0 < t < τ , and also f ∈ C 2+α (Ωτ ) for all 0 < τ < T , where Ωτ = {(x, t) ∈ Rn × [0, τ ] : f (x, t) > 0}. Therefore there exists a number 0 < τ0 < T such that f (·, τ0 ) is smooth on the closure of its support and satisfies the nondegeneracy condition f (·, τ0 ) + |Df (·, τ0 )|2 ≥ c0 > 0 with c0 = c/2. In addition, by Theorem 6.1 the function f (·, τ0 ) is root-concave on its support. Hence we can apply Theorem 1.1 to conclude that f must remain smooth up to the interface for all 0 < t < T , proving the desired result. We finish the paper with the proof of Theorem 1.2. Proof of Theorem 1.2 Let us approximate f 0 by a sequence of functions fk0 which are compactly supported, smooth on the closure of their support Ω¯ k , and satisfy the gradient estimate 0 Df ≤ C on Rn , k the nondegeneracy estimate 2 fk0 + Dfk0 ≥ c

on Ωk ,

the lower bound on the Laplacian fk0 ≥ −K

on Rn

in the distributional sense, and the root concavity estimate Dij2 fk0 ≤ 0 on Ωk for some positive constants c > 0 and C > 0 which are independent of k. We can choose such a sequence fk0 so that f0k → fk uniformly on Rn . According to Theorem 1.1, for each k ∈ N the solution fk of equation (1.1) with initial data fk0 is smooth up to the interface. In addition, each fk (·, t) is root-concave on its support for all t > 0 and satisfies the following estimates proven in Section 4: |Dfk | ≤ C

on Rn

326


and

on Rn

fk ≥ −K

in the distributional sense and 1 1 r+ |Dfk |2 ≥ ce−Kt fk + t + K +r 2

on Ωk .

Let P = (x0 , t0 ) be a point on the interface of f with t0 > 0. Denoting by Qδ (x0 , t0 ) the cylinder Qδ (x0 , t0 ) = |x − x0 | ≤ δ, t0 − δ 2 ≤ t ≤ t0 , one can show that the arguments of the proof of Theorem 3.7 imply the following claim. claim There exist numbers δ > 0, c0 > 0, and a unit direction νx0 such that Dfk · νx0 ≥ c0

on Qδ (x0 , t0 ) ∩ {fk > 0}

for all k sufficiently large. Let us assume without loss of generality that the unit direction νx0 in the above claim is the unit vector νx0 = en parallel to the xn -axis. Then we can perform the local coordinate change (4.3) and apply Theorem 4.1 as in the proof of Lemma 5.2 to conclude that Dfk C γ (Qδ,k ) ≤ C for some γ > 0, with Qδ,k = Qδ (x0 , t0 ) ∩ {fk > 0}, and that C is independent of k. Therefore, using the Schauder estimate Theorem 4.3, we conclude that fk C 2+β (Q s

δ,k )

≤C

(6.5)

for some β < γ , where again C is independent of k. On the other hand, since fk0 → f 0 uniformly, one can show by standard arguments (see the proof of Theorem 6.1) that fk → f uniformly on compact sets of Rn × (0, ∞). In particular, fk → f on ˜ δ = Qδ (x0 , t0 ) ∩ {f > 0}. By (6.5) we have Q f C 2+β (Q˜ ) ≤ C, s

δ

2+β ˜ which implies that f ∈ Cs (Q δ ). Hence by Theorem 4.5 f is smooth up to the interface, showing the desired result.


327

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symmetric porous medium equation, Comm. Partial Differential Equations 20 (1995), 1217–1240. MR 96c:35074 ´ D. G. ARONSON and P. BENILAN , Régularité des solutions de l’équation des milieux n poreux dans R , C. R. Acad. Sci. Paris Ser. A-B 288 (1979), 103–105. MR 82i:35090 D. G. ARONSON and L. A. CAFFARELLI, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), 351–366. MR 85c:35042 L. A. CAFFARELLI and A. FRIEDMAN, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), 361–391. MR 82a:35096 ´ L. A. CAFFARELLI, J. L. VAZQUEZ and N. I. WOLANSKI, Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation, Indiana Univ. Math. J. 36 (1987), 373–401. MR 88k:35221 L. A. CAFFARELLI and N. I. WOLANSKI, C 1,α regularity of the free boundary for the N-dimensional porous media equation, Comm. Pure Appl. Math. 43 (1990), 885–902. MR 91h:35332 P. DASKALOPOULOS and R. HAMILTON, The free boundary for the n-dimensional porous medium equation, Internat. Math. Res. Notices 1997, 817–831. MR 98m:35227 , Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc. 11 (1998), 899–965. MR 99d:35182 H. KOCH, Non-Euclidean singular integrals and the porous medium equation, University of Heidelberg, Habilitation thesis, 1999, http://www.iwr.uniheidelberg.de/groups/amj/koch.html P. SACKS, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), 387–409. MR 84d:35081

Daskalopoulos and Lee Department of Mathematics, University of California at Irvine, Irvine, California 92697-3875, USA Hamilton Department of Mathematics, Columbia University, New York, New York 10027, USA

ON THE QUANTUM COHOMOLOGY OF A SYMMETRIC PRODUCT OF AN ALGEBRAIC CURVE AARON BERTRAM and MICHAEL THADDEUS

Abstract The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example, as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [(3/4)g, g − 1). Otherwise, they still give a formula modulo third-order terms. Explicit generators and relations are also given unless d is in [(4/5)g − 3/5, g − 1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically, a formula of J. Harris and L. Tu for the Chern numbers of determinantal varieties. The case of d = g − 1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with A. Givental’s work is also explained. Quantum cohomology is a novel multiplication on the cohomology of a smooth complex projective variety, or even a compact symplectic manifold. It can be regarded as a deformation of the ordinary cup product, defined in terms of the Gromov-Witten invariants of the manifold. Since its introduction in 1991, there has been enormous interest in computing quantum cohomology for various target spaces. Attention has focused on homogeneous spaces, complete intersections, surfaces, and, of course, on Calabi-Yau 3-folds, where it is a key part of mirror symmetry. However, to the authors’ knowledge, no one has yet studied the quantum cohomology of a symmetric product of a smooth curve. This is strange because the problem is attractive from several points of view. First, though quantum cohomology is clearly a fundamental invariant of a variety DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 2, Received 23 February 2000. Revision received 28 July 2000. 2000 Mathematics Subject Classification. Primary 14N35, 14H51. Bertram’s work supported by National Science Foundation grant number DMS-95-00865 Thaddeus’s work supported by National Science Foundation grant number DMS-95-00964

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or a symplectic manifold, it has been explicitly computed in relatively few cases: just some of those mentioned above. A more specific motivation comes from a link with Seiberg-Witten theory. For a complex curve C, the Seiberg-Witten-Floer cohomology of the real 3-manifold C ×S 1 is isomorphic to the cohomology of the dth symmetric product Cd (see [7], [24], [25]), and it is expected to carry a natural product that corresponds to the quantum product (see [6], [22], [26]). The index d depends on the spin-c structure on C × S 1 chosen in Seiberg-Witten theory. In the ordinary Seiberg-Witten theory, this isomorphism only holds when d < g − 1, where g is the genus of C. However, higher symmetric products enter the picture if the Seiberg-Witten functional is perturbed in the manner of C. Taubes [30] and of J. Morgan, Z. Szabó, and Taubes [24]. The quantum cohomology of a symmetric product is therefore the base ring of the “quantum category” in Seiberg-Witten theory, as introduced by S. Donaldson (see [23, §10.4]). On the other hand, there is also a link with Brill-Noether theory and the stratification of Cd by the special linear series. This makes it feasible to compute the quantum cohomology using algebraic geometry. To carry out the necessary enumerative computations on the strata, crucial use is made of a generalization of Porteous’s formula, due to Harris and Tu [16]. There are several kinds of quantum cohomology; this paper is concerned only with the “quantum cohomology algebra” (see [12]) or “little quantum cohomology” (see [10]) of Cd . This is a ring generated by the rational cohomology together with one deformation parameter for each generator of H2 (Cd ; Z). Its structure constants are given by 3-point Gromov-Witten invariants. Not quite all of these invariants are calculated herein, but in a sense most of them are. More precisely, the following results are proved. • The number of possible deformation parameters is the second Betti number of Cd , which is fairly large. Nevertheless, the quantum product is shown to depend nontrivially only on a single parameter q (see Proposition 2.4). • Explicit formulas are given for the coefficients of q (see Corollary 4.3) and q 2 (see Corollary 5.4) in the quantum product. • All the terms in the quantum product are computed for Cg−1 (see Corollary 6.2(ii)). • The coefficient of q e in the quantum product for Cd is shown to vanish if d < g − 1 and e > (d − 3)/(g − 1 − d) (see Corollary 2.8), or if d > g − 1 and e > 1 (see Corollary 6.2(i)). Schematically, the coefficients vanish in the regions marked A and B in Figure 1. A follows from a straightforward dimension count; B is more subtle. Only in the regions under the hyperbola, or under the line e = 1, are there nonzero coefficients. Putting these results together completely determines the quantum product on Cd in all cases except d ∈ [(3/4)g, g − 1).

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QUANTUM COHOMOLOGY OF A SYMMETRIC PRODUCT

e

A

B

1 g−1

d

Figure 1 Instead of seeking to characterize the quantum product completely, one can alternatively ask just for generators and relations for the quantum ring. This is a weaker question because the quantum relations do not determine the additive isomorphism between the quantum and classical rings. It is answered in all cases except d ∈ [(4/5)g − 3/5, g − 1), even using only the first order terms (see Proposition 7.2 and the discussion following). In some cases, one encounters the curious fact that the quantum and classical rings are isomorphic but only by an automorphism that is not the identity. The organization of the paper is straightforward. Section 1 recalls some basic facts on quantum cohomology. Section 2 recalls some basic facts on symmetric products of a curve, shows why only one deformation parameter is involved, and explains the vanishing in region A. Section 3 introduces the Brill-Noether methods that are used to compute the 3-point Gromov-Witten invariants. In particular, the result of Harris and Tu mentioned above is reviewed. Using this, the degree-1 invariants are computed in Section 4, and the degree-2 invariants are computed in Section 5. Section 6 is devoted to the vanishing of higher-degree invariants in region B and to the computation for d = g − 1. Section 7 explains how to find generators and relations for the quantum ring. Finally, Sections 8 and 9 are essentially appendices: Section 8 outlines a remarkable connection with Givental’s work on the rational curves on a quintic 3-fold, while Section 9 is concerned with the first two homotopy groups of Cd , explaining Proposition 2.4 from the point of view of symplectic topology. We note a few conventions. Cohomology of a space is with rational coefficients unless otherwise mentioned, and cohomology of a sheaf is over the curve C unless otherwise mentioned. If n! appears in the denominator of some expression and n < 0, this means that the whole expression vanishes, that is, 1/n! = 0 for n < 0.

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1. Preliminaries on quantum cohomology Let X be a smooth complex projective variety with fixed polarization. E. Witten [32] introduced two rings associated to X, the big and little quantum cohomology. The big quantum cohomology was used by M. Kontsevich [18] to count rational curves in the plane. However, there are very few spaces for which it has been characterized in full. Most work, including that of Givental and B. Kim [12], [13] and the whole of the present paper, is concerned with the little quantum cohomology. Definition 1.1 A stable map is a morphism φ from a complete nodal curve to X such that {ψ ∈ Aut | φψ = φ} is finite. An n-pointed stable map is similar, except that n distinct smooth points x1 , . . . , xn ∈ are also chosen, and the automorphisms ψ are required to fix each xi . The arithmetic genus g = pa () and the homology class e = φ∗ [] ∈ H2 (X; Z) are discrete invariants of a stable map. When these are fixed, a fundamental theorem asserts the existence of projective moduli spaces M g (X, e) of stable maps and M g,n (X, e) of n-pointed stable maps, as well as a forgetful morphism fe : M g,n (X, e) → M g (X, e) and an evaluation morphism eve : M g,n (X, e) → Xn (see W. Fulton and R. Pandharipande [10] for details). The subscript e is suppressed when there is no danger of confusion. Also, this paper is concerned with only the case g = 0, n = 1. It is most accurate to regard these moduli spaces as stacks rather than schemes. They are stratified by smooth substacks on which the dimension of the deformation space is constant. Since each stable map has finitely many automorphisms, these strata are Deligne-Mumford stacks. This paper, however, adopts a more naïve point of view, regarding the moduli spaces as schemes stratified by subschemes with a natural orbifold structure. As pointed out by the referee, this is a bit tricky since smooth Deligne-Mumford stacks need not be orbifolds in the optimal algebraic sense of being locally a quotient of a smooth affine scheme by a finite group. So, strictly speaking, the word orbifold should henceforth be taken to mean a smooth Deligne-Mumford stack, and an orbifold vector bundle should mean a vector bundle over such a stack. Fortunately, in the only place where an explicit calculation with orbifolds is carried out, namely, the proof of Theorem 5.2, the space in question is an orbifold in the optimal sense mentioned above. The moduli space M 0 (X, e) has expected dimension dim X − 3 + c1 (T X) · e. It is not generally of this dimension or even equidimensional. But it is endowed with a natural virtual fundamental class, an equivalence class of algebraic cycles


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[M 0 (X, e)]vir ∈ A∗ (M 0 (X, e)) having the expected dimension. The general construction of this virtual class involves the deformation theory of stable maps and is rather complicated. It has been carried out by several authors (see [4], [19], [29]). We use only the following three basic facts. proposition 1.2 (i) If TX1 (φ), TX2 (φ) are the first-order deformation and obstruction spaces of the map φ, and if T 0 (), T 1 () are the first-order endomorphism and deformation spaces of the curve , then there is a natural exact sequence 0 −→ T 0 () −→ H 0 , φ ∗ T X −→ TX1 (φ) −→ T 1 () −→ H 1 , φ ∗ T X −→ TX2 (φ) −→ 0. (ii) On any reduced locus where the first-order obstruction spaces of the map have constant dimension, the virtual class is the Euler class of the orbifold vector bundle formed by these spaces. In particular, if the first-order obstructions vanish, then the virtual class is simply the orbifold fundamental class. (iii) The forgetful morphism f is flat, and vir vir M 0,n (X, e) = f ∗ M 0 (X, e) . Proof See the work of Z. Ran [27], J. Li and G. Tian [19], K. Behrend [3], Behrend and B. Fantechi [4], and Behrend and Yu. Manin [5]. Example 1.3 If X = Pr , then, from the long exact sequence of 0 −→ O −→ O (1)r+1 −→ T Pr −→ 0, it follows that H 1 (, φ ∗ T Pr ) = 0 for a curve of arithmetic genus zero. Hence the obstructions vanish and M 0 (Pr , e) is an orbifold of the expected dimension r − 3 + e(r + 1). Definition 1.4 For a1 , . . . , an ∈ H ∗ (X; Q), the n-point degree-e Gromov-Witten invariant is defined as vir ∗ ∗ πi ai M 0,n (X, e) , a1 , . . . , an e = eve i

where πi :

Xn

→ X is the projection on the ith factor.

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In practice, we always evaluate Gromov-Witten invariants using the following equivalent formulation. Suppose that M˜ α are a collection of orbifold resolutions of the closures of the smooth strata Mα of M 0 (X, e) such that M˜ α ×M 0 (X,e) M 0,1 (X, e) are also orbifold resolutions of the strata of M 0,1 (X, e). Let f˜α and evα be the liftings of fe and eve to these fibered products. Choose a cycle representing the virtual class of M 0 (X, e), and let [M˜ α ]vir be the cycle whose projection to the closure M α consists of all components of the virtual cycle supported in M α but not in the closure of any smaller stratum. The rational equivalence classes of the individual cycles [M˜ α ]vir might depend on the choice of the representative, but this does not impair our arguments. proposition 1.5 With the above notation, a1 , . . . , an e =

α

i

f˜α ∗ ev∗α ai

M˜ α

vir

.

In particular, if M 0 (X, e) is an orbifold of the expected dimension, ∗ a1 , . . . , an e = f∗ ev ai M 0 (X, e) . i

Proof First of all, the forgetful morphism fe factors through a birational morphism to the n

nth fibered power M 0,1 (X, e)M 0 (X,e) , which is also flat over M 0 (X, e). An evaluation map to Xn is still defined on this space, so it suffices to perform the computation here. It is convenient to denote the disjoint union of the M˜ α by M˜ 0 , its fibered product over M 0 (X, e) with M 0,1 (X, e) by M˜ 0,1 , and the forgetful and evaluation maps on M˜ 0,1 by f˜ and ev. Then the proposition follows immediately from Proposition 1.2(iii) and the fact that the diagram n ev n M˜ 0,1 M˜ M˜ 0,1 Xn 0 f˜

M˜ 0

f˜

M˜ 0n

is a fiber square, where the top and bottom of the square are diagonal embeddings. A fundamental result states that Gromov-Witten invariants are deformation invariants (see [4], [8], [19]). Also, they satisfy the following properties.


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lemma 1.6 (i) Gromov-Witten invariants are symmetric on classes of even degree, antisymmetric on classes of odd degree. (ii) If e = 0 and a1 ∈ H 0 or H 1 , then a1 , . . . , an e = 0. (iii) If e = 0 and a1 ∈ H 2 , then a1 , . . . , an e = (a1 · e)a2 , . . . , an e . Proof See, for example, Behrend [3] and Behrend-Manin [5]. The quantum product has coefficients in the following ring. Definition 1.7 The Novikov ring is the subring of Q[[H2 (X; Z)]] consisting of formal power series

e e∈H2 λe q , where q is a formal variable, λe ∈ Q, and e ∈ H2 | λe = 0, ω · e ≤ κ is finite for all κ ∈ Q, ω being the class of a fixed polarization on X. As an example, if H2 ∼ = Z, then ∼ = Q[[q]][q −1 ]. One extends the Gromov-Witten invariants linearly to H ∗ (X; ). Definition 1.8 Let γi be a basis for H ∗ (X; Q), and let γ i be the dual basis with respect to the Poincaré pairing. For a, b ∈ H ∗ (X; ), the little quantum product is

q e γi a, b, γ i e . a∗b = e

i

Alternatively, one can define the product without choosing a basis or mentioning the Gromov-Witten invariants explicitly, as q e ev∗ f ∗ f∗ ev∗ a · f∗ ev∗ b , a∗b = e

where f is restricted to the virtual cycle. This reveals that the cup product has in some sense been transferred from X to the moduli space of stable maps. Since M 0,1 (X, 0) = M 0 (X, 0) = X, the quantum product equals the cup product modulo q. Furthermore, it follows from Lemma 1.6(ii) that the quantum product with any element of H 1 equals the cup product. A fundamental theorem asserts that the little quantum product is associative (see [4], [8], [19], [28]).

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Definition 1.9 The little quantum cohomology QH ∗ (X) is defined to be the ring additively isomorphic to H ∗ (X; ) but with the little quantum product as multiplication. The expected complex dimension of M 0 (X, e) is dim X −3+c1 (T X)·e, so QH ∗ (X) is graded if q e is given degree 2c1 (T X) · e. 2. Symmetric products of a curve We begin by recalling some basic facts on the cohomology of symmetric products of a curve. Good references are E. Arbarello, M. Cornalba, P. Griffiths, and Harris [1] and I. Macdonald [20]. Let C be a smooth projective curve of genus g, and let Cd be the dth symmetric product, which is a smooth projective variety of dimension d. It can be regarded as the moduli space of effective divisors D on C of degree |D| = d. Accordingly, there exists a universal divisor ( ⊂ Cd × C having the obvious property. The Poincaré dual of ( is a class in H 2 (Cd × C; Z), which determines a map µ : H∗ (C; Z) → H ∗ (Cd ; Z). Let ei be a basis of H1 (C; Z) for which the intersection form is the standard symplectic form. Then define ξi = µ(ei ) ∈ H 1 (Cd ) and η = µ(1) ∈ H 2 (Cd ). It is also convenient to define σi = ξi ξi+g for i ≤ g and θ =

g i=1 σi . There is a morphism ι : Cd−1 → Cd given by D → D + p; it is an embedding, and its image is a divisor in Cd . Indeed, this divisor is exactly (|Cd ×p , so it is the Poincaré dual to η. There is also a natural morphism AJ : Cd → Jacd (C), the Abel-Jacobi map, taking a divisor D to the line bundle O (D). Since every nonzero section of a line bundle determines a divisor, and vice versa up to scalars, every fiber AJ−1 (L) is a projective space, namely, PH 0 (L). Indeed, for d ≥ 2g − 2, Cd is the projectivization of a vector bundle over Jacd , as follows. Fix p ∈ C, and let L be a Poincaré line bundle over Jacd ×C, normalized such that L |Jacd ×p = O . Let π : Jacd ×C → Jacd be the projection, and let U = π∗ L . Then U is a vector bundle of rank d − g + 1 by the Riemann-Roch theorem, and there is a natural isomorphism Cd = PU . If O (1) is the twisting sheaf of this projective bundle, the embedding ι satisfies ι∗ O (1) = O (1) for any d > 2g − 2. So, for d < 2g − 2, it is reasonable to define O (1) = ι∗ O (1) by descending induction. For any d ≥ 0 and any D ∈ Cd , the fiber of O (−1) at D is then naturally isomorphic to the space of sections of L |AJ(D)×C vanishing at D, where L is again a Poincaré bundle normalized as above. To construct the isomorphism, one simply multiplies by the appropriate power of the natural section of O (p) vanishing at p.


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proposition 2.1 (i) The twisting sheaf O (1) of PU is isomorphic to O (Cd−1 ), so c1 (O (1)) = η. (ii) There is a natural isomorphism L (1) = O ((), where L is short for AJ∗ L . (iii) If ei is viewed as an element of H 1 (Jacd ) ∼ = H1 (C), then ξi = AJ∗ ei . (iv) If 1 is a theta-divisor on Jacd , then AJ∗ c1 (O (1)) = θ. Proof On PU = Cd , evaluation at p gives a natural homomorphism of sheaves O (−1) → L |Jacd ×p = O . This vanishes precisely on Cd−1 , so O (1) = O (Cd−1 ). To prove (ii), first assume d > 2g − 2. Now, for any L ∈ Jacd , the restriction of O (() to PH 0 (L) × C is L(1) because L(1) has a section with the universal property. Hence L (1) ⊗ O (−() is trivial on the fibers of the projection Cd × C → Jacd , which is locally trivial and hence flat for d > 2g − 2. Therefore L (1) ⊗ O (−() is the pullback of some line bundle on Jacd (see R. Hartshorne [17, Chap. III, Ex. 12.4]). But the restriction of O (() to Cd × p is O (Cd−1 ) = O (1) by (i), and the restriction of L to Cd × p is O by construction, so this line bundle is trivial. The case d ≤ 2g − 2 follows from this one by descending induction since the embedding ι : Cd−1 2→ Cd satisfies ι∗ L = L (p), ι∗ O (1) = O (1), and ι∗ O (() = O (()(p). Parts (iii) and (iv) then follow from (ii) together with well-known formulas for c1 (L ) and c1 (O (1)) (see Griffiths and Harris [14, §§2.6 and 2.7]). A presentation of the cohomology ring of Cd was given by Macdonald [20]. theorem 2.2 (Macdonald) The cohomology ring H ∗ (Cd ; Z) is generated by ξi and η with relations (η − σi ) ξj ξk+g , 0 = ηr i∈I

j ∈J

k∈K

where I, J, K ⊂ {1, . . . , g} are disjoint and r + 2|I | + |J | + |K| ≥ d + 1.

(2.1)

For d > 2g − 2, these relations are generated by the single relation 0 = ηd−2g+1

g

(η − σi ).

(2.2)

i=1

For d ≤ 2g − 2, they are generated by those for which r = 0 or 1 and equality holds in (2.1). (Macdonald’s paper contains a small error: it is asserted that r = 0 is enough in the

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last statement. However, the method of proof clearly requires r = 1 as well, and it is already necessary in the case d = 2.) corollary 2.3 The subring of H ∗ (Cd ) invariant under all monodromies of C through smooth curves is generated by η and θ . Proof Certainly the monodromy invariant part of H ∗ (Jacd ) is generated by θ. Indeed, since the monodromy surjects on Aut H ∗ (C) = Sp(2g, Z) and since H ∗ (Jacd ) is the exterior algebra on H 1 (C)∗ , this means simply that the symplectic form and its powers are the only alternating forms invariant under the symplectic group. Macdonald’s result shows that H ∗ (Cd ) is generated by η as an algebra over H ∗ (Jacd ). For d > 2g − 2, η is monodromy invariant by Proposition 2.1(i); for d < 2g − 2, it is still monodromy invariant since the embedding i : Cd−1 2→ Cd satisfies i ∗ η = η. We now turn to the quantum product on Cd . First, notice that since σi ∈ H 1 (Cd ), the quantum product with ξi equals the cup product by Lemma 1.6(ii). Hence the quantum product is completely determined by thevalues of ηu ∗ ηv for u, v ≥ 0. Also, notice that for d > 1, h2 (Cd ) = 2g + 1. This is an alarmingly large 2 number of deformation parameters, but in fact only one parameter is nontrivially involved in the quantum product for the following reason. proposition 2.4 If ; ∈ H2 (Cd ; Z) is the homology class of any line in any fiber AJ−1 (L), then the quantum product on H ∗ (Cd ; ) preserves the subring H ∗ (Cd , Q[[q ; ]]). Proof Since an abelian variety has no rational curves whatsoever, every genus-zero stable map to Cd has image contained in a fiber of AJ. But, for any line ; in any fiber of AJ, clearly η · ; = 1; while ν · ; = 0 for any class ν pulled back from Jacd . It follows that all such lines are homologous, so every genus-zero stable map has image homologous to a nonnegative multiple of ;. Remarks 2.5 (i) Though it is not needed in the sequel, a topological version of this statement remains true: the Hurewicz homomorphism π2 (Cd ) → H2 (Cd ; Z) has rank 1 for d > 1, so the multiples of ; are the only spherical classes. This is proved in Section 9. (ii) This subring determines the whole quantum product by -linearity, so from now


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on attention is focused on it alone. In particular, the choice of polarization on Cd used to define is immaterial. By abuse of notation, q ; is henceforth denoted simply by q, · · · ; by · · · 1 , and so on. (iii) Macdonald [20] also shows that c1 (T Cd ) = (d − g + 1)η − θ; consequently, the degree of q ∈ QH ∗ (Cd ) is 2c1 (T Cd ) · ; = 2(d − g + 1). In particular, it is negative for d < g − 1 and zero for d = g − 1. The latter case resembles that of a Calabi-Yau manifold: although the canonical bundle is not trivial, its restriction to every rational curve is trivial. Example 2.6 Take g = 2, d = 2. Then, by the Riemann-Roch theorem, every line bundle in Jac2 has one section, except the canonical bundle, which has two. The Abel-Jacobi map therefore collapses exactly one rational curve E = PH 0 (K). It is therefore precisely the blow-down of E (see [17, Chap. V, Cor. 5.4]). The Poincaré dual of E is easily seen to be θ − η. Since E is the only rational curve on C2 , M 0 (C2 , [E]) is a point and M 0,1 (C2 , [E]) = E. Therefore η ∗η = η2 +q(θ −η), which completely characterizes the quantum product. In some cases the vanishing of the 3-point invariants, and hence of certain terms in the quantum product, follows immediately from a dimension count. proposition 2.7 For d < g − 1 and e > (d − 3)/(g − 1 − d), a, b, ce = 0 for all a, b, c ∈ H ∗ (Cd ). Proof The expected dimension of M 0 (Cd , e) is d − 3 + e(d − g + 1), which is negative in this case. The result follows from Proposition 1.2(iii). Indeed, not only the 3-point invariants but also all higher-point invariants vanish in this range by the same argument. corollary 2.8 For d < g − 1 and e > (d − 3)/(g − 1 − d), Coeff q e a ∗ b = 0. In particular, for d < g/2 + 1, the quantum product is simply the ordinary cup product. It is also relatively easy to show that the 3-point invariants vanish for d > 2g − 2 and e > 1. Indeed, this follows from the vanishing of the higher degree equivariant 3-point invariants of projective space. However, we do not pursue this now, as it is subsumed in Theorem 6.1(i).

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3. The Brill-Noether approach To compute Gromov-Witten invariants for Cd for d ≤ 2g − 2, we must understand in detail how the fibers of the Abel-Jacobi map fit together. More precisely, we must understand the enumerative geometry of the strata where the dimension of the fiber is constant. This is the subject matter of Brill-Noether theory. Some wellknown definitions and results in the theory are recalled in Definition 3.2, Lemma 3.3, Theorem 3.4, and Corollary 3.5. We also make crucial use of a formula of Harris and Tu [16] for the Chern numbers of kernel and cokernel bundles on determinantal varieties. Harris and Tu prove slightly more than the main result they state; moreover, their statement contains a sign error (the very first + sign in the paper should be a −). In the form we need, the result is the following. theorem 3.1 (Harris-Tu) Let M be a complex manifold, let E and F be locally free sheaves of ranks m and n, and let f ∈ H 0 (M, Hom(E, F )). Let S be the universal subbundle over Gr k E, and let f˜ ∈ H 0 (Gr k E; Hom(S, F )) be the induced map. Suppose f˜ intersects the zerosection transversely in a variety Mk . If x1 , . . . , xk are the Chern roots of S ∗ on Mk , j j j then any characteristic number νc11 (S)c22 (S) · · · ckk (S)[Mk ], where ν ∈ H ∗ (M), can be calculated using the formal identity cn−m+k+i1 cn−m+k+1+i1 · · · cn−m+2k−1+i1 cn−m+k−1+i cn−m+k+i2 2 νx1i1 x2i2 · · · xkik [Mk ] = ν [M], .. .. . . c cn−m+k+ik n−m+1+ik where cα = cα (F − E). Definition 3.2 The moduli space Grd of (possibly incomplete) linear systems on C of dimension r and degree d can be constructed just as in the above theorem, taking M = Jacd . Fix any reduced divisor P on C of large degree, and let L be the Poincaré line bundle on Jacd ×C. Then the natural map L (P ) → OP ⊗ L (P ) pushes forward to a map E → F of locally free sheaves on Jacd such that, for any r, Grd is precisely the locus Mk of the theorem for k = r + 1. The tautological subbundle over Gr r+1 E restricts to a bundle Udr over Grd whose projectivization PUdr admits a canonical map τ : PUdr → Cd . Its image is referred to as Cdr , and the subscheme AJ(Cdr ) of Jacd is referred to as Wdr . Most of these definitions are discussed at greater length by Arbarello, Cornalba, Griffiths, and Harris [1].


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lemma 3.3 The total Chern classes of the bundles E and F defined above are c(E) = exp(−θ) and c(F ) = 1. Proof The Poincaré bundle is normalized such that, for some p ∈ C, L |Jacd ×p = O . Since F is just a sum of bundles deformation equivalent to this, c(F ) = 1. As for c(E), this can be calculated using the Grothendieck-Riemann-Roch theorem (see Arbarello, Cornalba, Griffiths, and Harris [1, Chap. VIII, §2]). Since Gromov-Witten invariants are deformation invariants, the curve C may be taken to be general, and we assume this henceforth. This allows us to use one of the central results of Brill-Noether theory. theorem 3.4 (Gieseker) Let C be a general curve. For any effective divisor D on C, the natural map H 0 (O (D)) ⊗ H 0 K(−D) −→ H 0 (K) is injective. Proof See D. Gieseker [11]. corollary 3.5 For a general curve C, Grd , and hence Wdr \Wdr+1 and Cdr \Cdr+1 , are smooth of dimension ρ, ρ, and ρ + g, respectively, where ρ is the Brill-Noether number g − (r + 1)(g − d + r). Proof See Arbarello, Cornalba, Griffiths, and Harris [1, Chap. V, Th. 1.6]. A further useful consequence of Gieseker’s result is the following. lemma 3.6 (i) At L ∈ Wdr \Wdr+1 , there is a natural isomorphism NWdr / Jacd = Hom(H 0 (L), H 1 (L)). (ii) At D ∈ Cdr \Cdr+1 , there is a natural isomorphism 0 H (O (D)) 1 , H (D) . NCdr /Cd = Hom D

342


Proof Part (i) follows easily from [1, Chap. IV, Prop. 4.2(i)]. Part (ii) follows in the same way from [1, Chap. IV, Lem. 1.5], provided that the composition of natural maps H 0 O (D) ⊗ H 0 K(−D) −→ H 0 (K) −→ H 0 K ⊗ OD has kernel D ⊗ H 0 (K(−D)). But the latter map has kernel exactly H 0 (K(−D)), so this follows from Gieseker’s theorem. proposition 3.7 For general C, the reduced induced subscheme of the moduli space M 0 (Cd , e) is a disjoint union of orbifolds M 0 (Cd , e)red =

r e

Mr,i ,

r=1 i=1

consisting of the stable maps whose image spans a linear system of dimension i contained in a complete linear system of dimension r (or greater than or equal to r if r = e). The closure of each stratum has a resolution M˜ r,i that is the M 0 (Pi , e)-bundle associated to the tautological subbundle over Gr i+1 Udr .

Proof The resolutions M˜ r,i can be viewed as moduli spaces of triples consisting of a linear system of dimension r, a projective subspace of dimension i, and a stable map to that subspace. By Corollary 3.5, they are orbifolds. The open subset where the map spans the i-dimensional subspace is a subvariety of M 0 (Cd , e). These subvarieties partition M 0 (Cd , e) because a rational curve of degree e in projective space spans at most an e-dimensional subspace. Since every complete linear system has dimension greater than or equal to d − g, Mr,i = ∅ if r < d − g. Otherwise Mr,i = ∅, and Mr,i−1 is in its closure. Hence the irreducible components of M 0 (Cd , e)red are the closures of Mr,r for min(d − g, e) ≤ r ≤ e. proposition 3.8 At any point of Mr,r , M 0 (Cd , e) is reduced. Before the proof, we note a few observations that are useful again later. lemma 3.9 For any stable map φ : → Cd ,

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(i) the natural map TC1d (φ) → T 1 () of deformation spaces is surjective; and (ii) if PH 0 (L) is the complete linear system containing φ(), the obstruction space TC2d (φ) is naturally isomorphic to H 1 (, φ ∗ NPH 0 (L)/Cd ). Proof Any stable map has image in some Pr = PH 0 (L) ⊂ Cd , so there is a natural diagram of deformation spaces H 0 , φ ∗ T Pr TP1r (φ) 0 T 0 () T 1 ()

0

T 0 ()

H 0 , φ ∗ T Cd

TC1d (φ)

T 1 ()

By Example 1.3, H 1 (, φ ∗ T Pr ) = 0, so by Proposition 1.2(i) the last arrow in the first row is surjective, which proves (i). Then Proposition 1.2(i) also implies that TC2d (φ) = H 1 (, φ ∗ T Cd ), and the long exact sequence of 0 −→ φ ∗ T Pr −→ φ ∗ T Cd −→ φ ∗ NPr /Cd −→ 0 completes the proof of (ii). Proof of Proposition 3.8 It suffices to show that, for φ ∈ Mr,r , the deformation space TC1d (φ) has dimension equal to that of Mr,r itself. Now Mr,r is an M 0 (Pr , e)-bundle over an open set in Grd , so dim Mr,r = dim M 0 (Pr , e) + dim Grd . On the other hand, the number of deformations of φ as a map to Pr equals dim M 0 (Pr , e) since the obstructions vanish. By Lemma 3.9(i), the number of deformations as a map to Cd exceeds this by dim H 0 , φ ∗ T Cd − dim H 0 , φ ∗ T Pr = dim H 0 , φ ∗ NPr /Cd . Hence it suffices to show dim H 0 (, φ ∗ NPr /Cd ) = dim Grd . Because of the definition of Mr,r , there are two cases depending on whether or not r = e. If r < e, the image of φ is in Cdr \Cdr+1 , which is a Pr -bundle over the open set Wdr \Wdr+1 ⊂ Grd . On any fiber Pr = PH 0 (L) of this bundle, there is an exact sequence 0 −→ TL Wdr ⊗ O −→ NPr /Cd −→ NCdr /Cd −→ 0. Hence it suffices to show that H 0 (, φ ∗ NCdr /Cd ) = 0. By Lemma 3.6(ii), there is a short exact sequence on Pr : 0 −→ NCdr /Cd −→ O ⊗ Hom H 0 (L), H 1 (L) −→ O (1) ⊗ H 1 (L) −→ 0.

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By the definition of Mr,r , the image of φ spans PH 0 (L). Hence the natural map H 0 (L)∗ → H 0 (, φ ∗ O (1)) is injective, so H 0 , O ⊗ Hom H 0 (L), H 1 (L) −→ H 0 , φ ∗ O (1) ⊗ H 1 (L) is also injective and H 0 (, φ ∗ NCdr /Cd ) = 0 as desired. If r = e, then the image of φ spans a linear system Pr that may not be complete. Let PH 0 (L) be the complete linear system containing it, and let ; be its projective dimension. Then there are three short exact sequences on Pr : 0 −→ NPr /PH 0 (L) −→ NPr /Cd −→ NPH 0 (L)/Cd −→ 0, 0 −→ TL Wd; ⊗ O −→ NPH 0 (L)/Cd −→ NC ; /Cd −→ 0, d 0 1 0 −→ NC ; /Cd −→ Hom H (L), H (L) ⊗ O −→ H 1 (L) ⊗ O (1) −→ 0, d

the last by Lemma 3.6(ii). But NPr /PH 0 (L) ∼ = O ;−r (1) since Pr ⊂ PH 0 (L) is a pro1 1 jective subspace. Since H (, O ) = H (, φ ∗ O (1)) = 0, all three of the long exact sequences contain only H 0 -terms. Hence dim H 0 (, φ ∗ NPr /Cd ) can be expressed in terms of dim H 0 (, φ ∗ O (1)), which is r + 1 since φ spans Pr , and the ranks of the bundles above. A little high-school algebra gives the desired result.

4. The degree-1 invariants proposition 4.1 As schemes, M 0 (Cd , 1) = G1d and M 0,1 (Cd , 1) = PUd1 . Proof The identification of sets is clear from Proposition 3.7, and the moduli space is reduced by Proposition 3.8. theorem 4.2 For u, v, w ≥ 0,

ηu , ηv , θ g−m ηw

1

=

u−1 g! m m − , m! g−d +i+v g−d +i i=0

where m = 2g − 2d − 1 + u + v + w. For example, consider the case when the expected dimension ρ = g − 2(g − d + 1) of G1d is zero, which occurs when g = 2d−2. Then η, η, η1 counts the number of points 2d−2 in G1d . According to Theorem 4.2, this is 2d−2 d−1 − d−2 , which agrees with the 2d−2 Catalan number (1/d) d−1 computed by G. Castelnuovo (see [1, Chap. V, (1.2)]).


345

Proof Since the moduli space is smooth of the expected dimension, the Gromov-Witten invariants can be calculated using Proposition 1.5. The evaluation map is precisely τ : PUd1 → Cd , as defined in Definition 3.2. By Proposition 2.1(i), τ ∗ η = c1 (O (1)), and hence f∗ τ ∗ ηu is the Segre class su−1 (Ud1 ) (see, e.g., Fulton [9, §3.1]). If x1 and x2 are the Chern roots of (Ud1 )∗ , then the Segre classes are the complete symmetric polynomials k x1i x2k−i . sk = i=0

So the 3-point invariant is

u

v

η ,η ,θ

g−m w

η

1

=θ

g−m

u−1 v−1 w−1 i=0 j =0 k=0

The Harris-Tu formula implies that

i+j +k u+v+w−3−i−j −k 1 x2 Gd .

x1

p u+v+w−3−p

θ g−m x1 x2

G1d = θ g−m (γp − γp+1 )[Jacd ],

where γp = cg−d+1+p (F − E) · cg−d+u+v+w−2−p (F − E) θm (g − d + u + v + w − 2 − p)!, = (g − d + 1 + p)! the last equality by Lemma 3.3. The sum over k therefore telescopes to yield

ηu , ηv , θ g−m ηw

1

= θ g−m

v−1 u−1

(γi+j − γi+j +w )[Jacd ].

i=0 j =0

But there is another symmetry, namely, γp = γu+v+w−3−p ; applying this to the second term and canceling gives v−1 u−1

γi+j − γi+j +w =

i=0 j =0

v−1 u−1

γi+j − γ(u−1−i)+(v−1−j )−1

i=0 j =0

=

v−1 u−1

γi+j − γi+j −1

i=0 j =0

=

u−1

γi+v−1 − γi−1 .

i=0

Substituting this and using θ g [Jacd ] = g! then gives the answer as stated.

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Of course, by the same method, one could easily derive a formula for 3-point invariants involving arbitrary elements of H ∗ (Cd ). But these are enough to characterize the linear term of the quantum product. corollary 4.3 For u, v ≥ 0, u

v

η ∗η = η

u+v

+q

u−1 i=0

θ g−d+i+v θ g−d+i u−1−i v+u−1−i η η +O q 2 . − (g − d + i + v)! (g − d + i)!

Proof By the definition of quantum product, the coefficient of q in ηu ∗ ηv is µ if and only if for all ν ∈ H ∗ (Cd ), ηu , ηv , ν1 = µν[Cd ]. However, Gromov-Witten invariants, and hence the quantum product, are deformation invariant. Since η and its powers are monodromy invariant by Corollary 2.3 whenever C lies in a family of smooth curves, each coefficient of ηu ∗ ηv must be as well. Again by Corollary 2.3, this means that it is in the subring generated by η and θ. This subring is the algebraic part of H ∗ (Cd ) for C general (see Arbarello, Cornalba, Griffiths, and Harris [1, Chap. VIII, §5]). In particular, it satisfies Poincaré duality; hence it suffices to take ν in the subring as well. This circumvents the cumbersome task of dealing with arbitrary monomials in the ξi . It is easy to check against Theorem 4.2 that the coefficient stated above satisfies the required condition. One simply applies the convenient formula   g! if i ≤ g, i d−i θ η [Cd ] = (g − i)!  0 if i > g, which follows from Proposition 2.1(i) and Theorem 2.2 using the fact that Cd−1 ⊂ Cd is the Poincaré dual to η. The required identity follows term-by-term from a comparison of the coefficient above with the Gromov-Witten invariant in Theorem 4.2. 5. The degree-2 invariants The degree-2 invariants can also be calculated using ideas from Brill-Noether theory. Here things are considerably more complicated; in particular, the virtual class comes into play. However, some delightful cancellations make the computations tractable. In the decomposition of Proposition 3.7, the moduli space M 0 (Cd , 2) has three strata, M11 , M22 , and M12 . If d ≥ g + 2, M11 is empty, but this does not affect the results. Of these strata, M22 has the expected dimension, while M11 exceeds it by g+1−d. The third, M12 , is in the closure of M22 and has less than the expected dimension.


347

The virtual class is therefore a sum of cycles pushed forward from M˜ 11 and M˜ 22 . By Proposition 3.8, the former is simply the ordinary fundamental class of M 22 . The Gromov-Witten invariant therefore can be computed as in Proposition 1.5: vir M˜ 11 + M˜ 22 . f˜∗ ev∗ ak a1 , a2 , a3 2 = k

proposition 5.1 For all a1 , a2 , a3 ∈ H ∗ (Cd ),

f˜∗ ev ak ∗

M˜ 22 = 0.

k

Proof On M˜ 22×M 0 (Cd ,2)M 0,1 (Cd , 2), the evaluation maps factor through the P2 -bundle PUd2 , where any cohomology class can be expressed as a polynomial over H ∗ (G2d ) of degree less than or equal to 2. So, on M˜ 22 , f˜∗ ev∗ ai is in the submodule H ≥deg ai −4 (G2d ) · H ∗ (M˜ 22 ). It certainly vanishes on the fundamental class for dimensional reasons

unless deg ai = 2 dim M˜ 22 + 6 = 6d − 4g + 4. But then the product is in the submodule H ≥6d−4g−8 (G2d ) · H ∗ (M˜ 22 ), which is zero since dim G2d is the BrillNoether number ρ = 3d − 2g − 6. The above is really a special case of the more general statement that the equivariant 3-point invariants of projective space vanish in degree greater than 1. We now attack the virtual class [M˜ 11 ]vir , or rather its pushforward to G1d , where the invariant is calculated. For simplicity, Ud1 is denoted U in the rest of this section. theorem 5.2 Let p : M˜ 11 → G1d be the projection. Then p∗ [M˜ 11 ]vir is the Poincaré dual to the degree-2(g − 1 − d) part of (1/8) exp θ . 1 + c1 (U )/2 Proof First, it suffices to work only on p : M11 → Wd1 \Wd2 because the codimension of the missing locus in G1d is g + 2 − d, which is greater than g − 1 − d. Take the short exact sequence 0 −→ O −→ O (() −→ O( (() −→ 0

348


on (Cd1 \Cd2 ) × C, and push forward to a long exact sequence on the first factor: 0 −→ R 0 π ∗ O −→ R 0 π ∗ O (() −→ R 0 π ∗ O( (() −→ R 1 π ∗ O −→ R 1 π ∗ O (() −→ 0. The third nonzero term is T Cd , and the image of the second is the tangent space to the pencils. The quotient is therefore the vector bundle N whose fibers along each pencil are the normal spaces to that pencil in Cd . So there is a short exact sequence on Cd1 \Cd2 , 0 −→ N −→ R 1 π ∗ O −→ R 1 π ∗ O (() −→ 0, and hence a long exact sequence 0 −→ H 0 , φ ∗ N −→ H 1 (C, O ) −→ H 0 , φ ∗ R 1 π ∗ O (() −→ H 1 , φ ∗ N −→ 0. By Lemma 3.9(ii), the obstruction space TC2d (φ) is nothing but the last term of this sequence. Moreover, by Proposition 1.2(ii), the virtual class on M11 is exactly the Euler class of the orbifold vector bundle whose fiber at φ : → Cd is this obstruction space. How can the terms in this long exact sequence be described better? Well, note that N also fits into the exact sequence on Cd1 \Cd2 : 0 −→ T Wd1 −→ N −→ NC 1 /Cd −→ 0. d

By Lemma 3.6(ii), NC 1 /Cd restricted to the pencil PH 0 (L) is O (−1) ⊗ H 1 (L), so d

H 0 (, φ ∗ NC 1 /Cd ) vanishes and hence H 0 (, φ ∗ N) is simply TL Wd1 . On the other d hand, if O (1) on Cd denotes the relative hyperplane bundle of the Abel-Jacobi map and if L is a Poincaré line bundle pulled back from Jacd ×C, then, on Cd × C, O (() = L (1) by Proposition 2.1(ii). Hence, on Cd1 \Cd2 , (R 1 π)∗ O (() is the tensor product V (1), where V = (R 1 π)∗ L is a vector bundle pulled back from Wd1 \Wd2 . The term H 0 (φ ∗ (R 1 π )∗ O (()), which appeared above, is therefore the tensor product V ⊗ H 0 (, φ ∗ O (1)). Putting it all together yields an exact sequence 0 −→ NWg1 / Jacd −→ V ⊗ H 0 , φ ∗ O (1) −→ H 1 , φ ∗ N −→ 0 for any φ ∈ M11 . The first two terms clearly have dimension independent of φ, so they determine an orbifold vector bundle on M11 , and hence so does the third term. Now consider the orbifold structure on M11 . It is an M 0 (P1 , 2)-bundle over 1 Wd \Wd2 . In fact, a degree-2 stable map to a line is characterized by its base points,


349

∼ P2 and M11 = P Sym2 U ∗ where U = U 1 . But every so as schemes M 0 (P1 , 2) = d stable map in M11 has an involution, so as an orbifold M11 is a global quotient by Z2 of a double cover, say, Mˆ 11 , branched over the bundle of conics PU ∗ ⊂ P Sym2 U ∗ . The involution splits H 0 (, φ ∗ O (1)) into ±1-eigenspaces. The +1-eigenspace consists of those sections pulled back from the pencil, so the corresponding bundle is just U ∗ . The −1-eigenspace is generated by the square root of the section of O (2) on the pencil vanishing on the branch points of the double cover, so the corresponding bundle is an orbifold line bundle on M11 , coming from a Z2 -equivariant line bundle on Mˆ 11 whose tensor square is the pullback of O (−1) from M11 . Call this orbifold bundle O (−1/2). Now it follows from Lemma 3.6(i) that the normal bundle NWg1 / Jacd is V ⊗ U ∗ ,

and its image in V ⊗ H 0 (, φ ∗ O (1)) is clearly in the +1-eigenspace. This splits the exact sequence mentioned above. Hence the orbifold bundle with fiber H 1 (, φ ∗ N) is none other than V (−1/2), and the virtual class is the Euler class of this. It only remains to push forward the virtual class to G1d . If ξj are the Chern roots of V and if h is the hyperplane class of P Sym2 U ∗ , then the virtual class is j (ξj − h/2). This pushes forward to g−d−1 1 1 n+2 − cg−1−d−n (V ) sn Sym2 U ∗ , 2 2 n=0

where si denotes the Segre class. The extra factor of 1/2 appears because of the orbifold structure on M11 . The Chern roots of Sym2 U ∗ are 2x1 , 2x2 , and x1 + x2 . The factors of 2 cancel, and one gets the degree-2(g − d − 1) part of (1/8)c(V ) . (1 − x1 )(1 − x2 )(1 − (x1 + x2 )/2) Now a miraculous cancellation. The (1 − x1 )(1 − x2 ) in the denominator is just c(U ). But U and V are exactly the kernel and cokernel of the map of bundles E → F defined in Definition 3.2. Hence, by Lemma 3.3, c(F ) 1 c(V ) = = = exp θ, c(U ) c(E) exp(−θ) and plugging this in yields the stated formula. theorem 5.3 For u, v, w ≥ 0,

350


ηu , ηv , θ d+1−m ηw =

2 n

g−1−d

n=0 p=0

n g! n p 2 (g − 1 − d − n)!(m + n)!

·

u−1 i=0

m+n m+n − , g−d +i+v+p g−d +i+p

where m = 2g − 2d − 1 + u + v + w. Proof The proof is parallel to that of Theorem 4.2. By Proposition 5.1, vir ∗ a1 , a2 , a3 2 = f˜∗ ev ak M˜ 11 . k

Conveniently, M˜ 11 = P Sym2 U ∗ is isomorphic to the closure M 11 . Moreover, 2 ∗ 11 ) is simply the fibered product P Sym U ×G1 PU . In contrast to M11 , a

f −1 (M

d

generic stable map in f −1 (M 11 ) has no involution; it is rigidified by the marked point. The evaluation map f −1 (M 11 ) → Cd factors through PU ; hence f∗ ev∗ ηu restricted to M 11 is a class pulled back from G1d , namely, 2 su−1 (U ), where si is the ith Segre class. The factor of 2 appears because of the orbifold structure on M 11 . In terms of the Chern roots x1 , x2 of U ∗ , this is 2 su−1 (U ) = 2

u−1

x1i x2u−1−i .

i=0

Expanding the formula of Theorem 5.2 in terms of x1 and x2 yields the cap product

p∗ M˜ 11

vir

g−1−d n p n−p 1 n (−1)n θ g−1−d−n x1 x2 ∩ G1d . = n 2 (g − 1 − d − n)! p 8 n=0 p=0

Consequently,

ηu , ηv , θ g−n−m ηw =

g−1−d

2 n u−1 v−1 w−1

n=0 p=0 i=0 j =0 k=0

p+i+j +k n−p+u+v+w−3−i−j −k 1 x2 n θ d−4−m x1 Gd . n 2 (g − 1 − d − n)! p

As in the proof of Theorem 4.2, one now applies the Harris-Tu formula in

351


Theorem 3.1, telescopes the sum over k, and cancels using the additional symmetry to obtain the stated result. corollary 5.4 For u, v ≥ 0, u

v

η ∗η =η

u+v

+q +q

u−1

θ g−d+i+v θ g−d+i ηu−1−i − ηv+u−1−i (g − d + i + v)! (g − d + i)!

i=0 g−1−d 2

n n

n=0 p=0

×

p

2n (g

1 − 1 − d − n)!

u−1 2g−2d+1−n+i+v+p u+n−i−p−3 η θ

(g − d + i + v + p)!

i=0

−

θ 2g−2d+1−n+i+p ηu+v+n−i−p−3 (g − d + i + p)!

+ O q3 .

Proof The proof is similar to that of Corollary 4.3. 6. The higher degree invariants for d ≥ g − 1 Examining the formulas of Sections 4 and 5 reveals that the degree-2 invariants vanish for d > g − 1 and that they equal the degree-1 invariants for d = g − 1. The result below shows that this is the case for all higher degree invariants. theorem 6.1 For all e > 1 and all a1 , a2 , a3 ∈ H ∗ (Cd ), (i) a1 , a2 , a3 e = 0 when d > g − 1; (ii) a1 , a2 , a3 e = a1 , a2 , a3 1 when d = g − 1. Proof The virtual class is some algebraic cycle class of degree d − 3 + e(d − g + 1). Choose a cycle representing this class; it decomposes uniquely as a sum of cycles

vir vir is supported on the closure of M r,i r,i [Mr,i ] , where every component of [Mr,i ] vir but not on any smaller stratum. Then [Mr,i ] is in the image of the pushforward from the resolution M˜ r,i , so its contribution to the Gromov-Witten invariant a1 , a2 , a3 e can be computed on M˜ r,i as vir ∗ ˜ f∗ ev ak M˜ r,i . k

352


Now, on M˜ r,i , all three of the evaluation maps factor through PUdr . The pullback of ai to this bundle can be written as a polynomial in the hyperplane class over H ∗ (Grd ) of degree less than or equal to r. Hence f˜∗ ev∗ ai is in the submodule H ≥deg ai −2r Grd · H ∗ M r,i .

Certainly a1 , a2 , a3 e = 0 on dimensional grounds unless k deg ak = 2d + 2e(d − g + 1). In that case the product k f˜∗ ev∗ ak is in the submodule H ≥2d+2e(d−g+1)−6r Grd · H ∗ M r,i . But the dimension of Grd is the Brill-Noether number ρ = g − (r + 1)(g − d + r), and some high-school algebra shows that d + e(d − g + 1) − 3r − ρ = (e − r)(d − g + 1) + r(r − 1). This is positive for all d > g − 1 unless e = r = 1, and also for d = g − 1 unless r = 1. Hence the ideal H ≥2d+2e(d−g+1)−6r (Grd ) vanishes in those cases. This immediately implies part (i), and it shows that, when d = g − 1, only the single stratum M11 contributes nontrivially to the 3-point invariants. We therefore turn to the contribution of this stratum, which consists of e-fold covers of pencils on Cg−1 . The closure M 11 is isomorphic to M˜ 11 , which is a bundle over G1d with fiber M 0 (P1 , e). The evaluation map factors through PUd1 : f −1 M 11

PUd1

Cd

π

f

M 11

τ

p

G1d

The morphism f −1 (M 11 ) → M 11 ×G1 PUd1 is finite of degree e, so, for a ∈ d d ), p∗ f∗ ev∗ a = e · π∗ τ ∗ a.

H ∗ (C

The Gromov-Witten invariant can be expressed as vir ∗ p∗ f∗ ev ak p∗ M˜ 11 ; a1 , a2 , a3 e = k

but since G1d has dimension g − 4, exactly that of the virtual cycle, p∗ [M˜ 11 ]vir ∈ H2g−8 (G1d ) must be a scalar multiple of [G1d ]. On the other hand, by Proposition 4.1, G1d = M 0 (Cd , 1) and PUd1 = M 0 (Cd , 1).

353


Hence

∗

π∗ τ ak

k

G1d = a1 , a2 , a3 1 .

So, to prove part (ii), it suffices to show that p∗ [M˜ 11 ]vir = 1/e3 [G1d ]. One may work on any single fiber of p; choose one over Wd1 . The moduli space is an orbifold in a neighborhood of this fiber, so Proposition 1.2(ii) applies. Hence we need to compute the Euler class of the orbifold bundle whose fiber at a map φ is the obstruction space TC2d (φ). By Lemma 3.9(ii), TC2d (φ) = H 1 (, φ ∗ NP1 /Cd ). From the short exact sequence 0 −→ NP1 /C 1 −→ NP1 /Cd −→ NC 1 /Cd −→ 0, d

d

together with the isomorphism NP1 /C 1 ∼ = O ρ , it follows that d

H , φ NP1 /Cd = H 1 , φ ∗ NC 1 /Cd . 1

∗

d

Cdr

at a divisor D is naturally isomorAs seen in Lemma 3.6(ii), the normal space to phic to Hom(H 0 (O (D))/D, H 1 (O (D))). In this case dim H 0 = dim H 1 = 2, so on P1 the normal bundle NC 1 /Cd is isomorphic to Hom(O (1), O ⊕ O ) = O (−1) ⊕ d O (−1). So, on M 0 (P1 , e), we want to know the Euler class of the orbifold bundle whose fiber at a map φ is H 1 (φ ∗ O (−1) ⊕ φ ∗ O (−1)). Very felicitously, this is exactly the number computed to be 1/e3 , using some nontrivial combinatorics, in the work of P. Aspinwall and D. Morrison [2], Manin [21], and C. Voisin [31]. In their work, the motivation was to compute Gromov-Witten invariants for Calabi-Yau 3-folds containing rational curves with the normal bundle O (−1) ⊕ O (−1); the present case 1 of rational is in a sense a relative version of this since Cg−1 contains the family Cg−1 curves whose normal bundle restricts to O (−1) ⊕ O (−1) on every curve. In any case, this completes the proof. corollary 6.2 For all u, v ≥ 0, (i) when d > g − 1, u

v

η ∗η =η

u+v

+q

u−1 i=0

θ g−d+i+v θ g−d+i u−1−i u+v−1−i ; η η − (g − d + i + v)! (g − d + i)!

(ii) when d = g − 1, u

v

η ∗η =η

u+v

u q θ i+v u−i θ i v+u−i + − η . η 1−q (i + v)! i! i=1

354


Proof The proof is similar to that of Corollary 4.3. 7. Presentation of the quantum ring The results of Corollaries 2.3, 4.3, 5.4, and 6.2 have given an explicit quantum multiplication table for Cd , provided d ∈ [(3/4)g, g − 1). It is natural to look for a presentation of the quantum cohomology ring as well. Such a presentation contains less information than the multiplication table because it does not specify the additive isomorphism between QH ∗ (Cd ) and H ∗ (Cd ; ). We are able to determine it even in the slightly more general case d ∈ [(4/5)g − 3/5, g − 1). To go this far requires only the degree-1 invariants and the results of Section 6; in principle one could continue as far as (6/7)g − 5/7 by using the degree-2 invariants, but this becomes cumbersome. proposition 7.1 The rings QH ∗ (Cd ) are generated over by Macdonald’s generators η and ξi , and there is a complete set of relations uniquely determined by the property that it reduces mod q to Macdonald’s relations. Proof These facts are well known for quantum cohomology generally in the case deg q > 0. The first statement is proved by a simple induction on the degree of the cup product of an arbitrary collection of classical generators. Just take a similar monomial where all the cup products are replaced by quantum products. The difference between the two monomials is a multiple of q, so the coefficient has lower degree and by induction it can be expressed as a quantum product of the classical generators. This shows that the classical generators generate the quantum ring as well. To extend the classical relations to quantum relations, first take a classical relation, and replace all the cup products by quantum products. This quantum expression may not be a quantum relation, but it is in the ideal q because it reduces to a classical relation. So express it as q times a classical monomial, replace the cup products in this monomial by quantum products, and subtract the result from the quantum expression. This difference is now in q 2 . Proceed inductively; when deg q > 0, the coefficient of a high power of q eventually is in H 0 but by induction on the power of θ rather than q. How do we find the quantum relations alluded to in Proposition 7.1? Macdonald’s classical relations from Theorem 2.2 can be expressed as r+|I |−α η sα (σi ) ξj ξk+g , (7.1) 0= α

where sα is the elementary symmetric polynomial of degree α. To find explicit quantum relations reducing to these mod q, the following result is useful. We adopt the notation ∗ for a quantum product and η∗u for a quantum power. proposition 7.2 For u ≥ 0, ηu equals (i) n θj η∗(n−j ) η∗u − q j!

for d > g, where n = u − d + g − 1;

j =0

(ii) (η + q)

∗u

−q

u−1

(η + q)∗(u−1−j )

j =0

θj j!

for d = g;

(iii) (η + r)∗u − r

u−1

(η + r)∗(u−1−j )

j =0

θj (j + 1)!

for d = g − 1, where r = q

(iv) η

∗u

−q

u−1 j =0

η∗j

θ n−j θ g−d + quη∗(u−1) + O q2 (n − j )! (g − d)!

g for < d ≤ g − 1, where n = u − d + g − 1. 2

θ ; (1 − q)

356


Proof For d > g, by Corollary 6.2(i), η ∗ ηv = ηv+1 + q

θ g−d+v . (g − d + v)!

Part (i) follows inductively by taking the quantum product of both sides with η. Similarly, for d = g, θv (η + q) ∗ ηv = ηv+1 + q , v! and, for d = g − 1, θv , (η + r) ∗ ηv = ηv+1 + r (v + 1)! so parts (ii) and (iii) also follow inductively. Part (iv) follows from Corollary 4.3 by an easy induction. For d ≥ g − 1 or d < (4/5)g − 3/5, it is now easy to obtain a complete set of quantum relations that are explicit if somewhat inelegant. Just plug the formulas of Proposition 7.2 into (7.1). When (2/3)g + 1/3 ≤ d < (4/5)g − 3/5, terms of higher order in q appear in the quantum product. However, these do not affect the relations since, by the last statement in Theorem 2.2, the latter are homogeneous of degree d or d + 1, too large to contain a power of q 2 . The same is true for the linear terms in q when (1/2)g + 1 ≤ d < (2/3)g − 1/3, leading to the amusing phenomenon that, in that range, the quantum and classical rings are isomorphic but only by a map that is not the identity. Although the general quantum relation is no thing of beauty, there are a few exceptions. corollary 7.3 (i) For d > 2g − 2, η

∗(d−2g+1)

∗

g ∗

(η − σi ) = q.

i=1

(ii) For g < d ≤ 2g − 2, g ∗

(η − σi ) = qη∗(2g−1−d) .

i=1

(iii) For d = g,

g ∗ i=1

(η − σi + q) = q(η + q)∗(g−1) .


357

Proof The first relation follows directly from Theorem 2.2 and Proposition 7.2(i) since the coefficient of q in the latter is −1 for u = d − g + 1 and zero for u < d − g + 1. To prove (ii) and (iii), note first that, by Theorem 2.2, the relation 0=

g

(η − σi )

i=1

holds in the classical ring for d ≤ 2g − 2. This is equivalent to 0=

g j =0

(−1)j ηg−j

θj . j!

One plugs in the formulas of Proposition 7.2(i) and (ii) for ηu , and one notes that the double sum in the second term telescopes using the binomial theorem. 8. Relation with Givental’s work In fact, Corollary 7.3(ii) and (iii) were known to the authors before any of the GromovWitten invariants, and they were the starting point of the investigation. With a little ingenuity, they can be read off from formulas in the wonderful paper of Givental [12], which is concerned chiefly with proving that the Gromov-Witten invariants of the quintic 3-fold satisfy the Picard-Fuchs equation. We sketch an outline of the connection. At the heart of Givental’s paper is a “quantum Lefschetz hyperplane theorem” explaining how the Gromov-Witten invariants of a variety are related to those of a hyperplane section. This allows him to compute a generating function for the GromovWitten invariants of complete intersections. He shows in [12, Cor. 6.4] that any differential operator annihilating this generating function determines a relation in the quantum cohomology. The symmetric products of a curve do not appear in any natural way as complete intersections in a projective space. But they do appear as complete intersections in a

projective bundle. Indeed, for any d ≤ 2g − 2, choose a reduced divisor P = pi of degree 2g − 1 − d. Then D → D + P gives an embedding Cd 2→ C2g−1 , whose image is a complete intersection of divisors in the linear system of O (1). In particular, for d ≤ 2g − 2, Cd embeds in C2g−1 , which is a Pg−1 -bundle over Jac2g−1 . Givental’s methods are perfectly adapted to studying this more general case. Indeed, his formulas for Gromov-Witten invariants are derived as special cases of formulas for equivariant Gromov-Witten invariants. These can be regarded as universal formulas for Gromov-Witten invariants of complete intersections in projective bundles because they are defined as relative Gromov-Witten invariants for complete intersections in the universal projective bundle over the classifying space.

358


Givental works with the group (C× )n+1 acting on Pn . The classifying space is then (CP∞ )n+1 , and the universal bundle is a direct sum of line bundles. This would appear to be a problem because C2g−1 is not the projectivization of a direct sum of line bundles. However, a splitting principle argument shows that all of Givental’s equivariant formulas extend word for word to the action of GL(n, C). The classifying space is then an infinite Grassmannian, and any projective bundle at all is pulled back from some map to a Grassmannian, even in the algebraic category. Thus a formula in equivariant cohomology determines a formula in the cohomology of C2g−1 or, more properly, in the cohomology of the spaces of stable maps to C2g−1 . This is perhaps surprising since no group is acting on C2g−1 . However, the group is present as the maximal torus of the structure group of the projective bundle C2g−1 → Jac2g−1 ; one regards equivariant cohomology as giving universal formulas in the cohomology of all such bundles. Another apparent problem is that the equivariant methods treat only GromovWitten invariants for classes e ∈ H2 killed by the projection to the base, namely, Jac2g−1 . But luckily, as we have seen, these are the only nonzero invariants of Cd . The symmetric product Cd is an equivariant complete intersection in C2g−1 of type (1, 1, . . . , 1). Such an intersection in a single projective space would just be a linear subspace, but it can be less trivial equivariantly, that is to say, in families. Givental’s results therefore apply with r = 2g − 1 − d, l1 = · · · = lr = 1. The case d > g is covered by Givental’s [12, §9]. The equivariant quantum potential satisfies the differential equation shown below his [12, Theorem 9.5]. As instructed in [12, Cor. 6.4], we make the substitutions d/dt = η (the relative hyperplane class), et = q, = 0 to obtain a quantum relation. We also substitute λi = σi since by Theorem 2.2 these are the Chern roots of the bundle whose projectivization is C2g−1 , and λ$i = 0 since these come from an additional equivariance we are not using. The result is precisely Corollary 7.3(ii)! Similarly, the case d = g is covered by [12, §10]. The differential equation satisfied by the equivariant quantum potential S $ is not explicitly stated, but it is (D − λi )S $ = l1 · · · lr et lj D − λ$j + m S $ , i

j,m

et

where D = d/dt + l1 · · · lr as in [12, Cor. 10.8]. Making the same substitutions as before transforms D to η + q and this equation to Corollary 7.3(iii). Finally, the case d = g − 1 is analogous to the Calabi-Yau case, which is covered by [12, §11]. Again the differential operator determines a relation in degree 2g, but now this is vacuous since the real dimension of Cg−1 is only 2g − 2. This is a familiar occurrence in Givental’s work. The Picard-Fuchs equation, for example, has degree 4, so it gives a quantum relation in H 8 of the quintic 3-fold, which is of course trivial.


359

Nevertheless, the Picard-Fuchs equation on the quintic does contain valuable information, enough to determine the 3-point invariants at genus zero and hence the virtual number of rational curves of all degrees. On Cg−1 the corresponding invariants were all worked out in Theorem 6.1, and they are all determined by the lines using the Aspinwall-Morrison formula. It is like a Calabi-Yau with no higher degree rational curves. But it certainly ought to be possible to recover these invariants for Cg−1 , or even Cd for d ≥ g − 1, by calculating Givental’s quantum potential in this case. A more daunting project would be to extend these methods to Cd for d < g − 1, where the results of this paper are incomplete. Givental remarks that the corresponding case for complete intersections remains unsolved, but it is in a sense “less interesting” since nonzero invariants appear only in finitely many degrees. For symmetric products, however, this is clearly the most interesting and mysterious case. 9. The homotopy groups of symmetric products of curves In this section, which plays the role of an appendix, we first compute the fundamental group of a symmetric product; then we show that the Hurewicz homomorphism π2 → H2 has rank 1 for d > 1. These results are not needed anywhere else, but they clarify the noncontribution of other homology classes, from the point of view of symplectic topology. For d > 2g − 2, the symmetric product is a projective bundle over the Jacobian, so it has fundamental group H1 (C; Z). On the other hand, for d = 1, it is of course π1 (C). What happens in between? The simplest possible thing, it turns out. We are grateful to Michael Roth for supplying the following theorem and its proof; but see also A. Grothendieck [15]. theorem 9.1 For d > 1, π1 (Cd ) = H1 (C; Z). Indeed, the Abel-Jacobi map induces an isomorphism on fundamental groups. Proof Choose a basepoint p ∈ C, and let i : C → C d be given by x → (x, p, . . . , p). Also let ψ : C d → Cd be the quotient by the symmetric group. We show, first, that π1 (ψi) surjects on π1 (Cd ), second, that its image is abelian, and third, that the kernel is exactly the commutator subgroup. Choose as a basepoint in Cd the divisor d · p. Since the big diagonal has real codimension 2, any loop can be perturbed such that it meets the big diagonal only at the basepoint. It then lifts unambiguously to a loop on C d . Hence π1 (ψ) is surjective. But any loop on C d is homotopic to a composition of loops on the various factors. Since ψ is symmetric, each of these loops can be replaced with the corresponding

360


loop on the first factor, that is, on the image of i, without changing π1 (ψ) of the composite. Hence π1 (ψi) is surjective. Let γ1 , γ2 be two loops in π1 (C). To show that their images in Cd commute, first include them in C d via i. Then note that γ2 can be transferred to the second factor without changing its image in Cd ; then, however, it commutes with γ1 in π1 (C d ) = π1 (C)d . Finally, to show that ψi kills only loops in the commutator, compose ψi with the Abel-Jacobi map. The resulting map C → Jacd is itself just the composition of the Abel-Jacobi map with the identification Jac1 ∼ = Jacd induced by p. But it is well known that the Abel-Jacobi map induces an isomorphism H1 (C; Z) = H1 (Jac1 ; Z) (see, e.g., Griffiths and Harris [14, Chap. 2, §7]). theorem 9.2 For d > 1, the Hurewicz homomorphism π2 (Cd ) → H2 (C; Z) has rank 1. Proof It is easy to see that the rank is at least 1; for example, take C to be hyperelliptic. There is then a 1-dimensional linear system ; on Cd , which is a 2-sphere, nontrivial in rational homology because η · ; = 1. To show it is at most 1, note that by the above theorem the universal cover C˜ d is the fibered product Cd ×Jacd H 1 (C, O ), where the vector space H 1 (C, O ) is the universal cover of Jacd . Now the pushforward H2 (Cd ; Q) → H2 (Jacd ; Q) has 1-dimensional kernel. This follows, for example, from Proposition 2.1(iii) and Theorem 2.2. Since the map C˜ d → Cd → Jacd also factors through the contractible space H 1 (C, O ), the induced map on H2 is zero, and so the map H2 (C˜ d ) → H2 (Cd ) has rank at most 1. But by the Hurewicz isomorphism and the homotopy exact sequence, H2 (C˜ d ) = π2 (C˜ d ) = π2 (Cd ), and the map to H2 (Cd ) is exactly the Hurewicz homomorphism. Acknowledgments. We wish to thank Arnaud Beauville, Olivier Debarre, Tom Graber, Michael Hutchings, Rahul Pandharipande, Michael Roth, Bernd Siebert, Ravi Vakil, and Angelo Vistoli for very helpful conversations and advice. We also wish to thank the Institut Mittag-Leffler for its warm hospitality and extraordinary atmosphere, which inspired the present work.

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Bertram Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA Thaddeus Department of Mathematics, Columbia University, New York, New York 10027, USA

A METHOD FOR PROVING Lp -BOUNDEDNESS OF SINGULAR RADON TRANSFORMS IN CODIMENSION 1 MICHAEL GREENBLATT

Abstract Singular Radon transforms are a type of operator combining characteristics of both singular integrals and Radon transforms. They are important in a number of settings in mathematics. In a theorem of M. Christ, A. Nagel, E. Stein, and S. Wainger [3], Lp boundedness of singular Radon transforms for 1 < p < ∞ is proven under a general finite-type condition using the method of lifting to nilpotent Lie groups. In this paper an alternate approach is presented. Geometric and analytic methods are developed which allow us to prove Lp -bounds in codimension 1 under a curvature condition equivalent to that of [3]. We restrict consideration to the important case where the hypersurfaces are graphs of C ∞ -functions. Our methods do not involve the Fourier transform, lifting, or facts about Lie groups. This might prove useful in extending our work to related problems. 1. Setup We assume that we have an operator T taking Schwartz functions on Rn+1 to Schwartz functions on Rn+1 given by f t + S(t, x, y), y K(t, x, y) dy. (1.1) Tf (t, x) = Rn

Rn ,

t ∈ R. We assume that K(t, x, y) is a distribution supported on a Here x, y ∈ compact set U and is equal to a function at points where x = y. K(t, x, y) is to be viewed as a singular integral kernel in x and y, so we stipulate that for 0 ≤ |α| ≤ 1, K(t, x, y) satisfies α α ∂ K(t, x, y) < M|x − y|−n−|α| ∂ K(t, x, y) < M|x − y|−n−|α| , x y α ∂ K(t, x, y) < M|x − y|−n . (K1) t

We also need a cancellation condition on K. Although the methods presented here can be applied to K’s satisfying more general cancellation conditions, for the purposes DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 2, Received 16 July 1999. Revision received 11 September 2000. 2000 Mathematics Subject Classification. Primary 42B20; Secondary 42B35.

363

364

MICHAEL GREENBLATT

of this paper we stipulate that for all > 0, K satisfies K(t, x, y) dy , K(t, x, y) dx < M. |y−x|=

|y−x|=

(K2)

Here the integrals are taken with respect to surface measure. Thus Tf (t, x) is the integral of f on the graph of y → (t + S(t, x, y), y) with respect to the distribution K(t, x, y). We have that S(t, x, x) = 0, so that (t, x) is on this surface. The conditions (K1)–(K2) are a way of describing K(t, x, y) as being a singular integral kernel; the operator at (t, x) is a singular integral over a surface containing (t, x), “centered at (t, x),” and these surfaces may vary with (t, x). The curvature condition is a way of saying that the surfaces “change to finite order” in at least one direction tangent to the surfaces. We now turn to our assumptions on S. We will define a positive integer d below, the order of degeneracy of T . We assume that S is a C ∞ -function satisfying S(t, x, x) = 0, α β γ ∂ ∂ ∂t S(t, x, y) < M, |∂t S| < 1 . sup x y 2 (t,x,y)∈U,|α|+|β|+|γ |≤d+1

(S1)

The curvature condition is a finite-type condition equivalent to that of [3]. There are a few ways to state it; a convenient definition for the purposes of this paper is as follows. At each (t0 , x0 ), we consider the function ttS0 ,x0 (x, y) defined to be the solution tt∗0 ,x0 (x, y) to tt∗0 ,x0 (x, y) + S tt∗0 ,x0 (x, y), x, y − t0 − S(t0 , x0 , y) = 0. This function exists and is as smooth as S is by the implicit function theorem. Define S (t , x ) = ∂ α ∂ β t ∗ (x , x ). The definition of the curvature condition used in τα,β 0 0 x y t0 ,x0 0 0 this paper is: S (t0 , x0 ) = 0. ∀(t0 , x0 ) ∈ U ∃|α| ≥ 1 and |β| ≥ 1 such that τα,β

(S2)

By compactness we can assume that for our d,

S (t0 , x0 ) ≥ . ∃ > 0∃|α|, |β| ≥ 1 with |α| + |β| ≤ d such that ∂xα ∂yβ τα,β

(S2 )

Assume that d has been chosen to be minimal. To help see what this means, consider the coordinate change given by Bt0 ,x0 : (t, y) → (t − S(t0 , x0 , y), y). This is evidently a smooth coordinate change with a Jacobian determinant near 1 taking (t0 , x0 ) to itself. Thus, if Ct0 ,x0 denotes the operator taking a function f to f ◦ Bt0 ,x0 , Lp -boundedness of T is equivalent to T Ct0 ,x0 . However, Ct−1 T Ct0 ,x0 is also a singular Radon Lp -boundedness of Ct−1 0 ,x0 0 ,x0 transform (although now K(t, x, y) is replaced by something which may not satisfy

365

SINGULAR RADON TRANSFORMS

(K2)) with S-function given by S1 (t, x, y) = S(t0 , x0 , x) + S(t + S(t0 , x0 , x), x, y) − S(t0 , x0 , y). Note that S1 has the property that S1 (t0 , x0 , y) = 0 for all y. We see that the curvature condition at (t0 , x0 ) is just ∃|α| ≥ 1, |β| ≥ 1 with |α| + |β| ≤ d such that ∂xα ∂yβ S1 (t0 , x0 , x0 ) ≥ . (S2 ) Equation (S2 ) is equivalent to the curvature condition of [3], as is shown in that paper. This statement of the curvature condition also appears in [18] (see also [11]). In the ˜ − y) for some S) ˜ the curvature translation-invariant case (where S(t, x, y) = S(x condition appearing here is equivalent to S being of finite type. This is the main result. main theorem Suppose that T is as in equation (1.1) that K satisfies (K1)–(K2), and that S satisfies (S1)–(S2 ). Then there exists an r > 0 depending on , n, d, and M such that if K is supported in a set of the form {(t, x, y) : |t − t0 |, |x − y| < r}, then T is bounded on each Lp for 1 < p < ∞ with bound depending on , n, d, p, and M. Note In what follows, any constant C, C , C , C0 , C1 , C2 , and so on, or δ, δ , δ , δ0 , δ1 , δ2 , and so on, that appears denotes a positive constant depending only on M, d, n, , and p. Sometimes we use the notation C several times within a single argument to denote several such constants. The main theorem suffices to prove Lp -boundedness of our original singular Radon transform; we just chop up the function K into two parts. The first part Ko is supported on {(t, x, y) : |x − y| > r}, and the resulting singular Radon transform To is easily seen to be bounded on any Lp since Ko is now a bounded compactly supported function: p To f (t, x)p dt dx = n f t + S(t, x, y), y Ko (t, x, y) dy dt dx R f t + S(t, x, y), y p dt dx dy. ≤C U

But since |∇t S| < 1/2, in the t-y integration we may make the coordinate change (t , y ) = (t + S(t, x, y), y), and the above is at most p f (t , y ) dt dy < Cf pp . C R2n+2

Thus this first part of our singular Radon transform is bounded on Lp . The second part can be written as the sum of at most Cr −(2n+1) kernels, each of which is supported in a set of the form {(t, x, y) : |t − t0 |, |x − y| < r}, and thus the

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MICHAEL GREENBLATT

theorem applies to the resulting singular Radon transforms. Hence the norm of the original singular Radon transform on Lp is bounded in terms of , n, d, M, and p. One successful method of studying singular Radon transforms has proceeded along the following lines. Start with the translation-invariant situation on a homogeneous nilpotent Lie group, where proving Lp -boundedness is less complicated due to the fact that T is a convolution operator. The idea is to then reduce the study of an arbitrary T to one that is almost translation-invariant on a nilpotent Lie group by a lifting technique. Applying the lifted operator to a function independent of certain coordinates is essentially equivalent to applying the unlifted operator to a “slice” of this function. This method was developed by several workers, culminating in [3]. In [3], Lp -boundedness of singular Radon transforms is proven for all 1 < p < ∞ in any codimension, under a natural curvature condition that reduces in the translationinvariant case to the submanifolds that are integrated over being of finite type. Although the method of [3] was successful, the use of lifting to Lie groups made the proof indirect, and for the sake of applications a more direct approach was sought. How this might be brought about is suggested by a common thread in many of the papers written on the subject, including the papers of the previous paragraph, the papers using a pseudodifferential operator approach such as [8] and [12], the direct approach in [1], and even some of the earlier results such as [20] and the pioneering work of E. Fabes [6]. In all of these a natural metric is associated to a given singular Radon transform, and a naturally associated singular integral with respect to this metric, call it T˜ , is defined. In most cases the difference T − T˜ is considered, and by a variety of means appropriate Lp -boundedness is proved. Therefore, to prove a general theorem of this kind by a direct approach it would be reasonable to expect to need to be able to define such a metric in a general nontranslation-invariant setting. Clues on how to do this appear in a paper by D. Phong and Stein [13] (announced in [12]). The approach of [13] involves writing Tf (t, x) as eiλ,t+S(t,x,y) K(t, x, y)fˆ(λ, y) dλ dy. Rn+m

Here t ∈ Rm , x, y ∈ Rn , and fˆ denotes Fourier transform in the first component only. In [6], T is then written as the sum of two parts: the “inside” portion where |x − y| < C1 |λ|1/2 and the “outside” portion where |x −y| > C2 |λ|1/2 . The “outside” portion is dealt with via a T T ∗ argument, while the “inside” portion is viewed as eiλ,t a(t, λ)fˆ(λ, ·) dλ, (1.2) Rn+m

where a(t, λ) is an oscillatory integral operator taking functions of y to functions of x given by


a(t, λ)f (x) =

Rn

367

eiλ,S(t,x,y) φ |λx − y|2 K(t, x, y)f (y) dy.

So in this sense the “inner” portion is a pseudodifferential operator with an operatorvalued symbol, and in [13] pseudodifferential operator methods are used to prove L2 -boundedness. The results in [13] are in the nondegenerate case d = 2 in codimension 1, when S satisfies a condition called rotational curvature at every point (t, x). This means that if at some (t, x) we are in coordinates such that S(t, x, y) = 0 for all y, the Hessian of S in x and y has full rank. It is clear from the arguments in [13] that the results can be extended to some higher codimension situations when S has an analogous nondegeneracy condition. This is taken a large step further in the thesis of S. Cuccagna [4], where the the Hessian of S is merely required to have rank at least 1 with appropriate generalizations to higher codimensions. That setup eventually led to the setup of this paper, in the following fashion. Let γ0 (t) be a radial nonnegative bump function on R supported on |t| < 1, and equal to 1 for |t| < 1/2. Let γ = γ0 (t) − 2γ0 (2t), and γ1 = γ0 (t) − γ0 (2t). Rewrite (1.2) as eiλ,t+S(t,x,y) γˆ 2−i 2−2j λ γ1 2j (x − y) K(t, x, y)fˆ(λ, y) dλ dy. Rn+1

If we let j = i + 2j , this becomes eiλ,t+S(t,x,y) γˆ 2−j λ γ1 2−i/2 2j /2 (x − y) K(t, x, y)fˆ(λ, y) dλ dy. Rn+1

This is basically the same as eiλ,t+S(t,x,y) γˆ 2−j λ γ1 2−i |λx − y|2 K(t, x, y)fˆ(λ, y) dλ dy. Rn+1

In Cuccagna’s thesis [4], when i = 0, the sum of these operators over j is considered (i.e., the γˆ disappears), and a 1/2-1/2 pseudodifferential operator argument is used analogous to the “inside” operators mentioned above. As for the rest of the operator, for a fixed j he considers the sum over i > 0 of these operators (i.e., there is no decomposition in i); call it Tj . Almost-orthogonality is considered, and then for a fixed j uniform estimates |Tj | < C are proved using a T T ∗ argument with an appropriate integration by parts that uses the nondegeneracy condition. Although not done this way in [4], one way to prove these uniform estimates is to write Tj as the sum of Ti,j ’s, as above, and to prove exponential decay in i of Ti,j . This approach to the L2 -estimates in [4], suggested to me by E. Stein, grows out of arguments in [16] where an analogous decomposition is used for analyzing oscillatory integral operators with polynomial phase (see also [10] for generalizations). This was the setup of an earlier rendition of this paper, where operators Ui,j were defined as

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MICHAEL GREENBLATT

Ui,j =

Rn+1

eiλ,t+S(t,x,y) γˆ 2−j λ γ 2−i 2j σ (t, x; |x−y|) K(t, x, y)fˆ(λ, y) dλ dy.

Here σ (t, x; |x − y|) is a “height” factor adapted to the geometry of T that we define below. For i > 0 almost orthogonality was again used over the different j ’s, and an argument like that of Section 5 was used to show exponential norm decay in i. For i = 0 a separate argument somewhat resembling that of Section 4 was used. However, it became clearer as time went on that these Fourier transform definitions were not natural in some respects, and they were eventually supplanted by the operators used in this paper. It should be noted that this type of decomposition T = i,j Ti,j is also done in [2] and [5] for the (translation-invariant) operators considered in those papers. 2. Definition of balls and setup of proof We now define a system of non-Euclidean balls. They are used not only to apply the Calderón-Zygmund theory toward proving our Lp -estimates but also to make relevant estimates used in the L2 theory. Simpler non-Euclidean balls were used for the Lp theory in [4] and [13]; with the help of some suggestions from Stein, they were the inspiration for the non-Euclidean balls and the function σ (t, x; r) used here. It also would not be surprising if the metric used here were equivalent to the one obtained by applying the machinery of [9] to the vector fields associated to a singular Radon transform in [3] (see also [1], [2], [4], [17], [20] for other examples of such metrics used in related problems). Define σ S (t, x; r) by  σ S (t, x; r) = 

S (t, x)(cr)|α|+|β| τα,β

2

1/2 

.

0 0, we define −1 i −1 t + S(t, x, y) − s 2i σ t, x, 2−j γ 2 σ t, x; 2−j Ti,j f (t, x) = Rn+1

369


× γ1 2j (x − y) K(t, x, y)f (s, y) ds dy.

(2.1)

T0,j is defined the same way with i = 0, except that the γ is replaced by γ0 . We define Ti = j Ti,j . Thus T = i≥0 Ti . Thus Ti,j is Ti with K(t, x, y) localized to |x − y| ∼ 2−j . If we write Kj (t, x, y) = K(t, x, y)γ1 (2j (x − y)), then Kj satisfies the same estimates that K does and in addition is supported in |x − y| ∼ 2−j , so we can rewrite (3.1) as −1 i −1 2i σ t, x; 2−j γ 2 σ t, x; 2−j t +S(t, x,y)−s Kj (t, x,y)f (s,y) ds dy. Rn+1

For the purposes of our arguments Ti f (t, x) could have been defined as 2i σ (t, x; |x − y|)−1 γ 2i σ (t, x; |x − y|)−1 t + S(t, x, y) − s Rn+1

(2.1 )

× K(t, x, y)f (s, y) ds dy

(with γ replaced by γ0 if i = 0) instead of writing Ti as a sum over j and fixing the radius in the σ ’s on a fixed Ti,j . However, for the arguments appearing here a dyadic decomposition over these annuli would be necessary anyhow, and in addition the analysis becomes slightly less cumbersome with a fixed radius in the σ ’s. But in the form (2.1 ) it is perhaps more apparent that T0 is basically a singular integral operator with a metric deriving from the balls defined above and that Ti is in some sense a hybrid of T0 and the original operator T ; with increasing i, Ti may be viewed as getting “closer” to T , but with an additional cancellation condition arising from the fact that γ = 0. One way of proving L2 -boundedness of many singular integrals, such as Riesz transforms, involves the use of almost-orthogonality, and due to the resemblance of the Ti to singular integral operators, we are able to use almost-orthogonality here too. We see that we have Ti,j T ∗ , T ∗ Ti,j < C23i/2 2−|j1 −j2 |/2 . (2.2) 1 i,j2 2,2 2 2,2 i,j1 In addition, we show for i > 0 that there is a δ > 0 such that as an operator on L2 , Ti,j < C2−δi . As a result, using (2.2), we have

Ti,j T ∗ , T ∗ Ti,j < min C23i/2 2−|j1 −j2 |/2 , C2−2δi < C 2−δ |j1 −j2 |−δ i . 1 i,j2 2 i,j1

We can then apply almost-orthogonality to conclude that Ti ≤ C 2−δ i . Summing over i gives T < C . We prove the uniform L2 -bounds on the Ti,j by applying the following variant of ∗ , whose simple proof is included for convenience. Schur’s test to the kernel of Ti,j Ti,j

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MICHAEL GREENBLATT

lemma 2.1 Suppose that T (f )(x) = K(x, y)f (y) dy, and suppose that |K(x, y)| dx < C1 , |K(x, y)|dy < C2 . Then as an operator on L2 , T < (C1 C2 )1/2 . Proof For any functions f and g, Tf, g = K(x, y)f (y)g(x) dy dx

1/4 1/4 C1 C2 f (y) g(x) dy dx = K(x, y) C1 C2 1/2 1 C2 ≤ |f (y)|2 |K(x, y)| dy dx 2 C1 1 C1 1/2 |g(x)|2 |K(x, y)| dy dx + 2 C2 1 1/2 2 2 |f (y)| dy + |g(x)| dx . ≤ (C1 C2 ) 2

Taking the supremum over all f and g with L2 -norm 1 gives us the lemma. ∗ satisfies the conditions of Lemma 2.1 for approWe show that the kernel of Ti,j Ti,j priate C1 and C2 by a rather complicated integration by parts argument involving the geometric and analytic constructs of this paper. A lemma of M. Christ similar to the Van der Corput lemma is used in conjunction with the curvature condition and the ¯ −δi for all i and j . fact that γ = 0. We then have that Ti,j 2,2 ≤ C2 As for the Lp theory for the singular Radon transforms of this paper, the generalization of the Calderón-Zygmund theorem for non-Euclidean balls shows that on any Lp , 1 < p < ∞, we have Ti ≤ C (i + 1). Thus by simple interpolation with the L2 situation we have Ti ≤ C 2−δ i . Summing then gives T < C . It should be pointed out that L2 and Lp estimates such as the above were proved for the analogous operators in [2]. That paper considers singular Radon transforms over odd translation-invariant homogeneous curves on a homogeneous nilpotent Lie group. The (T T ∗ )n arguments are used in conjunction with a fair amount of Lie group analysis to prove the corresponding estimates. The main novelties of this paper are in the approach used here for the L2 -estimates of Section 5 (e.g., the functions t ∗ and τα,β ) and the geometric properties generated by this approach, such as the definition of the balls of Section 2 and the special coordinate changes used to make S(t, x, y) = 0 for all y and a fixed (t, x). It is worth

371


stressing that the arguments presented here are direct real-variable arguments; neither the Fourier transform, lifting, nor facts about Lie groups are used. 3. Geometric lemmas The following two facts about our non-Euclidean balls allow us to apply the CalderónZygmund method for our Lp -estimates, and the second fact is also very important for our L2 -estimates. fact 1 Let m be Lebesgue measure: m(B S (t, x; 2r)) ≤ 2n+d+1 m(B S (t, x, r)). fact 2 There exists C such that B S (t0 , x0 ; r) ∩ B S (t1 , x1 ; r) = ∅ −→ B S (t1 , x1 ; r) ⊂ B S (t0 , x0 ; Cr). For the proof of Fact 1 we use the following simple lemma. lemma 3.1 If r2 ≥ r1 , then (r2 /r1 )2 σ (t, x; r1 ) ≤ σ (t, x; r2 ) ≤ (r2 /r1 )d σ (t, x; r1 ). Proof Because 2 ≤ |α| + |β| ≤ d, we have the following two inequalities: 2 r2 S S (t, x)(cr2 )|α|+|β| , τα,β (t, x)(cr1 )|α|+|β| ≤ τα,β r1 r d 2 S S |α|+|β| ≤ τα,β (t, x)(cr1 )|α|+|β| . τα,β (t, x)(cr2 ) r1 Squaring and adding these over α and β gives us the lemma. Proof of Fact 1 The proof of Fact 1 is then as follows:   S m B (t, x; 2r) = m  (s, y) : |y − x| < 2r, |s − t − S(t, x, y)| 



2 1/2  S . < (t, x)(2cr)|α|+|β| τα,β  1≤|α|,|β|≤d

By Lemma 3.1 this expression is at most 2d times

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MICHAEL GREENBLATT

  m  (s, y) : |y − x| < 2r, s − t − S(t, x, y) 
0 such that if (t1 , x1 ) ∈ B S (t0 , x0 ; r), then δσ (t0 , x0 ; r) < σ (t1 , x1 ; r) < 1/δσ (t0 , x0 ; r). In what follows we write σ (t0 , x0 ; r) ∼ σ (t1 , x1 ; r) as a shorthand for the conclusion of the above lemma. Proof We first consider the case where S(t0 , x0 , y) = 0 for all y. We differentiate the relation t ∗ (t1 , x1 , x, y) + S(t ∗ (t1 , x1 , x, y), x, y) − t1 − S(t1 , x1 , y) = 0 with respect to y and x, obtaining −1 (3.1a) Sy (t ∗ , x, y) − Sy (t1 , x1 , y) , ∂y t ∗ = − 1 + St (t ∗ , x, y) −1 (3.1b) ∂x t ∗ = − 1 + St (t ∗ , x, y) Sx (t ∗ , x, y). Differentiating these expressions with respect to x and y at most d times, substituting (3.1) whenever ∂y t ∗ or ∂x t ∗ appear, we get −1 α β ∗ ∂x ∂y S t , x, y + Eα,β t ∗ , t1 , x1 , x, y . (3.2) ∂xα ∂yβ t ∗ = − I + St t ∗ , x, y Here Eα,β (t ∗ , t1 , x1 , x, y) is a constant times a product of functions of the following types:

373


(1) (I + St (t ∗ , x, y))−1 ; α (2) ∂tα1 ∂xα2 ∂y 3 S(t ∗ , x, y) for |α1 | + |α2 | + |α3 | ≤ d; (3) ∂yα4 S(t ∗ , x, y) − ∂yα4 S(t1 , x1 , y) for |α4 | ≤ d. α At least one of the members of this product must be of the form ∂xα2 ∂y 3 S(t ∗ , x, y) for α2 ≤ α and α3 ≤ β with at least one of the inequalities being strict, or of the form ∂yα4 S(t ∗ , x, y) − ∂yα4 S(t1 , x1 , y) for α4 < β. The above characterization can be proved inductively in a straightforward way; to avoid a technical but unenlightening argument the proof is omitted here. If we set t1 = t0 and y = x = x1 , then t ∗ = t1 = t0 , and (3.2) becomes ∂xα ∂yβ t ∗ t0 , x1 , x1 , x1 = ∂xα ∂yβ S(t0 , x1 , x1 ) + Eα,β t0 , t0 , x1 , x1 , x1 . (3.3) And since one of the factors in each of the products mentioned above must be of the α form ∂xα2 ∂y 3 S(t0 , x1 , x1 ) for appropriate α2 and α3 , or of the form ∂yα4 S(t0 , x1 , x1 ) − α4 ∂y S(t0 , x1 , x1 ) = 0, we have α β Eα,β t0 , t0 , x1 , x1 , x1 ≤ C ∂ ∂ S(t0 , x1 , x1 ). x y (0,0)≤(α ,β )0,|α |+|β |≤d

x

y

α β ∗ ∂ ∂ t (t0 , x1 , x1 , x1 )(cr)|α|+|β| . x y

(3.8) Thus if first c is chosen appropriately large, and r is appropriately small, we may


conclude that 0.99

∂xα ∂yβ t ∗ (t0 , x1 , x1 , x1 )(cr)|α|+|β| ∂ α ∂ β S(t0 , x0 , x0 )(cr)|α|+|β| < x y ∂ α ∂ β t ∗ (t0 , x1 , x1 , x1 )(cr)|α|+|β| . < 1.01 x y

375

(3.9)

Equation (3.9) is true for any x1 with |x1 − x0 | < r, including x0 itself. So since σ (t0 , x1 ; r) is within a factor of C of the right-hand sum, we conclude that there is a δ with δ < σ (t0 , x1 ; r)/σ (t0 , x0 ; r) < δ −1 for all such x1 . Note also that by setting x0 = x1 in (3.9) one can see the equivalence of curvature conditions (S2 ) and (S2 ). Next, note that for (t1 , x1 ) ∈ B S (t0 , x0 ; r) we have τ S (t1 , x1 ) − τ S (t0 , x1 )(cr)|α|+|β| σ (t1 , x1 ; r) − σ (t0 , x1 ; r) < C α,β α,β < C t1 − t0 r 2 < C r 2 σ (t0 , x0 ; r). Thus, as long as r 2 < (1/2)C 2 , we get 1/2 < σ (t1 , x1 ; r)/σ (t0 , x1 ; r) < 2, and hence we may conclude that there is a δ with δ < σ (t1 , x1 ; r)/σ (t0 , x0 ; r) < δ −1 . So we are done with Lemma 3.2 in the special case that S(t0 , x0 , y) = 0 for all y. For the general case we replace S by S1 (t, x, y) = S(t0 , x0 , x)+S(t+S(t0 , x0 , x), x, y) − S(t0 , x0 , y) to insure that S1 (t0 , x0 , y) = 0. The above argument proves the lemma for S1 and B S1 (t0 , x0 ; r), and the invariance properties of B S (t, x; r) under this coordinate change (see the discussion prior to the proof of this lemma) prove the result for B S (t0 , x0 ; r). It is true that the bounds obtained depend on bounds on the derivatives of S1 and not S, but the coordinate change above has Jacobian near 1, and as a result the bounds obtained really do depend on the bounds on derivatives of S. Hence we are finished with the proof of Lemma 3.2. lemma 3.3 There is a C such that if (t, x) ∈ B S (t0 , x0 ; r) and |y −x0 | < 3r, then |t +S(t, x, y)− t0 − S(t0 , x0 , y)| ≤ Cσ (t0 , x0 ; r). Proof As in the proof of Lemma 3.2, the invariance properties of the σ ’s and B’s under special coordinate changes allow us to assume that S(t0 , x0 , y) = 0. Our task is to prove that there exists a C such that if |t − t0 | < σ (t0 , x0 ; r), |x − x0 | < r, and |y − x0 | < 3r, then |t + S(t, x, y) − t0 | < Cσ (t0 , x0 ; r). But expanding S(t0 , x, y) in x and y about y = x = x0 gives S(t0 , x, y) ≤ C0 ∂xα ∂yβ S(t0 , x0 , x0 )r |α|+β| + C1 |r|d+1 < C2 σ (t0 , x0 ; r). |α|+|β|≤d

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MICHAEL GREENBLATT

Therefore if |t| < σ (t0 , x0 ; r), we have t +S(t, x, y)−t0 ≤ |t −t0 |+ S(t, x, y)−S(t0 , x, y) + S(t0 , x, y) ≤ C3 σ (t0 , x0 ; r). So we are done with the proof of Lemma 3.3. We can now proceed to the proof of Fact 2. Proof of Fact 2 Suppose B S (t0 , x0 ; r) ∩ B S (t1 , x1 ; r) is nonempty, and (t, x) ∈ B S (t0 , x0 ; r) ∩B S (t1 , x1 ; r). Then |x − x0 |, |x − x1 | < r (and thus |x0 − x1 | < 2r) and t − S(t1 , x1 , x) − t1 ≤ σ (t1 , x1 ; r), (3.10) t − S(t0 , x0 , x) − t0 ≤ σ (t0 , x0 ; r). Thus by Lemma 3.2 we have σ (t0 , x0 ; r) ∼ σ (t, x; r) ∼ σ (t1 , x1 ; r), and by Lemma 3.3 whenever |y − x0 |, |y − x1 | < 3r, we have t + S(t, x, y) − t0 − S(t0 , x0 , y) ≤ Cσ (t0 , x0 ; r), t + S(t, x, y) − t1 − S(t1 , x1 , y) ≤ Cσ (t1 , x1 ; r). Therefore we have t1 + St1 , x1 , y) − t0 − S(t0 , x0 , y) ≤ Cσ (t0 , x0 ; r) + Cσ (t1 , x1 ; r) ≤ C1 σ (t0 , x0 ; r).

(3.11)

Now (3.11) holds whenever |y − x0 |, |y − x1 | < 3r, in particular, when |y − x1 | < r. If (s, y) ∈ B(t1 , x1 ; r), then (s, y) satisfies |y − x1 | < r, |s − t1 − S(t1 , x1 , y)| ≤ σ (t1 , x1 ; r), and we have s − t0 − S(t0 , x0 , y) ≤ σ (t1 , x1 ; r) + C1 σ (t0 , x0 ; r) ≤ C2 σ (t0 , x0 ; r)

(3.12)

≤ σ (t0 , x0 ; C3 r). We use Lemma 3.1 for the last inequality. Furthermore, |y−x0 | ≤ |y−x1 |+|x1 −x0 | < 3r (assuming C3 ≥ 3). We conclude that (s, y) ∈ B S (t0 , x0 ; C3 r). Fact 2 is hence proven. Lastly, we repeatedly use this little lemma. lemma 3.4 For all (t, x) and for all i and j , (1) we have |∂ti σ (t, x; r)| ≤ Cr 2 ;

377


(2) if (t0 , x0 ) is such that S(t0 , x0 , y) = 0 for all y, and (t1 , x1 ) ∈ B(t0 , x0 ; r), then |∂xj σ (t1 , x1 ; r)| ≤ (C/r)σ (t0 , x0 ; r). Proof We have (σ (t, x; r))2 = 0 0, the proof in the case i = 0 again ∗ T being identical once the function γ has been replaced by γ0 . The kernel of Ti,j i,j2 , 1


381

call it Q(s1 , y1 , s2 , y2 ), is given by −1 −1 i −1 22i σ t, x; 2−j2 σ t, x; 2−j1 γ 2 σ t, x; 2−j2 t + S(t, x, y2 ) − s2 −1 t + S(t, x, y1 ) − s1 Kj1 (t, x, y1 )Kj2 (t, x, y2 ) dx dt. × γ 2i σ t, x; 2−j1 (4.6) (When i = 0, replace γ by γ0 .) Again we might as well assume that j1 ≤ j2 . Analogous to before, our objective is to show that |Q(s1 , y1 , s2 , y2 )| ds1 dy1 < C23i 2j1 −j2 . Once more |Q(s1 , y1 , s2 , y2 )| ds2 dy2 < C is a simple matter, and Lemma 2.1 then completes the proof of this lemma. As before, we perform a convenient coordinate change, this time to insure S(s2 , y2 , y) = 0 for all y. To this end we consider Q(s1 +S(s2 , y2 , y1 ), y1 , s2 , y2 ) and change variables in (4.6) from t to t = t − S(s2 , y2 , x). As in the previous proof, we have replaced S by S1 (t, x, y) = S(s2 , y2 , x)+S(t +S(s2 , y2 , x), x, y)−S(s2 , y2 , y), the σ ’s for S have been replaced by the σ ’s for S1 , and we have replaced the kernels Kj1 (t, x, y) and Kj2 (t, x, y) by Kj 1 (t, x, y) = Kj1 (t + S(s2 , y2 , x), x, y) and Kj 2 (t, x, y) = Kj2 (t + S(s2 , y2 , x), x, y), respectively. Thus in the following we may replace Q(s1 , y1 , s2 , y2 ) by Q (s1 , y1 , s2 , y2 ) = Q(s1 + S(s2 , y2 , y1 ), y1 , s2 , y2 ), S(t, x, y) by S1 (t, x, y), Kj1 (t, x, y) by Kj 1 (t, x, y), and Kj2 (t, x, y) by Kj 2 (t, x, y). In analogy to (4.3a) and (4.3b), note that when the integrand in (4.6) is nonzero, (s1 , y1 ) ∈ B(t, x; 2−j1 ) and (s2 , y2 ) ∈ B(t, x; 2−j2 ). So by Fact 2, (t, x) ∈ B(s1 , y1 ; C2−j1 ) ∩ B(s2 , y2 ; C2−j2 ), and then by Fact 2 again we have (s1 , y1 ) ∈ B(s2 , y2 ; C 2 2−j1 ). So since S1 (s2 , y2 , y) = 0 for all y, we have |t − s2 | < C σ (s2 , y2 ; 2−j2 ) and |s1 − s2 | < C σ (s2 , y2 ; 2−j1 ). Our strategy here is not all that different from that of Lemma 4.1. We would like to fix (t, x) at (s2 , y2 ) in (4.6). Correspondingly, we define the function G(s1 ,y1 ,s2 ,y2 ,t,x) t , x −1 −1 −1

t + S(t, x, y2 ) − s2 = σ t , x; 2−j2 σ t , x ; 2−j1 γ 2i σ t , x; 2−j2 −1 t + S t , x , y1 − s1 Kj 1 t , x , y1 Kj 2 t , x, y2 . ×γ 2i σ t , x ; 2−j1 We then write Q = Q + Q , where −1 −1 σ s2 , y2 ; 2−j1 Q (s1 , y1 , s2 , y2 ) = 22i σ s2 , x; 2−j2 −1 × γ 2i σ s2 , x; 2−j2 t + S(t, x, y2 ) − s2 −1 s2 + S(s2 , y2 , y1 ) − s1 × γ 2i σ s2 , y2 ; 2−j1 × Kj 1 (s2 , y2 , y1 )Kj 2 (s2 , x, y2 ) dx dt.

(4.7)

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MICHAEL GREENBLATT

And Q s1 , y1 , s2 , y2 ≤

22i G(s1 ,y1 ,s2 ,y2 ,t,x) (t, x) − G(s1 ,y1 ,s2 ,y2 ,t,x) (s2 , y2 ) dx dt. (4.8)

As before, we may use the cancellation condition to estimate Q . In the dt integration of (4.7), do a change of variables t = t + S(t, x, y2 ) − s2 . Since S(t, x, x) = 0, ∂t S(t, x, x) = 0, and this change of variables has Jacobian 1 + 0(|x − y2 |). Furthermore, Kj 1 (s2 , y2 , y1 )Kj 2 (s2 , x, y2 ) = Kj1 (s2 + S(s2 , y2 , x), y2 , y1 )Kj2 (s2 + S(s2 , y2 , x), x, y2 ) = Kj1 (s2 , y2 , y1 )Kj2 (s2 , x, y2 ) + O(|x − y2 |−n+1 ) using (K1). As a result Q may be written as the sum of two terms. The first is −1 −1 i −1 σ s2 , y2 ; 2−j1 γ 2 σ s2 , x; 2−j2 t Q1 s1 , y1 , s2 , y2 = 22i σ s2 , x; 2−j2 i −1 s2 + S(s2 , y2 , y1 ) − s1 × γ 2 σ s2 , y2 ; 2−j1 × Kj1 (s2 , y2 , y1 )Kj2 (s2 , x, y2 ) dx dt . Integrating with respect to t then x, the cancellation condition (K2) gives us Q s1 , y1 , s2 , y2 < C2i 2−j2 σ s2 , y2 ; 2−j1 −1 1 −1 × γ 2i σ s2 , y2 ; 2−j1 s2 + S s2 , y2 , y1 − s1 × Kj1 s2 , y2 , y1 . Integrating with respect to s1 then y1 then shows |Q1 | ds1 dy1 < C2−j2 , an estimate better than the one we need. Our second term Q2 is bounded in absolute value by −1 −1 i −1 C σ s2 , x; 2−j2 σ s2 , y2 ; 2−j1 γ 2 σ s2 , x; 2−j2 t |y2 −y1 |∼2−j1 ,|y2 −x|∼2−j2

−1 × 22i |x − y2 |γ 2i σ s2 , y2 ; 2−j1 s2 + S s2 , y2 , y1 − s1 × |y2 − y1 |−n x − y2 |−n dx dt . However, |x − y2 | ∼ 2−j2 , so when this expression is integrated with respect to s1 and y1 , we again get something bounded by C2−j2 . We conclude that |Q (s1 , y1 , s2 , y2 )| ds2 dy2 < C2−j2 . We thus may direct our attention to Q . Analogous to the proof of Lemma 4.1, we observe that since B(s2 , y2 ; C2−j2 ) = {(s, y) : |s − s2 | < σ (s2 , y2 ; C2−j2 ), |y − y2 | < C2−j2 }, we have that

383


G(s

(t, x) − G(s1 ,y1 ,s2 ,y2 ,t,x) (s2 , y2 ) ∂t G(s ,y ,s ,y ,t ,x ) t , x < Cσ s2 , y2 ; 2−j2 sup 1 1 2 2

1 ,y1 ,s2 ,y2 ,t,x)

(t ,x )∈B(s2 ,y2 ;2−j2 )

+ C2

−j2

sup (t ,x )∈B(s2 ,y2 ;2−j2 )

∇x G(s

1 ,y1 ,s2 ,y2 ,t

,x )

t , x .

To estimate this expression we examine the derivative on a term-by-term basis. The details, not hard, are omitted. The facts about (t , x ) ∈ B(s2 , y2 ; 2−j2 ) being used are the following: (1) for r = 2−j1 and 2−j2 , σ (t , x ; r) ∼ σ (s2 , y2 ; r); (2) σ (t , x ; 2−j2 ) ≤ 2j1 −j2 σ (t , x ; 2−j1 ); (3) for r = 2−j1 , 2−j2 , ∂t σ (t , x ; r) < Cr 2 and ∇x σ (t , x ; r) < (C/r)σ (s2 , y2 ; r); (4) that γ (t) and γ (t) are supported on |t| < 1; (5) the estimates (K1). The conclusion is that G(s ,y ,s ,y ,t,x) (t, x) − G(s ,y ,s ,y ,t,x) (s2 , y2 ) 1 1 2 2 1 1 2 2 −1 −1 < C2i 2j1 −j2 σ s2 , y2 ; 2−j1 σ s2 , y2 ; 2−j2 |x − y1 |−n |x − y2 |−n . As a result, Q s1 , y1 , s2 , y2 ds1 dy1 < |t−s2 | δ0 σ t1 , x1 ; 2−j . 2

1

Since the multiindices of order d − 1 are spanned by the multiindices of the form ∂zd−1 , where ∂z denotes l αl ∂xl + l βl ∂yl (i.e., ∂zd−1 denotes differentiation in the same direction d − 1 times), we can find a yk and a ∂z = l αl ∂xl + l βl ∂yl with |αl |2 + |βl |2 = 1 such that |α |+|β | |α |+|β |−1 ∂yk t ∗ t1 , x1 , x2 , y x =y=x c2−j > δ1 σ t1 , x1 ; 2−j . ∂z 2

1

|α |+|β |−1 ∂yk t ∗ (t1 , x1 , x2 , y) If we expand ∂z 1−j |x2 − x1 |, |y − x1 | < 2 , we have

in x2 and y about x2 = y = x1 , for

|α |+|β |−1 ∂yk t ∗ t1 , x1 , x2 , y − ∂z|α |+|β |−1 ∂yk t ∗ t1 , x1 , x1 , x1 ∂z α β ∗ ≤C ∂x2 ∂y t t1 , x1 , x1 , x1 2−j (|α|+|β|−|α |−|β |) |α |+|β | δ2 σ (t1 , x1 ; 2−j ) for all |x2 − x1 |, |y − x1 | < 22−j . In what follows, for notational convenience we write σ1 = σ (t1 , x1 ; 2−j ) and σ2 = σ (t2 , x2 ; 2−j ). We split (5.2) into 2 parts, P1 t1 , x1 , x2 , y = 22i σ2−1 σ1−1 γ0 2i/4 2−j σ1−1 ∂yk t ∗ γ 2i σ2−1 L(t2 − t ∗ − s) × γ 2i σ1−1 s Kj (t1 , x1 , y)Kj (t2 , x2 , y) dy ds and where P2 (t1 , x1 , t2 , x2 ) is the same except the bump function γ0 is replaced by 1 − γ0 . We make use of the following Van der Corput–type lemma of Christ (see [2]). lemma 5.1 If I ⊂ R is an interval and f : I → R is a function with |f (q) | ≥ 1 , then the set {t : |f (t)| ≤ 2 } has measure less than or equal to Cq (2 /1 )1/q . Proceeding to the analysis of P1 , integrating with respect to t2 then s, we get P1 t1 , x1 , t2 , x2 dt2 dx2 < C γ0 2i/4 2−j σ1−1 ∂yk t ∗ t1 , x1 , x2 , y Kj (t1 , x1 , y)Kj (t2 , x2 , y) dy dx2

< C 22nj γ0 2i/4 2−j σ1−1 ∂yk t ∗ t1 , x1 , x2 , y dy dx2 . |y−x1 |,|x2 −x1 | δ2 σ1 (c2−j )−q−1 . Thus by Lemma 5.1, along any ray in direction z we have γ0 (2i/4 2−j σ1−1 ∂yk t ∗ ) = 0 on a set of measure at most C2−i/4q 2−j . Thus integrating (5.3) first in the z-direction and then in the other 2n − 1 directions, we get that (5.3) is greater than C2−i/4q . We are thus done with the analysis of P1 . We now proceed to the analysis of P2 . We integrate by parts in the yk direction.


387

To this end we write P2 = P2 + P2 , where P2 t1 , x1 , t2 , x2 ∗ 2i −1 −1 (∂yk L) t2 − t (t1 , x1 , x2 , y) − s (1 − γ0 ) 2i/4 2−j σ1−1 ∂yk t ∗ = − 2 σ2 σ1 ∗ L∂yk t (t1 , x1 , x2 , y) i −1 × γ 2 σ2 L t2 − t ∗ (t1 , x1 , x2 , y) − s × γ 2i σ1−1 s Kj (t1 , x1 , y)Kj (t2 , x2 , y) dy ds and where P2 t1 , x1 , t2 , x2 ∗ 2i −1 −1 ∂yk L t2 − t (t1 , x1 , x2 , y) − s (1 − γ0 ) 2i/4 2−j σ1−1 ∂yk t ∗ = 2 σ2 σ1 ∗ L∂yk t (t1 , x1 , x2 , y) i −1 i −1 × γ 2 σ1 s γ 2 σ2 t2 − t ∗ (t1 , x1 , x2 , y) − s

× Kj (t1 , x1 , y)Kj (t2 , x2 , y) dy ds.

But |P2 (t1 , x1 , t2 , x2 )| dt2 dx2 is easy to estimate; we utilize the facts that |(t2 − t ∗ (t1 , x1 , x2 , y) − s)| < 2−i σ1 and |(1/∂yk t ∗ (t1 , x1 , x2 , y))| < 2i/4 2−j σ1−1 . We get P t1 , x1 , t2 , x2 dt2 dx2 2 < C2−(3/4)i 2−j 22i σ2−1 σ1−1 γ 2i σ2−1 L t2 − t ∗ − s × γ 2i σ1−1 s Kj (t1 , x1 , y)Kj (t2 , x2 , y) dy ds dt2 dx2 , which is less than C2−(3/4)i 2−j < C2−(3/4)i after an integration successive with respect to t2 , s, x2 , and y. In P2 (t1 , x1 , t2 , x2 ) we integrate by parts. We integrate the ∂yk [L(t2 − t ∗ (t1 , x1 , x2 , y)−s)]γ (2i σ2−1 L(t2 −t ∗ (t1 , x1 , x2 , y)−s)) which for some function δ is equal to 2−i σ2 ∂yk δ(2i σ2−1 L(t2 − t ∗ (t1 , x1 , x2 , y) − s)) because γ = 0. After our integration by parts, we have that P t1 , x1 , t2 , x2 dt2 dx2 2 < 22i σ2−1 σ1−1 2−i σ2 δ 2i σ2−1 L t2 − t ∗ (t1 , x1 , x2 , y) − s γ 2i σ1−1 s (1 − γ0 ) 2i/4 2−j σ1−1 ∂yk t ∗ (t1 , x1 , x2 , y) Kj (t1 , x1 , y)Kj (t2 , x2 , y) × ∂yk L∂yk t ∗ (t1 , x1 , x2 , y) dy ds dx2 dt2 . (5.4)

388

MICHAEL GREENBLATT

We examine the yk derivative on a term-by-term basis and bound the resulting terms. The first term is (1 − γ0 ) 2i/4 2−j σ1−1 ∂yk t ∗ (t1 , x1 , x2 , y) Kj (t1 , x1 , y)Kj (t2 , x2 , y) ∂y L−1 k ∂yk t ∗ (t1 , x1 , x2 , y) i/4 −j −1 C(1 − γ0 ) 2 2 σ1 ∂yk t ∗ (t1 , x1 , x2 , y) Kj (t1 , x1 , y)Kj (t2 , x2 , y) < ∂yk t ∗ (t1 , x1 , x2 , y) −1 < C2i/4 2−j σ Kj (t1 , x1 , y)Kj (t2 , x2 , y) 1

< C2i/4 2−j σ1−1 22nj . The next term is

i/4 −j −1 ∗ L ∂y 1 (1 − γ0 ) 2 2 σ1 ∂yk t Kj (t1 , x1 , y)Kj (t2 , x2 , y) k ∗ ∂yk t 2 ∗

∂yk yk t i/4 −j −1 ∗ < C (1 − γ0 ) 2 2 σ1 ∂yk t Kj (t1 , x1 , y)Kj (t2 , x2 , y) ∗ 2 (∂yk t ) < C 2i/2 2−2j σ1−2 ∂y2k yk t ∗ Kj (t1 , x1 , y)Kj (t2 , x2 , y) < C 2i/2 2−2j σ1−2 2−2nj ∂y2k yk t ∗ . However, expanding ∂y2k yk t ∗ (t1 , x1 , x2 , y) in x2 and y about x2 = y = x1 in the bynow-usual fashion gives us |∂y2k yk t ∗ (t1 , x1 , x2 , y)| < C 22j σ1 , and the above is less than C 2i/2 σ1−1 22nj . Our next term is −1 L ∂yk t ∗ ∂yk (1 − γ0 ) 2i/4 2−j σ1−1 ∂yk t ∗ Kj (t1 , x1 , y)Kj (t2 , x2 , y) −1 < L ∂yk t ∗ 2i/4 2−j σ1−1 ∂y2k yk t ∗ γ0 2i/4 2−j σ1−1 ∂yk t ∗ × Kj (t1 , x1 , y)Kj (t2 , x2 , y) −2 < C ∂y2k yk t ∗ ∂yk t ∗ γ0 2i/4 2−j σ1−1 ∂yk t ∗ Kj (t1 , x1 , y)Kj (t2 , x2 , y). So exactly as above we have that this term is less than C2i/2 σ1−1 22nj . Our final two terms, where the derivative lands on either Kj (t1 , x1 , y) or Kj (t2 , x2 , y), are bounded by −1 C ∂yk t ∗ (1 − γ0 ) 2i/4 2−j σ1−1 ∂yk t ∗ 2(2n+1)j < C 2i/4 2−j σ1−1 2(2n+1)j . We conclude that

−1 ∂yk L∂yk t ∗ (1 − γ0 ) 2i/4 2j σ1−1 ty∗k Kj (t1 , x1 , y)Kj (t2 , x2 , y) < C 2i/2 σ1−1 2−2nj .


389

Substituting this in (5.4), we get that (5.4) is bounded by a constant times 23i/2+2nj (σ1 )−2 δ 2i σ2−1 L t2 −t ∗ −s γ 2i σ1−1 s dy ds dx2 dt2 . |y−x1 |,|y−x2 | 2, gives us Ti∗ f p ≤ C 2−δ i f p . Adding these up over i gives us Tf p ≤ C f p , and we have the main theorem. Recall that Ti f = j Ti,j f , where −1 i −1 γ 2 σ t, x, 2−j t + S(t, x, y) − s Ti,j f = 2i σ t, x, 2−j × Kj (t, x, y)f (s, y) ds dy. If

−1 −1 t +S(t, x, y)−s Kj (t, x, y), Li,j (t, x, s, y) = 2i σ t, x, 2−j γ 2i σ t, x, 2−j Ti has kernel Li = j Li,j (t, x, s, y). lemma 6.1 There exists k > 1 such that if (t , x ) ∈ B(t, x; r), then Li (t, x, s, y) − Li (t , x , s, y) ds dy ≤ C(i + 1). k r) (s,y)∈B(t,x,2 /

(6.1a)

If (s , y ) ∈ B(s, y; r), then k r) (t,x)∈B(s,y,2 /

Li (t, x, s, y) − Li (t, x, s , y ) dt dx ≤ C(i + 1).

(6.1b)

Proof The proofs of (6.1a) and (6.1b) are similar, so we only prove the slightly more difficult inequality (6.1a) here. The left-hand side of (6.1a) is at most

390

MICHAEL GREENBLATT

j

−1 i −1 i γ 2 σ t, x; 2−j 2 σ t, x; 2−j k r) (s,y)∈B(t,x,2 /

× t + S(t, x, y) − s Kj (t, x, y) −1 i −j −1 − 2i σ t , x ; 2−j γ 2 σ t ,x ;2

× t + S(t , x , y) − s Kj (t , x , y) ds dy.

(6.2)

Let j0 be such that 2−j0 −1 ≤ r ≤ 2−j0 . If the expression γ (2i σ (t, x; 2−j )−1 (t + S(t, x, y) − s)) appearing in (6.2) is nonzero, we must have (s, y) ∈ B(t, x; 2−j ). But the integral is being taken only over (s, y) ∈ / B(t, x, 2k r), so we conclude that if i −j −1 j > j0 , γ (2 σ (t, x; 2 ) (t + S(t, x, y) − s)) = 0 on the domain of integration. Similarly, if we have j > j0 and γ (2i σ (t , x ; 2−j )−1 (t + S(t , x , y) − s)) = 0, then (s, y) ∈ B(t , x ; 2−j ), so by Fact 2, (t , x ) ∈ B(s, y; C2−j ) ⊂ B(s, y; C2−j0 ). Since (t , x ) ∈ B(t, x; 2−j0 ), applying Fact 2 again gives us (s, y) ∈ B(t, x; C 2 2−j0 ). This is impossible on our domain of integration if k is chosen appropriately large since we are integrating over (t, x) ∈ / B(s, y, 2k r). So in (6.2) we may assume that i −j −1 γ (2 σ (t , x ; 2 ) (t + S(t , x , y) − s)) = 0 only if j ≤ j0 as well. Hence all the terms in (6.2) are zero for j > j0 . We now do the usual coordinate change to make S(t, x, y) = 0 for all y: rewrite t as t + S(t, x, x ), and change variables in s to rewrite s as s − S(t, x, y). As a result B(t, x, 2k r) becomes the rectangle {(s, y) : |s − t| < σ (t, x, 2k r), |y − x| < 2k r}. Equation (6.2) is bounded by −1 i −1 t + S(t, x, y) − s Kj (t, x, y) γ 2 σ t, x; 2−j 2i σ t, x; 2−j j ≤j0 −1 i −j −1 − 2i σ t , x ; 2−j γ 2 σ t ,x ;2 × t + S(t , x , y) − s Kj (t , x , y) ds dy. (6.3) Define F (t , x ) = 2i σ (t, x; 2−j )−1 γ (2i σ (t, x; 2−j )−1 (t + S(t, x, y) − s))Kj (t, x, y). By the mean value theorem and Lemma 3.1, the integrand in the j th term of (6.3) is at most C times ∂t F t , x + 2−j0 ∇x F t , x . sup sup σ t, x; 2−j0 (t ,x )∈B(t,x;2−j0 )

(t ,x )∈B(t,x;2−j0 )

Differentiating the product then using Lemmas 3.1 and 3.4, equation (3.9), and the fact that σ (t , x ; 2−j ) ∼ σ (t, x; 2−j ), we get that for (t , x ) ∈ B(t, x; 2−j0 ),


391

∂t F t , x < C22i σ t, x; 2−j −2 2j n , ∇x F t , x < C22i σ t, x; 2−j −1 2j (n+1) . As a result the integrand of the j th term in (6.3) is at most C22i+j n

−1 −1 σ (t, x; 2−j0 ) 2i+j n j −j0 +2 2 σ t, x; 2−j < C 2j −j0 +2i+j n σ t, x; 2−j . −j 2 σ (t, x; 2 )

But this integrand is being integrated over an s set of measure less than C2−i σ (t, x; 2−j ) and a y set of measure less than C2−j n , and we conclude that the j th term of (6.3) is less than C2i 2j −j0 . However, each term in (6.3) is less than C. (To see this, integrate with respect to s first.) So (6.3) is less than Ci + j ≤j0 −i 2i 2j −j0 < C (i + 1). We are thus finished with the proof of Lemma 6.1. By the Calderón-Zygmund method applied to our non-Euclidean balls, (6.1a) proves Ti f p ≤ C(i +1)f p for 2 ≤ p < ∞, and (6.1b) proves Ti f p ≤ C(i +1)f p for 1 < p ≤ 2. By the discussion preceeding the proof of Lemma 6.1, the proof of the main theorem is complete. 7. Extensions and generalizations There are a few natural directions in which generalizations of the arguments presented here might be attempted. First, there is the issue of extending the arguments here to situations where the submanifolds being integrated over have codimension m > 1. The author has recently found a method of proving Lp -boundedness of singular Radon transforms that works for arbitrary codimension. Again the method is direct and does not involve Fourier transform, lifting, or Lie groups. Although inspired by this paper, the methods of the full codimension paper are rather divergent and do have a few disadvantages. For example, proving maximal analogues of the theorems of this paper is a routine matter in our current set-up; the same cannot be said for the full codimension paper, to the author’s knowledge. Next, there is the question of finding a natural generalization of conditions (K1)– (K2). It is hoped that an argument like that of the T (1) theorem can replace the arguments of Section 4 and thus extend the arguments here to a general class of K(t, x, y); the author thanks E. Stein for pointing this possibility out. There has been some work in this direction (see, e.g., the thesis of D. Potinton [15]). One might also examine the possibility of extending the arguments of this paper to situations where the curvature condition is not satisfied. Such situations have been considered before, often with a convexity assumption on the submanifolds being integrated over (see, e.g., [19], [21]). Finally and more generally, there is the question

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MICHAEL GREENBLATT

of to what extent the methods being developed here apply to the study of more general Radon transforms and related oscillatory integral operators. This issue appears to be substantially more difficult than the ones treated here. Much work has been done on such problems; to give a sampling we mention [7], [14], [17], and [18]. Acknowledgments. I would like to thank my advisor Elias M. Stein for the many useful suggestions he made throughout the course of my doctoral research, which resulted in this paper. I am certain it could never have been done without his generous assistance. I would also like to thank the referees for their helpful suggestions and for alerting me to relevant earlier work.

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[6] [7]

[8]

[9] [10] [11]

J. BOURGAIN, “A remark on the maximal function associated to an analytic vector

field” in Analysis at Urbana (Urbana, Ill., 1986–1987), London Math. Soc. Lecture Note Ser. 137, Cambridge Univ. Press, Cambridge, 1989, 111–132. MR 90h:42028 M. CHRIST, Hilbert transforms along curves, I: Nilpotent groups, Ann. of Math. (2) 122 (1985), 575–596. MR 87f:42039a M. CHRIST, A. NAGEL, E. STEIN, and S. WAINGER, Singular and maximal radon transforms: Analysis and geometry, Ann. of Math. (2) 150 (1999), 489–577. MR 2000j:42023 S. CUCCAGNA, Sobolev estimates for fractional and singular Radon transforms, J. Funct. Anal. 139 (1996), 94–118. MR 98f:42010 J. DUOANDIKOETXEA and J. RUBIO DE FRANCIA, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541–561. MR 87f:42046 E. FABES, Singular integrals and partial differential equations of parabolic type, Studia Math. 28 (1966/1967), 81–131. MR 35:4601 A. GREENLEAF, A. SEEGER, and S. WAINGER, On X-ray transforms for rigid line complexes and integrals over curves in R4 , Proc. Amer. Math. Soc. 127 (1999), 3533–3545. MR 2001a:44002 A. GREENLEAF and G. UHLMANN, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal. 89 (1990), 202–232. MR 91i:58146 A. NAGEL, E. STEIN, and S. WAINGER, Balls and metrics defined by vector fields, I: Basic properties, Acta Math. 155 (1985), 103–147. MR 86k:46049 Y. PAN, Uniform estimates for oscillatory integral operators, J. Funct. Anal. 100 (1991), 207–220. MR 93f:42034 D. H. PHONG, “Singular integrals and Fourier integral operators” in Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, 1991), Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, 1995, 286–320. MR 95k:42026


[12]

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boundary value problems, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), 7697–7701. MR 86j:42024 , Hilbert integrals, singular integrals, and Radon transforms, I, Acta Math. 157 (1986), 99–157. MR 88i:42028a , The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), 105–152. MR 98j:42009 D. POTINTON, dissertation, Princeton University, 1998. F. RICCI and E. STEIN, Harmonic analysis on nilpotent groups and singular integrals, I: Oscillatory integrals, J. Funct. Anal. 73 (1987), 179–194. MR 88g:42043 A. SEEGER, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), 685–745. MR 94h:35292 , Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (1998), 869–897. MR 99f:58202 A. SEEGER, S. WAINGER, J. WRIGHT, and S. ZIESLER, Classes of singular integral operators along curves and surfaces, Trans. Amer. Math. Soc. 351 (1999), 3757–3769. MR 99m:42029 E. STEIN and S. WAINGER, Maximal functions associated to smooth curves, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 4295–4296. MR 54:3290 S. WAINGER, J. WRIGHT, and S. ZIESLER, Singular integrals associated to hypersurfaces: L2 theory, J. Funct. Anal. 167 (1999), 148–182. MR 2001b:42018

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA; [email protected]

q-SERIES IDENTITIES AND VALUES OF CERTAIN L-FUNCTIONS ´ GEORGE E. ANDREWS, JORGE JIMENEZ-URROZ, and KEN ONO

Abstract As usual, define Dedekind’s eta-function η(z) by the infinite product η(z) := q

1/24

∞

1 − qn

q := e2π iz throughout .

n=1

In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout) η(24z) − q 1 − q 24 1 − q 48 · · · 1 − q 24n = η(24z)D(q) + E(q),

∞ n=0

where the series D(q) and E(q) are defined by ∞

1 q 24n D(q) = − + 2 1 − q 24n n=1

1 =− + 2

∞

d(n)q 24n

n=1

1 = − + q 24 + 2q 48 + 2q 72 + 3q 96 + · · ·, 2 ∞ 5 1 7 11 1 12 2 nq n = q − q 25 − q 49 + q 121 + · · ·. E(q) = 2 n 2 2 2 2 n=1

Here d(n) denotes the number of positive divisors of n. We obtain two infinite families of such identities and describe some consequences√for L-functions and partitions. For example, if χ2 is the Kronecker character for Q( 2), these identities can be used to show that DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 3, Received 17 July 2000. Revision received 21 September 2000. 2000 Mathematics Subject Classification. Primary 11P99; Secondary 11M99. Andrews’s work supported by the National Science Foundation. Jiménez-Urroz’s work supported by Comunidad de Madrid grant number PB 98-0067. Ono’s work supported by the National Science Foundation, the Alfred P. Sloan Foundation, and the David and Lucile Packard Foundation.

395

´ ANDREWS, JIMENEZ-URROZ, AND ONO

396

− 2e

−t/8

∞ n=0

1 − e−2t 1 − e−4t · · · 1 − e−2nt 1 + e−t 1 + e−3t · · · 1 + e−(2n+1)t ∞ −1 n tn = · L(χ2 , −2n − 1) · . 8 n! n=0

1. Introduction and statement of results As usual, define Dedekind’s eta-function η(z) by the infinite product η(z) := q

1/24

∞

1 − qn

q := e2π iz throughout .

(1.1)

n=1

In a recent paper, Zagier [Z, Theorem 2] proved that (note: empty products equal 1 throughout) η(24z) − q 1 − q 24 1 − q 48 · · · 1 − q 24n = η(24z)D(q) + E(q), (1.2)

∞ n=0

where the series D(q) and E(q) are defined by ∞

1 q 24n D(q) = − + 2 1 − q 24n n=1

1 =− + 2

∞

d(n)q 24n

n=1

1 = − + q 24 + 2q 48 + 2q 72 + 3q 96 + · · ·, 2 ∞ 1 12 5 1 7 11 2 E(q) = nq n = q − q 25 − q 49 + q 121 + · · ·. 2 n 2 2 2 2 n=1

Here d(n) denotes the number of positive divisors of n. This identity plays an important role in Zagier’s work on Vassiliev invariants in knot theory (see [Z]). Two other similar identities are known, and they were noticed by G. Andrews in connection with one of Ramanujan’s mock theta functions. In [A1] Andrews proved that ∞ M1 (q) η(48z) η(48z) − q 1 + q 24 1 + q 48 · · · 1 + q 24n = D(q) + , η(24z) η(24z) 2 n=0

(1.3)

q-IDENTITIES

397

∞ η(48z) 2 M1 (q) η(48z) q = − D q + , 24 72 24(2n+1) η(24z) η(24z) 2 1−q 1 − q ··· 1 − q n=0 (1.4) where M1 (q) is the mock theta function given by M1 (q) = q +

∞ n=1 25

=q +q

q 12n +12n+1 1 + q 48 · · · 1 + q 24n 2

1 + q 24

−q

49

+ 2q

73

(1.5)

− · · ·.

The q-series of the function M1 (q) was the focus of two extensive studies, [ADH] and [C]. Although M1 (q) is not the Fourier expansion of a modular form, these works show that √ the coefficients of M1 (q) are given by a Hecke character for the quadratic field Q( 6). In particular, M1 (q) enjoys nice properties that one expects for certain weight 1 cusp forms. For these reasons we refer to M1 (q) and M2 (q) (defined in (1.8)) as mock theta functions, although they do not exactly fit Ramanujan’s original definition (see [A2, p. 291]). In view of identities (1.2)–(1.4), it is natural to investigate the general behavior of q-series which are obtained by summing the iterated differences between an infinite product and its truncated products. Here we establish two general theorems that yield infinitely many such identities, and we illustrate how such identities are useful in determining the values at negative integers for certain L-functions. We employ the standard notation ∞ 1 − Aq j , (1.6) (A; q)n = 1 − Aq n+j j =0 and throughout we assume that |q| < 1 and that the other parameters are restricted to domains that do not contain any singularities of the series or products under consideration. theorem 1 We have ∞ (t; q)∞ n=0

(a; q)∞ =

−

(t; q)n (a; q)n

∞ (q/a; q)n (a/t)n n=1

(q/t; q)n

(t; q)∞ + (a; q)∞

∞ n=1

∞

q n t −1 qn + 1 − qn 1 − q n t −1 n=1

−

∞ n=0

∞

aq n t −1 tq n − 1 − tq n 1 − aq n t −1 n=0

.


398

theorem 2 We have ∞ (a; q)∞ (b; q)∞ n=0

(a; q)n (b; q)n − (q; q)∞ (c; q)∞ (q; q)n (c; q)n

∞ ∞ ∞ aq n (c/b; q)n bn (b; q)∞ (a; q)∞ q n . − − = (c; q)∞ (q; q)∞ 1 − qn 1 − aq n (a; q)n 1 − q n n=1

n=0

n=1

Many interesting specializations of these two theorems yield identities for modular forms that are eta-products (including identities√(1.2)–(1.4)). Here we highlight ten of these identities. First we fix notation. We let be the operator defined by ∞

∞ √ √ n a(n)q = na(n)q n . (1.7) n=0

n=0

It is easy to see that the series E(q) in (1.2) is given by √ (η(24z)) E(q) = . 2 In addition to the mock theta function M1 (q), we require the mock theta function M2 (q) defined by ∞

(−1)n q 24n −1 = −q 23 − q 47 − · · ·. (1.8) M2 (q) = 24 1 − q 72 · · · 1 − q 24(2n−1) 1 − q n=1 2

See [ADH] for a detailed study of this function. The ten eta-products F1 (z), F2 (z), . . . , F10 (z) we consider are of the form Fi (z) = ηai (δi z)ηbi (2δi z) with ai = 0. Obviously each Fi (z) is a modular form of weight (ai + bi )/2. For each Fi (z) we define quantities ci and fi (j ), which are not necessarily unique, for which Fi (z) = ci

∞

fi (j ).

(1.9)

j =1

These are listed in Table 1. For δ ∈ {1, 8, 24}, let dδ (n) be the divisor function defined by   1 if δ = 24, d(n) =     d|n    d (−1) if δ = 8, dδ (n) =  d|n     1 if δ = 1.    d|n odd

(1.10)

q-IDENTITIES

399

Table 1

i

Fi (z)

δi

ci

fi (j ) 24j

1

1/η(24z)

24

q −1

2

η(2z)/η2 (z)

1

1

3

η(8z)/η2 (16z)

8

q −1

4

η(48z)/η(24z)

24

q

5

η(48z)/η(24z)

24

q/ 1 − q 24

6

η(24z)/η(48z)

24

q −1

1 + q 24j 1/ 1 − q 24(2j +1) 1/ 1 + q 24j

7

η(24z)/η(48z)

24

q −1

1 − q 24(2j −1)

8

η(24z)

24

q

9

η2 (z)/η(2z)

1

1

10

η2 (16z)/η(8z)

8

q/ 1 − q 8

1 − q 24j 1 − qj / 1 + qj 1 − q 16j / 1 − q 16j +8

1/ 1 − q 1 + qj / 1 − qj 1 − q 16j −8 / 1 − q 16j

Also, for each i, define αi by

 − 1 αi = 2 0

if (ai + 2bi )δi = 24,

(1.11)

otherwise.

Notice that αi = −1/2 if and only if the order of vanishing of Fi (z) at ∞ is 1. The last quantity we require is γi , which is defined by 2 if i = 5, 7, γi = (1.12) 1 otherwise. theorem 3 If 1 ≤ i ≤ 10, then   ∞ n 1 Fi (z) − ci fi (j ) = 1 + Fi (z)Di (q) + Gi (q), δi n=0

j =1

where [•] denotes the greatest integer function,


400

Di (q) = αi +

∞

dδi (n)q δi γi n ,

n=1

and

  0 if i = 1, 2, 3,     M (q) 1   if i = 4, 5,   2 Gi (q) = 2M2 (q)  if i = 6, 7,   γi    2 √    − αi + (Fi (z)) if i = 8, 9, 10. δi

The three forms F1 (z), F2 (z), and F3 (z) have weight −1/2, and the four forms F4 (z), F5 (z), F6 (z), and F7 (z) have weight zero. The remaining three forms have weight 1/2. The series G4 (z), G5 (z), G6 (z), and G7 (z) are mock theta functions, whereas G8 (q), G9 (q), and G10 (q) are the half-derivatives of F8 (z), F9 (z), and F10 (z). In other words, the “error series” Gi (q) in Theorem 3 satisfy  1  0 if Fi (z) has weight − ,    2 Fi (z) −→ Gi (q) ∼ mock theta function if Fi (z) has weight zero,   1  √(F (z)) if Fi (z) has weight . i 2 Although these identities are elegant in their own right, they are also often useful in calculating the values of L-functions at negative integers. In particular, they lead to analogs of the classical result ∞

tn t =1+ , (−1)n+1 ζ (1 − n) · t e −1 (n − 1)! n=1

where ζ (s) is the Riemann zeta-function. In this direction Zagier used (1.2) to show that ∞ − e−t/24 1 − e−t 1 − e−2t · · · 1 − e−nt n=0

∞ tn 1 −1 n · L(χ12 , −2n − 1) · , = 2 24 n! n=0

where χ12 is the Dirichlet character with modulus 12 defined by   if n ≡ 1, 11 (mod 12),  1 χ12 (n) := −1 if n ≡ 5, 7 (mod 12),   0 otherwise.

(1.13)

q-IDENTITIES

401

Here we illustrate the generality of this phenomenon by proving the following theorems. theorem 4 As a power series in t, we have ∞ ∞ tn 1 1 − e−t 1 − e−2t · · · 1 − e−nt n n+1 = . (−1) −1 ·ζ (−2n−1)· 4 − · 4 n! 1 + e−t 1 + e−2t · · · 1 + e−nt n=0

n=0

In addition to ζ (s), we consider the Dirichlet L-function L(χ2 , s) :=

∞ χ2 (n) n=1

where χ2 (n) :=

   1

−1   0

ns

,

if n ≡ 1, 7

(mod 8),

if n ≡ 3, 5

(mod 8),

(1.14)

(1.15)

otherwise.

theorem 5 As a power series in t, we have ∞ 1 − e−2t 1 − e−4t · · · 1 − e−2nt −t/8 − 2e 1 + e−t 1 + e−3t · · · 1 + e−(2n+1)t n=0 ∞ tn −1 n = · L(χ2 , −2n − 1) · . 8 n! n=0

We also consider the Hecke L-function L(ρ, s) =

∞ a(n) n=1

ns

:=

√

a⊆Z[ 6]

χ(a)Na−s ,

(1.16)

√ √ where χ is the order-2 character of conductor 4(3+ 6) on ideals in Z[ 6] defined by   yx 12   if y is even, i x χ (a) := (1.17)   yx+1 12  if y is odd, i x √ when a = (x + y 6). If 1 ≤ r < 48 is an integer, then let Lr (ρ, s) be the partial L-function defined by


402

Lr (ρ, s) :=

n≡r (mod 48)

a(n) . ns

(1.18)

By the orthogonality of the Dirichlet characters modulo 48 and the analytic continuation of the associated twists of L(ρ, s), each Lr (ρ, s) has an analytic continuation to C. theorem 6 As a power series in t, we have − 2et/24

∞

1 − e−t 1 − e−3t · · · 1 − e−(2n−1)t

n=0

=

∞ tn −1 n · L23 (ρ, −n) + L47 (ρ, −n) · . 24 n! n=0

theorem 7 As a power series in t, we have − 2e−t/24

∞

1 − e−t 1 + e−2t · · · 1 + (−1)n e−nt

n=0

∞ tn −1 n = · L1 (ρ, −n) − L25 (ρ, −n) · . 24 n! n=0

In §2 we recall certain facts about q-series and basic hypergeometric series, and we prove Theorems 1 and 2. In §3 we prove Theorem 3, and in §4 we prove Theorems 4, 5, 6, and 7. In §5 we examine the partition-theoretic consequences of the identities for F1 (z) and F8 (z). In §6 we give a few more identities that are related to eta-products. The most interesting of these is

∞ 1 1 − η2 (24z) q 2 1 − q 24 2 1 − q 48 2 · · · 1 − q 24n 2 n=0

∞ 1 = 2 d(n) + m(n) q 24n , η (24z) n=1

where m(n) denotes the number√of middle divisors of n. A divisor is a middle divisor √ if it lies in the interval [ n/2, 2n). 2. Preliminaries and important facts The ten identities in Theorem 3 rely on Theorems 1 and 2. Here we prove Theorems 1 and 2, and we begin with an observation that follows from Abel’s theorem.

q-IDENTITIES

403

proposition 2.1 n Suppose that f (z) = ∞ n=0 α(n)z is analytic for |z| < 1. If α is a complex number for which ∞ (1) n=0 (α − α(n)) < +∞, (2) limn→+∞ n(α − α(n)) = 0, then ∞ d (1 − z)f (z) = α − α(n) . (2.1) lim − z→1 dz n=0

In all of our applications, f (z) has a pole of order 1 at z = 1, so the limz→1− can be replaced by a simple evaluation. We employ the standard notation of basic hypergeometric series (see [GR, pp. 3–4, 125]) ∞ (a0 ; q)j (a1 ; q)j · · · (ar ; q)j zj a0 , a1 , . . . , ar ; q, z = , (2.2) r+1 φr b1 , . . . , br (q; q)j (b1 ; q)j · · · (br ; q)j

r ψr

a1 , a2 , . . . , ar ; q, z = b1 , b2 , . . . , br

j =0 ∞

j =−∞

(a1 ; q)j (a2 ; q)j · · · (ar ; q)j zj . (b1 ; q)j (b2 ; q)j · · · (br ; q)j

We also require Heine’s transformation (see [GR, p. 9]) (b; q)∞ (az; q)∞ a, b; c/b, z; φ × φ q, z = q, b , 2 1 2 1 c az (c; q)∞ (z; q)∞ Ramanujan’s summation (see [GR, p. 126]) (q; q)∞ (b/a, q)∞ (az; q)∞ (q/az; q)∞ a; ψ , q, z = 1 1 b (b; q)∞ (q/a; q)∞ (z; q)∞ (b/az; q)∞

(2.3)

(2.4)

(2.5)

and the Rogers-Fine identity (see [F, p. 15]) (1 − t)

∞ (a; q)n t n n=0

(b; q)n

2 ∞ (a; q)n (atq/b; q)n 1 − atq 2n bn t n q n −n = . (b; q)n (tq; q)n

(2.6)

n=0

Throughout we assume that |q| < 1 and that the other parameters are restricted to domains that do not contain any singularities of the series or products under consideration. For succinctness of notation we define the differential operator + by +f (z) = f (1). Proof of Theorem 1 By Proposition 2.1 we have that ∞ ∞ (t; q)∞ (t; q)n zn (t; q)n − = +(1 − z) (a; q)∞ (a; q)n (a; q)n n=0

n=0

(2.7)


404

(by (2.2)–(2.6))

= +(1 − z) =+

−1 (t; q)n zn t; q, z − a (a; q)n n=−∞

1 ψ1

(q; q)∞ (a/t; q)∞ (tz; q)∞ (q/tz; q)∞ (a; q)∞ (q/t; q)∞ (zq; q)∞ (a/tz; q)∞ −(1 − z)

∞ (q/a; q)n (a/(tz))n n=1

(q/t; q)n

.

Differentiating this last expression with respect to z and then setting z = 1 yields the result. Proof of Theorem 2 By Proposition 2.1, (2.2), and (2.4), we have ∞ (a; q)∞ (b; q)∞ (a; q)n (b; q)n − (q; q)∞ (c; q)∞ (q; q)n (c; q)n n=0 a, b; = +(1 − z) 2 φ1 q, z c

∞ (c/b; q)n (z; q)n bn (b; q)∞ (az; q)∞ 1+ . =+ (c; q)∞ (zq; q)∞ (q; q)n (az; q)n n=1

Noting that +(z; q)n = −(q; q)n−1 when n > 0, we differentiate this last expression with respect to z and then set z = 1. This yields the result. 3. Proof of Theorem 3 In this section we prove each of the ten identities using the facts in §2. Case 1: F1 (z). This appears implicitly in [F, p. 14]. It is the instance of Theorem 2 where a = 0 and b = c. Case 2: F2 (z). This is the instance of a = −q and b = c in Theorem 2. Case 3: F3 (z). In Theorem 2, replace q by q 2 , then set b = c and a = q. Case 4: F4 (z). This result was proved in [A1, (1.4)]. It follows from Theorem 1 with t = −q when a → 0. Case 5: F5 (z). This result was proved in [A1, (1.5)]. In Theorem 2, replace q by q 2 ,

q-IDENTITIES

405

then set a = q 2 , b = 0, and c = q 3 . This yields, after multiplication by (1 − q)−1 ,

∞ 1 1 − q; q 2 ∞ q; q 2 n+1 n=0 2 ∞ (−1)n−1 q n +2n 1 = q; q 2 ∞ n=1 q 2 ; q 2 n 1 − q 2n 2 ∞ (−1)n−1 q n +2n 1 − q −n + q −n 1 = q; q 2 ∞ n=1 q 2 ; q 2 n 1 − q 2n

∞ 2 2 ∞ 1 (−1)n q n +n (−1)n−1 q n +n + , = q; q 2 ∞ n=1 q 2 ; q 2 n 1 + q n q 2 ; q 2 n 1 − q 2n n=1

which, by [F, p. 14, (12.42)], q q

∞ 1 1 1 qn , , −1; 2 lim 3 φ2 τ τ = q, −τ + − + , 2 1 − qn 2 q; q 2 ∞ τ →0 q; q 2 ∞ −q, −q n=1 which, by [GR, p. 241, (III.9)] with a = b = q/τ , c = −1, and d = e = −q,

q 1 , −τ, q; = q, −τ lim 3 φ2 τ 2 q; q 2 ∞ (−q; q)∞ τ →0 −q, −qτ

∞ 1 1 qn + − + 2 1 − qn q; q 2 ∞ n=1

2 ∞ ∞ 1 1 qn 1 q n +n − + + = . 2 2 (−q; q)n 2 1 − qn q; q n=0 n=1 This is the identity. Case 6: F6 (z). In Theorem 2, let a = q, b = 0, and c = −q. This yields ∞ n=0

1 1 − (−q; q)∞ (−q; q)n

2 ∞ q (n +n)/2 1 = (−q; q)∞ (q; q)n 1 − q n n=1 2 ∞ q (n +n)/2 1 − (−1)n−1 + (−1)n−1 1 , = (−q; q)∞ (q; q)n 1 − q n

n=1


406

which, by [F, p. 14, (12.42)], 2 ∞ ∞ q 2n +3n+1 qn 1 2 + = (−q; q)∞ (−q; q)∞ 1 − qn (q; q)2n+1 1 − q 2n+1 n=0 n=1 q q

∞ 2q 1 qn , , q; 2 2 3 = lim φ q , τ q , + τ τ 3 2 (−q; q)∞ 1 − qn (−q; q)∞ (1 − q)2 τ →0 q 3, q 3 n=1

which, by [GR, p. 241, (III.10)] with q replaced by q 2 followed by letting a = c = −q/τ , b = q, and d = e = q 3 , ∞ 2 2 m 2q q; q 2 ∞ q ; q mq = 2 (−q; q)∞ q 3 ; q 2 ∞ (1 − q)2 m=0

∞ 1 1 qn + − + 2 1 − qn q; q 2 ∞ n=1

∞ ∞ 2 2 m 1 qn 1 =2 q ; q mq + − + 2 1 − qn q; q 2 ∞ m=0 n=1

2 ∞ ∞ (−1)n−1 q n 1 qn 1 =2 − + + . 2 1 − qn q; q 2 n q; q 2 ∞ n=1 n=1 This is the identity for F6 (z). Case 7: F7 (z). In Theorem 1, replace q by q 2 , and then set a = 0 and t = q. Case 8: F8 (z). This is identity (1.2), and it is [Z, Theorem 2]. We include a proof for completeness. In Proposition 2.1, set α = (q; q)∞ and let α(n) := (q; q)n . This yields ∞ (q; q)n zn (q; q)∞ − (q; q)n = +(1 − z)

∞ n=0

n=0

∞ = −+ (q; q)n−1 q n zn

=−

(3.1)

n=1 ∞

(q; q)n−1 nq n .

n=1

Now it is immediate (because the partial sums equal the partial products) that 1−

∞ n=1

zq n (zq; q)n−1 = (zq; q)∞ ,

(3.2)

q-IDENTITIES

407

and applying + to (3.2), we find that −

∞

n



q +(zq; q)n−1 = −1 − (q; q)∞ −1 +

n=1

∞ j =1

 qj  . 1 − qj

(3.3)

Now set b = 0, a = z2 q, and t = z2 in (2.6). After simplification we obtain, upon multiplication by z, z − z3

∞

z2 q; q

n=1

Noting that 1+

∞

z2 q

n−1

+(zf (z2 ))

n

= z+

∞ 2 2 (−1)n q (3n −3n)/2 z6n−1 + q (3n +n)/2 z6n+1 . n=1

(3.4)

= f (1) + 2+f (z), we apply + to (3.4) and find that

n

(−1) (6n − 1)q

n=1

(3n2 −n)/2

∞ 2 + (−1)n (6n + 1)q (3n +n)/2 n=1

∞ = (q; q)∞ + 2+ 1 − z (zq; q)n−1 zn q n

n=1

∞ ∞ = (q; q)∞ − 2 (q; q)∞ − 1 − 2 (q; q)n−1 nq n − 2 q n +(zq; q)n−1

= 2 − (q; q)∞ + 2

n=1

∞

(q; q)∞ − (q; q)n

n=0





+ 2 −1 − (q; q)∞ −1 +

∞ j =1

n=1

 q j  1 − qj (3.5)

by (3.1) and (3.3). This is the identity. Case 9: F9 (z). In Theorem 2, let a = b = q and c = −q. This yields ∞ ∞ (q; q)∞ (q; q)∞ (−q; q)n−1 q n (q; q)n . = −2 − (−q; q)∞ (−q; q)n (−q; q)∞ (q; q)n 1 − q n n=0

(3.6)

n=1

Now in (2.6), set t = z, a = zq, b = −zq (cf. [F, p. 15, (14.31)]). This yields, after simplification, (1 − z)

∞ (zq; q)n zn n=0

Applying + to (3.7) yields

(−zq; q)n

=1+2

∞ n=1

n 2 − z2 q n .

(3.7)


408

4

∞

(−1)n nq n = +(1 − z)2 φ1 2

n=1

q, zq; q, z , −zq

which, by (2.4) with a = q, b = zq, and c = −zq, and also (3.6), (zq; q)∞ −1, z; q, zq (3.8) =+ 2 φ1 zq (−zq; q)∞   ∞ ∞ ∞ (q; q)∞  −q j (q; q)∞ (−q; q)n−1 q n qj  = − 2 − (−q; q)∞ 1 − qj 1 + qj (−q; q)∞ (q; q)n (1 − q n ) j =1

= −2

(q; q)∞ (−q; q)∞

∞ j =1

j =1

qj + 1 − q 2j

∞ n=0

n=1

(q; q)n (q; q)∞ − (−q; q)∞ (−q; q)n

.

Case 10: F1 0(z). In Theorem 2, replace q by q 2 , and then set a = b = q 2 and c = q 3 . This yields, after multiplication by (1 − q)−1 , 2 2

2 2 ∞ ∞ q ;q n q; q 2 n q 2n q 2; q 2 ∞ q ;q ∞ − . (3.9) =− q; q 2 ∞ q; q 2 ∞ n=1 q 2 ; q 2 n 1 − q 2n q; q 2 n+1 n=0 Now in (2.6), replace q by q 2 , then set a = zq and b = zq 2 (cf. [F, p. 16, (14.4)]). This yields, after simplification, ∞ ∞ zq; q 2 n zn q n 2 = zn q (n +n)/2 . (3.10) 2; q 2 zq n n=0 n=0 Applying + to (3.10), we find that ∞ n=1

nq

(n2 +n)/2

zq, q 2 ; 2 q , zq , zq 2

= + 2 φ1

which, by (2.4) with q replaced by q 2 and z replaced by zq with a = zq, b = q 2 , and c = zq 2 , 2 2 2 2 2 q ;q ∞ z q ;q ∞ z, zq; 2 2 2 φ1 2 2 q , q = + 2 2 z q zq ; q ∞ zq; q 2 ∞ z, zq; (3.11) = + q 2 ; q 2 ∞ (−zq; q)∞ 2 φ1 2 2 q 2 , q 2 z q 2 2 ∞ 2 2 ∞ q; q 2 n q 2n q ;q ∞ q ; q ∞ qj − = q; q 2 ∞ j =1 1 + q j q; q 2 ∞ n=1 q 2 ; q 2 n 1 − q 2n

q-IDENTITIES

409

2 2

∞ ∞ q ;q n q 2; q 2 ∞ (−1)j q j q 2 ; q 2 ∞ − = + . j 2 2 q; q ∞ j =1 1 − q q; q ∞ q; q 2 n+1 n=0

We use (3.9) in the last step above. 4. Proof of Theorems 4, 5, 6, and 7 In this section we prove Theorems 4, 5, 6, and 7. The proofs are similar to the proof of [Z, Theorem 3], and so we give a brief proof of Theorem 4 and sketches of the remaining cases. In each case it is well known that the relevant L-function has an analytic continuation to C (with the exception of a simple pole at s = 1 for ζ (s)) and a functional equation via a Mellin transformation. Proof of Theorem 4 The identity for F9 (z) in Theorem 3 is

∞ (1 − q) 1 − q 2 · · · 1 − q n F9 (z) − 2 · · · (1 + q n ) (1 + q) 1 + q n=0 ∞

√ n = 2F9 (z) d1 (n)q + 2 F9 (z) .

(4.1)

n=1

It is well known that F9 (z) = 1 + 2

∞ 2 (−1)n q n .

(4.2)

n=1

Notice that F9 (z) vanishes to infinite order as q → 1. Therefore we replace q by e−t with t 0. Now define coefficients c9 (n) and b9 (n) by the asymptotic expansions −

∞ (1 − e−t )(1 − e−2t ) · · · (1 − e−nt ) n=0

and

H9 e

(1 + e−t )(1 + e−2t ) · · · (1 + e−nt )

−t

=

∞ ∞ n −n2 t (−1) ne ∼ b9 (n)t n =4 n=1

∞

c9 (n)t n

(4.3)

n=0

as t 0.

n=0

Now we observe, by (4.1) and (4.2), that c9 (n) = b9 (n) for all n. On the other hand, ∞ ∞ ∞ 2 H9 e−t t s−1 dt = 4 (−1)n n e−n t t s−1 dt. 0

n=1

0

Replacing T by n2 t, we find that ∞ ∞ (−1)n H9 e−t t s−1 dt = 40(s) = 40(s) 41−s − 1 ζ (2s − 1). 2s−1 n 0 n=1

To compute the coefficients b9 (n), notice that

(4.4)


410

∞

∞

N −1

N −1

c9 (n) + F9 (s), s +n 0 0 n=0 n=0 (4.5) where F9 (s) is analytic for Re(s) > −N. The residue at s = −n of (4.4) is (−1)n 1+n b9 (n) = ·4 4 − 1 · ζ (−2n − 1). n! This completes the proof of Theorem 4. H9 e

−t

t

s−1

dt =

n

N

b9 (n)t + O(t ) t

Proof of Theorem 5 It is well known that F10 (z) =

∞

s−1

dt =

q (2n+1) . 2

(4.6)

n=0

It is easy to see that the identity in Theorem 3 for F10 (z) is equivalent to

∞ 1 − q 2 1 − q 4 · · · 1 − q 2n q 1/8 z F10 − × 3 1 − q 5 · · · 1 − q 2n+1 8 1−q 1 − q n=0

∞ ∞ z 1 1 2 n = F10 − + d8 (n)q + (2n + 1)q (2n+1) /8 . 8 2 2 n=1

(4.7)

n=0

Observe that F10 (z/8) vanishes to infinite order as q → −1. Define the coefficients c10 (n) and b10 (n) by the asymptotic expansions −e−t/8

∞ ∞ (1 − e−2t )(1 − e−4t ) · · · (1 − e−2nt ) c10 (n)t n = (1 + e−t )(1 + e−3t ) · · · (1 + e−(2n+1)t ) n=0

n=0

and

H10 e

−t

∞

∞

n=0

n=0

1 2 = (2n + 1)(−1)n(n+1)/2 e−t (2n+1) /8 ∼ b10 (n)t n 2

as t 0.

Therefore, by replacing q by −e−t with t 0 and ζ8 := eπ i/8 , (4.6) and (4.7) imply that c10 (n) = b10 (n) for all n and, on the other hand, that ∞ ∞ ∞ 1 2 H10 e−t t s−1 dt = (2n + 1)(−1)n(n+1)/2 e−(2n+1) t/8 t s−1 dt 2 0 0 n=0 ∞ ∞ 1 2 = χ2 (n)n e−n t/8 t s−1 dt (4.8) 2 0 n=1

1 = 8s 0(s)L(χ2 , 2s − 1). 2 The rest of the proof is identical to the proof of Theorem 4.

q-IDENTITIES

411

Proof of Theorem 6 By [C] it is known that the coefficients a(n) defining L(ρ, s) in (1.16) are defined by ∞

a(n)q n = M1 (q) + 2M2 (q).

(4.9)

n=1

In this case we use the identity for F7 (z) in Theorem 3: F7 (z) − q −1 1 − q 24 1 − q 72 · · · 1 − q 24(2n−1)

∞ n=0

= F7 (z)

∞

d(n)q

24n

(4.10) + M2 (q).

n=1

Moreover, notice in (1.8) that the nonzero coefficients of M2 (q) are supported on those exponents n ≡ 23 (mod 24). Now observe that F7 (z/24) vanishes to infinite order as q → 1. The rest of the argument is virtually identical to those above. Proof of Theorem 7 In this case we consider the identity for F4 (z) in Theorem 3: ∞

F4 (z) − q 1 + q 24 1 + q 48 · · · 1 + q 24n

n=0

∞ 1 1 24n d(n)q + M1 (q). = F4 (z) − + 2 2

(4.11)

n=1

Notice that F4 (z/24) vanishes to infinite order as q → −1. Arguing as before, we consider the asymptotic t-series expansion of −ζ24 e−t/24

∞

1 − e−t 1 + e−2t · · · 1 + (−1)n e−nt ,

n=0

where ζ24 = eπ i/24 . The rest of the proof is a routine exercise using ζ24 , the Mellin transform, and the fact that the nonzero coefficients of M1 (q) are supported on those exponents n ≡ 1 (mod 24). 5. Partition-theoretic consequences Recall that a partition of a nonnegative integer N is any nonincreasing sequence of positive integers whose sum is N. If p(N) denotes the number of partitions of N, then we have


412

∞ ∞ 1 1 = p(N )q 24N −1 = q −1 = q −1 + q 23 + 2q 47 + 3q 71 + · · ·. η(24z) 1 − q 24n N=0 n=1 (5.1) If pe (N ) (resp., po (N )) denote the number of partitions of N into an even (resp., odd) number of distinct parts, then Euler’s pentagonal number theorem asserts that the Fourier expansion of η(24z) ∈ S1/2 (00 (576), χ) is

η(24z) =

∞

pe (N )−po (N) q

24N+1

= q −q −q +· · · = 25

49

N=0

where

∞

χ(n)q n , (5.2) 2

n=1

  if n ≡ ±1 (mod 12),  1 χ (n) := −1 if n ≡ ±5 (mod 12),   0 otherwise.

(5.3)

As a consequence, for our identity for F1 (z) = 1/η(24z) we obtain the following partition-theoretic result, which is equivalent to the observation made by P. Erdös [E] in the beginning of his study of the asymptotics of p(N) by elementary means. theorem 5.1 If n is a positive integer, then let an (N) denote the number of partitions of N into parts not exceeding n. For every positive integer N we have (N + 1)p(N) =

N

p(N − n)d(n) +

n=1

N

an (N).

n=1

Proof Here we use the identity for F1 (z) = 1/η(24z) in the form

∞ 1 = P (q)D (q), P (q) − 2 · · · 1 − qn (1 − q) 1 − q n=0 where P (q) =

∞

p(n)q n =

n=0

D (q) =

∞

∞ n=1

d(n)q n .

n=1

If m is a nonnegative integer, then the series

1 , 1 − qn

(5.4)

q-IDENTITIES

P (q) −

413 ∞ 1 = q m+1 + bm (n)q n (1 − q) 1 − q 2 · · · 1 − q m n=m+2

(5.5)

for some sequence of integers bm (n). Now let N be a positive integer; then it is easy to see that the coefficient of N on the right-hand side of (5.4) is N

p(N − n)d(n).

n=1

Therefore, to prove the result it suffices to show that the coefficient of q N on the left-hand side of (5.4) is (N + 1)p(N) −

N

an (N).

(5.6)

n=1

By letting m = N as in (5.5), it is easy to see that the coefficient of q N on the left-hand side of (5.4) is the coefficient of q N in

N 1 . P (q) − (1 − q) 1 − q 2 · · · 1 − q n n=0 Claim (5.6) follows from the obvious fact that ∞

an (N )q n =

N=0

1 . (1 − q) 1 − q 2 · · · 1 − q n

This completes the proof. As with Theorem 5.1, Zagier’s identity for F8 (z) = η(24z) has an interesting partition-theoretic consequence that is similar in flavor to Euler’s pentagonal number theorem (5.2). We require some notation. If π is a partition, then let mπ denote its largest part. Moreover, if N is an integer, then let Se (N) (resp., So (N)) denote the set of partitions of N into an even (resp., odd) number of distinct parts. Using this notation, define the two partition functions Ae (N) and Ao (N) by Ae (N) := mπ , (5.7) π ∈Se (N)

Ao (N) :=

π ∈So (N)

theorem 5.2 If N is a positive integer, then

mπ .

(5.8)


414

N − 3k 2 + k Ae (N ) − Ao (N ) = (−1) d 2 k∈Z  3k 2 + k   (−1)k (3k) with k ≥ 0, if N =    2 3k 2 − k + with k ≥ 1, (−1)k (3k − 1) if N =    2   0 otherwise.

k

Proof If E (q) is the q-series defined by E (q) =

∞

1 − q n = 1 − q − q 2 + · · ·,

n=1

then we claim that ∞ E (q) − (1 − q) 1 − q 2 · · · 1 − q n n=0

=−

∞

nq (1 − q) 1 − q 2 · · · 1 − q n−1 .

(5.9)

n

n=1

As described above, we have that E (q) =

∞

1−q

n=1

n

=

∞

pe (n) − po (n) q n .

n=0

Since E (q) is the generating function for the number of partitions into an even number of distinct parts minus the number into an odd number of distinct parts, it is easy to see that E (q) − (1 − q) 1 − q 2 · · · 1 − q n is the generating function whose coefficient of q N is the number of partitions of N into an even number of distinct parts where at least one part exceeds n minus the number of partitions of N into an odd number of distinct parts where at least one part exceeds n. Consequently, each partition π into distinct parts is counted with multiplicity (−1)nπ mπ , where nπ is the number of parts in π. This implies (5.9). It is an easy exercise to deduce Theorem 5.2 from the identity in Theorem 1 for F8 (z) = η(24z). 6. Related results The uniform nature of the ten results proved in Theorem 3 makes it natural to group

q-IDENTITIES

415

those identities together. There are, however, a variety of results that can be deduced from Theorems 1 and 2. Here we record a number of further identities. theorem 6.1 √ √ Let m(n) denote the number of divisors of n in the interval [ n/2, 2n). Moreover, let de (n) (resp., do (n)) denote the number of even (resp., odd) divisors of n. The following identities are true:

∞ ∞ 1 1 1 − (6.1) d(n) + m(n) q 24n , = 2 2 2 η (24z) q 2 q 24 ; q 24 η (24z) n=0 n=1 n ∞ 2 ∞ η (2z) (−q; q)2n 4η2 (2z) − = do (2m + 1)q 2m+1 , (6.2) η4 (z) (q; q)2n η4 (z) n=0 m=0 8 16 2

∞ ∞ q ; q )n η2 (8z) η2 (8z) − (6.3) = (de (n) − 2do (n)) q 8n , η4 (16z) q 2 (q 16 ; q 16 2 η4 (16z) n=0 n=1 n ∞

2 ∞ ∞ 1 1 (−1)n−1 q 24n 1 48n − d(n)q + = , η(24z) q(q 24 ; q 24 )2n η(24z) 1 − q 48n n=0

n=1

∞

n=1

(6.4)

∞

q 2n − (−q)n η(2z) (−q; q)2n 2η(2z) − , = (q; q)2n η2 (z) η2 (z) 1 − q 4n n=0 n=1 8 16

∞ q ; q η(8z) 2n − η2 (16z) q q 16 ; q 16 2n n=0

∞ ∞ 8n η(8z) 8n2 1− q +2 de (n) − do (n) q = 2 . 4η (16z) n=−∞

(6.5)

(6.6)

n=1

Proof Here we prove these identities. Case 1: (6.1). Let a = b = 0 and c = q in Theorem 2. This yields ∞ n=0

1 1 − (q; q)2∞ (q; q)2n ∞

2 ∞ qn 1 (−1)n−1 q (n +n)/2 = . + 1 − qn 1 − qn (q; q)2∞ n=1

n=1

In order to complete the proof of (6.1), we must establish that

(6.7)


416 ∞

m(n)q n =

n=1

2 ∞ (−1)n−1 q (n +n)/2

1 − qn

n=1

.

(6.8)

To see (6.8), we first note that√m(n) equals d(n) minus the number of divisors in the √ two intervals [1, n/2) and [ 2n, n]. Hence we have that ∞

∞ qn − q (n)(2n) 1 + 2q n + 2q 2n + 2q 3n + · · · n 1−q n=1 n=1 2 ∞ ∞ n q 1 + q n q 2n − . = 1 − qn 1 − qn

m(n)q n =

n=1

∞

n=1

(6.9)

n=1

Next, in [GR, p. 242, (III.17)], set a = e = z, and let b, c, d, and f → +∞. This yields

2 2 ∞ ∞ (z; q)2n (1 − zq 2n )q 2n zn (z; q)n (−1)n q (n +n)/2 (q; q)∞ = 1+ . (zq; q)∞ (q; q)n (q; q)2n (1 − z) n=1

n=0

Hence we get 2 ∞ (−1)n−1 q (n +n)/2

n=1

1 − qn

=+

2 ∞ (z; q)n (−1)n q (n +n)/2

n=0

(q; q)n

2

∞ (z; q)2n 1 − zq 2n q 2n zn (q; q)∞ =+ 1+ (zq; q)∞ (q; q)2n (1 − z) n=1 2 ∞ ∞ 1 + q n q 2n qn = − 1 − qn 1 − qn =

n=1 ∞

(6.10)

n=1

m(n)q n ,

n=1

which proves (6.8) and therefore (6.1). Case 2: (6.2). To prove this identity, let a = b = −q and c = q in Theorem 2. The result now follows easily by combining the resulting Lambert series. Case 3: (6.3). To prove this identity, replace q by q 2 and set a = b = q and c = q 2 in Theorem 2. The result now follows easily by combining the resulting Lambert series. Case 4: (6.4). To prove this identity, replace q by q 2 and set a = b = 0 and c = q 2 in Theorem 2.

q-IDENTITIES

417

Case 5: (6.5). This identity follows from Theorem 2 by replacing q by q 2 and then setting a = −q, b = −q 2 , and c = q. Case 6: (6.6). Surprisingly, (6.6) is more intricate. Here we use Proposition 2.1 with q; q 2 2n α(n) = 2 2 . q ; q 2n Consequently (noting that +f (z) = (1/2)+f (z2 )), we find that

∞ q; q 2 2n q; q 2 ∞ − 2; q 2 2; q 2 q q ∞ 2n n=0 n ∞ 2 q; q z 2n = +(1 − z) 2 2 q ; q 2n n=0 ∞ n z q; q 2 n 1 + (−1)n 1 2 = + 1−z 2 2 q 2; q 2 n n=0 qz; q 2 ∞ − zq; q 2 ∞ 1 = + (1 + z) 2 2 + (1 − z) · 4 zq ; q ∞ − zq 2 ; q 2 ∞ ∞ 2 q; q 2 ∞ − q; q 2 ∞ 1 q; q 2 ∞ q 2n q 2n−1 = 2 2 − 2 2 + 2 2 − . 4 q ;q ∞ 1 − q 2n 1 − q 2n−1 − q ;q ∞ q ;q ∞ n=1

Identity (6.6) now follows by recalling from [A3, p. 21, (2.2.10)], with z = 1, that

∞ − q; q 2 ∞ q 2 ; q 2 ∞ 2 = qn . 2 2 2 q; q ∞ − q ; q ∞ n=−∞

Finally, we recall the following attractive and very elementary result on partitions into consecutive integers (cf. [L, p. 85, Problem 4]). proposition 6.2 The number of partitions of n into consecutive integers equals the number of odd divisors of n. Proof This result is easily deduced from the formula for the sum of an arithmetic progression. It is also directly deduced from the generating function identity (see [M, p. 28])


418 ∞ m=1

2 ∞ q 2m−1 q (m +m)/2 = . 1 − qm 1 − q 2m−1

m=1

From our proof of (6.1), we may easily deduce the following result for ce (n) (resp., co (n)), which is the number of partitions of n into an even (resp., odd) number of consecutive integers. theorem 6.3 For every positive integer n, we have co (n) − ce (n) = m(n). Proof This is immediate from (6.10): ∞ n=1

m(n)q n =

2 ∞ (−1)n−1 q (n +n)/2

n=1

1 − qn

=

∞

co (n) − ce (n) q n .

n=1

Acknowledgments. The authors thank Scott Ahlgren, Andrew Granville, Don Zagier, and the referee for a variety of useful comments and suggestions.

References [A1] [A2]

[A3] [ADH] [C] [E] [F] [GR]

G. E. ANDREWS, Ramanujan’s “lost” notebook, V: Euler’s partition identity, Adv.

Math. 61 (1986), 156–164. MR 87i:11137 , “Mock theta functions” in Theta Functions (Brunswick, Maine, 1987), Part 2, Proc. Sympos. Pure Math. 49, Amer. Math. Soc., Providence, 1989, 283–298. MR 90h:33005 , The Theory of Partitions, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1998. MR 99c:11126 G. E. ANDREWS, F. DYSON, and D. HICKERSON, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391–407. MR 89f:11071 H. COHEN, q-identities for Maass waveforms, Invent. Math. 91 (1988), 409–422. MR 89f:11072 ¨ , On an elementary proof of some asymptotic formulas in the theory of P. ERDOS partitions, Ann. of Math. (2) 43 (1942), 437–450. MR 4:36a N. J. FINE, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, Amer. Math. Soc., Providence, 1988. MR 91j:33011 G. GASPER and M. RAHMAN, Basic Hypergeometric Series, Encyclopedia Math. Appl. 35, Cambridge Univ. Press, Cambridge, 1990. MR 91d:33034

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[L]

W. J. LEVEQUE, Topics in Number Theory, Vol. 1, Addison-Wesley, Reading, Mass.,

[M] [Z]

P. A. MACMAHON, Combinatory Analysis, Chelsea, New York, 1960. MR 25:5003

1956. MR 18:283d D. ZAGIER, Vassiliev invariants and a strange identity related to the Dedekind

eta-function, to appear in Topology.

Andrews Department of Mathematics, Penn State University, University Park, Pennsylvania 16802, USA; [email protected] Jiménez-Urroz Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, España; [email protected] Ono Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA; [email protected]

¯ THE ∂-NEUMANN OPERATOR ON LIPSCHITZ PSEUDOCONVEX DOMAINS WITH PLURISUBHARMONIC DEFINING FUNCTIONS JOACHIM MICHEL and MEI-CHI SHAW

Abstract On a bounded pseudoconvex domain in Cn with a plurisubharmonic Lipschitz ¯ defining function, we prove that the ∂-Neumann operator is bounded on Sobolev (1/2)-spaces. 0. Introduction Let D be a bounded pseudoconvex domain in Cn with the standard Hermitian metric. ¯ The ∂-Neumann operator N for (p, q)-forms is the inverse of the complex Lapla¯ where ∂¯ is the maximal extension of the Cauchy-Riemann cian = ∂¯ ∂¯ ∗ + ∂¯ ∗ ∂, operator on (p, q)-forms with L2 -coefficients and ∂¯ ∗ is its Hilbert space adjoint. The ¯ existence of the ∂-Neumann operator for any bounded pseudoconvex domain follows ¯ The ∂-Neumann ¯ problem serves as from Hörmander’s L2 -existence theorems for ∂. a prototype for boundary value problems that are noncoercive and is of fundamental importance in the theory of several complex variables and partial differential equations. ¯ The ∂-Neumann problem has been studied extensively when the domain D has smooth boundary (see J. Kohn [21], [22] or H. Boas and E. Straube [4], [6] and ¯ the references within). In this paper we study the ∂-Neumann operator on a Lipschitz domain D when D has a plurisubharmonic defining function (see Theorem 1). s (D) denote Hilbert spaces of (p, q)-forms with H s (D)-coefficients. Their Let H(p,q) norms are denoted by s(D) for s ≥ 0. The principal result of this paper is the following theorem. theorem 1 Let D Cn be a bounded pseudoconvex Lipschitz domain with a defining func¯ tion that is plurisubharmonic in D. The ∂-Neumann operator N is bounded from DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 3, Received 30 September 1999. Revision received 22 August 2000. 2000 Mathematics Subject Classification. Primary 35N15; Secondary 32W05. Shaw’s work partially supported by National Science Foundation grant number DMS 98-01091.

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MICHEL AND SHAW

1/2

1/2

H(p,q) (D) to H(p,q) (D), where 0 ≤ p ≤ n and 1 ≤ q ≤ n − 1, and N satisfies the 1/2

following estimates: there exists C > 0 such that for any α ∈ H(p,q) (D), Nα1/2 (D) ≤ Cα1/2 (D),

(0.1)

where C depends only on the Lipschitz constant and the diameter of D but is independent of α. Theorem 1 is an easy consequence of the following two estimates: ∗ ∂¯ Nα (D) ≤ Cα1/2 (D), 1 ≤ q ≤ n, 1/2 ∂N ¯ α (D) ≤ Cα1/2 (D), 0 ≤ q ≤ n − 1. 1/2

(0.2) (0.3)

Estimate (0.2) was proved earlier by A. Bonami and P. Charpentier [7], [8] (see also Charpentier [9]) when the domain is smooth and bounded. When the domain is only Lipschitz with a plurisubharmonic defining function, they also proved that the operator ∂¯ ∗ N is bounded from H 1/2+ (D) to H 1/2 (D) for any > 0. In general, pseudoconvex domains do not have plurisubharmonic defining functions. If D has smooth boundary and has a C ∞ -plurisubharmonic defining function on bD, Boas ¯ and Straube [5] showed that the ∂-Neumann operator is bounded on Sobolev spaces s on H (D) for all s ≥ 0. On the other hand, D. Barrett [1] has proved that for any fixed s > 0 there exist some worm domains constructed by K. Diederich and ¯ J. Fornaess [13] such that the ∂-Neumann operator is not bounded from H s (D) to s ∞ H (D). In fact, N also does not preserve C -regularity on such worm domains (see M. Christ [10]). The worm domains are pseudoconvex domains with C ∞ -boundaries which do not have plurisubharmonic defining functions on bD. When bD is Lipschitz, there is a natural notion of strong pseudoconvexity (see (1.2)). Using the arguments of J. Michel and M.-C. Shaw [25], it is easy to −1/2 see that N can be extended as a bounded operator from H(p,q) (D) to the Sobolev 1/2

space H(p,q) (D) if D is Lipschitz and strongly pseudoconvex (see also G. Henkin, A. Iordan, and J. Kohn [17]). E. Straube [30] has extended the subelliptic estimates of N to domains with piecewise smooth boundaries of finite type. When the domain is the intersection of a strongly pseudoconvex domain with a strongly q-convex do¯ operator were proved by main, L2 -existence and subellipticity of the ∂-Neumann ¯ S. Vassiliadou [31]. The compactness of the ∂-Neumann operator on Lipschitz pseudoconvex domains is studied in Henkin and Iordan [16]. There has also been considerable work on the Dirichlet and classical Neumann problems on Lipschitz domains (see B. Dahlberg [11] and D. Jerison and C. Kenig [20]). We note that if the boundary is smooth, the regularity of the Bergman projection implies the regularity of the ¯ ∂-Neumann operator (see Boas and Straube [4]), and one can prove the regularity of

¯ THE ∂-NEUMANN OPERATOR

423

the Bergman projection by estimating the commutators (see Boas and Straube [5]). If the boundary is not sufficiently smooth (at least C 4 ), we cannot prove Theorem 1 by estimating the commutators of the tangential vector fields. Instead we use the ¯ and their duals. interplay between the boundary value of the operators ∂¯ ∗ N and ∂N This idea was due to B. Berndtsson [2] in attempting to generalize Carleson’s result to several variables. It was also used by Bonami and Charpentier [7], [8] (see also Charpentier [9]) in studying the Bergman projection and the canonical solutions for ∂¯ on domains with plurisubharmonic defining functions. Using some extensions of the basic a priori estimates of Morrey, Kohn, and L. Hörmander (see [18]), we first prove ¯ are bounded on H 1/2 (D) on domains with smooth that the operators ∂¯ ∗ N and ∂N (p,q) boundaries. Since the estimates depend only on the Lipschitz constant of the defining function, we can prove the estimates for Lipschitz domains by using approximation and some interpolation Sobolev spaces. The regularity of N follows easily since it ¯ . can be expressed in terms of ∂¯ ∗ N and ∂N The plan of this paper is as follows. In Section 1 we first recall some definitions and facts on Lipschitz pseudoconvex domains. In Section 2 we introduce some op¯ and derive their basic erators that are duals of the boundary values of ∂¯ ∗ N and ∂N properties. These operators have harmonic coefficients. In Section 3 some identities of the Morrey-Kohn-Hörmander type are derived. Theorem 1 is proved in Section 4. ¯ 1. Lipschitz domains and the ∂-Neumann problem 2n−1 → R be a function that satisfies the Lipschitz condition Let ψ : R ψ(x) − ψ x ≤ M x − x for all x, x ∈ R2n−1 .

(1.1)

A bounded domain D is called Lipschitz if near every boundary point p ∈ bD there exists a neighborhood U of p such that, after a rotation, D ∩ U = (x, x2n ) ∈ U | x2n > ψ(x) for some Lipschitz function ψ. The smallest M in which (1.1) holds is called the bound of the Lipschitz constant. By choosing finitely many balls {Ui } covering bD, the Lipschitz constant for a Lipschitz domain is the smallest M such that the Lipschitz constant is bounded by M in every ball Ui . A Lipschitz function is almost everywhere differentiable (see L. Evans and R. Gariepy [14] for a proof of this fact). Definition 1.1 A bounded Lipschitz domain D in Cn is said to have a plurisubharmonic Lipschitz defining function if (1) there exists a Lipschitz function ρ : Cn → R such that ρ < 0 in D, ρ > 0 outside D, and C1 < |dρ| < C2 on bD almost everywhere in a neighborhood of bD, where C1 , C2 are positive constants;

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(2)

n i,j =1

∂ 2ρ ti t¯j ≥ 0, ∂zi ∂ z¯ j

t = (t1 , . . . , tn ) ∈ Cn ,

where (2) is defined in the distribution sense. We note that (2) actually implies that the complex Hessian of ρ has measure coefficients. A bounded Lipschitz domain is called strongly pseudoconvex if there exists a Lipschitz defining function ρ in a neighborhood of D such that for some positive constant c > 0, n i,j =1

∂ 2ρ ti t¯j ≥ c|t|2 , ∂zi ∂ z¯ j

t = (t1 , . . . , tn ) ∈ Cn ,

(1.2)

where (1.2) is defined in the distribution sense. All convex Lipschitz domains have plurisubharmonic Lipschitz defining functions on D. Also, the transversal intersection of smooth domains with C 2 -plurisubharmonic defining functions ρi , i = 1, . . . , k, on Di is another example of Lipschitz domains with plurisubharmonic defining function on D. To see this, take ρ = max{ρ1 , . . . , ρk }. It is easy to see that ρ is Lipschitz and plurisubharmonic on D. ∞ (D) denote the space of smooth (p, q)-forms on D, that is, the restriction of Let C(p,q) smooth (p, q)-forms in Cn to D. Let (z1 , . . . , zn ) be the complex coordinates for Cn . ∞ (D) can be expressed as Then any (p, q)-form f ∈ C(p,q) fI,J dzI ∧ d z¯ J , f = I,J

where I = (i1 , . . . , ip ) and J = (j1 , . . . , jq ) are multi-indices and dzI = dzi1 ∧ · · · ∧ dzip , d z¯ = d z¯ j1 ∧ · · · ∧ d z¯ iq . The notation means the summation over strictly increasing multi-indices, and fI,J is defined for arbitrary I and J so that they are ∞ (D) → C ∞ antisymmetric. The operator ∂¯ : C(p,q) (p,q+1) (D) is defined by ¯ = ∂f

∂fI,J

I,J

k

∂ z¯ k

d z¯ k ∧ dzI ∧ d z¯ J .

We use L2(p,q) (D) to denote (p, q)-forms with L2 (D)-coefficients. We use ( , )D to denote the inner product in L2(p,q) (D), and when there is no danger of confusion, we drop the subscript D in the notation. If φ is a smooth function in D, we denote L2 (D, φ) the space of functions in D which are square integrable with respect to the density e−φ . The norm in L2(p,q) (D, φ) is defined by |f |2 e−φ dV , f ∈ L2(p,q) (D, φ), f 2φ = D


425

where dV is the Lebesgue measure. The inner product in L2(p,q) (D, φ) is defined by ( , )φ . The formal adjoint of ∂¯ under the usual L2 -norm is denoted by ϑ and defined by the requirement that ¯ (ϑf, g) = f, ∂g ∞ (D) with compact support in D. Therefore ϑ can be exfor all smooth g ∈ C(p,q) pressed explicitly by

ϑf = (−1)p−1

∂fI,j K dzI ∧ d z¯ K , ∂zj I,K

(1.3)

j

where |I | = p and |K| = q −1 are multi-indices. It is easy to check that ∂¯ 2 = ϑ 2 = 0. Let ϑφ be the formal adjoint of ∂¯ under the L2 (D, φ)-norm; that is, ¯ (1.4) (ϑφ f, g)φ = f, ∂g φ ∞ . We have the following relation befor every compactly supported g ∈ C(p,q−1) tween ϑ and ϑφ : (1.5) ϑφ f = eφ ϑe−φ f = ϑf − σ (ϑ, dφ)f,

where σ (ϑ, dφ) is the symbol of ϑ in the direction of dφ. We denote the maximal extension of ∂¯ in L2 (D) still by ∂¯ and by ∂¯ ∗ , its Hilbert space adjoint. We have the following complex: ∂¯

∂¯

∂¯ ∗

∂¯ ∗

L2(p,q−1) (D) L2(p,q) (D) L2(p,q+1) (D). We define = ∂¯ ∂¯ ∗ + ∂¯ ∗ ∂¯ : L2(p,q) (D) → L2(p,q) (D) such that

¯ ∈ Dom ∂¯ ∗ , ∂¯ ∗ ϕ ∈ Dom ∂¯ . Dom() = ϕ ∈ Dom ∂¯ ∩ Dom ∂¯ ∗ | ∂ϕ Then is a linear, closed, densely defined self-adjoint operator from L2(p,q) (D) to L2(p,q) (D). Following Hörmander’s L2 -estimates [18] for ∂¯ on any bounded pseudoconvex domains, one can prove that has closed range and Ker () = {0}. The ¯ ∂-Neumann operator N is the inverse of , and we have the following L2 -existence theorem of N on D. proposition 1.2 Let D be a bounded pseudoconvex domain in Cn , n ≥ 2. For each 0 ≤ p ≤ n and 1 ≤ q ≤ n, there exists a bounded N : L2(p,q) (D) → L2(p,q) (D) such that (i) Range(N ) ⊂ Dom(), and N = N = I on Dom();

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¯ (ii) for any α ∈ L2(p,q) (D), α = ∂¯ ∂¯ ∗ Nα ⊕ ∂¯ ∗ ∂Nα; (iii) if δ is the diameter of D, we have the following estimates: eδ 2 α, q 2 ∂Nα ≤ eδ α, ¯ q 2 ∗ ∂¯ Nα ≤ eδ α; q Nα ≤

(iv) we have ¯ ¯ 1 ≤ q ≤ n − 1, N ∂¯ = ∂N on Dom(∂), ∗ ∗ ∂¯ N = N ∂¯ on Dom(∂¯ ∗ ), 2 ≤ q ≤ n. This proposition follows from Hörmander’s L2 -existence theorems for ∂¯ (see Shaw ¯ [28, Proposition 2.3] for details). It is also proved in Shaw [28] that the ∂-Neumann 2 operator also exists when q = 0, and for any α ∈ L(p,0) (D), ∂Nα ≤ eδ 2 α. ¯

Let H s (D), s ≥ 0, be defined as the space of all u|D such that u ∈ H s (Cn ). We define the norm of H s (D)v by us(D) =

inf

U s(Cn ) .

U ∈H s (Cn ),U |D =u

s We define H(p,q) (D) to be the Hilbert space of (p, q)-forms with H s (D)-coefficients, and their norms are denoted by s(D) . Let H0s (D) be the completion of C0∞ (D)functions under the H s (D)-norm. If D is a Lipschitz domain, then C ∞ (D) is dense in H s (D) in the H s (D)-norm. If s ≤ 1/2, we also have that C0∞ (D) is dense in H s (D) η (see Grisvard [15, Theorem 1.4.2.4]). Thus H η (D) = H0 (D) if 0 ≤ η ≤ 1/2. We −η η use H (D) to denote the dual of H (D). For a detailed treatment of Sobolev spaces H s (D) for any real s, we refer the readers to J.-L. Lions and E. Magenes [24] for smooth domains and to P. Grisvard [15] for nonsmooth domains. The following two propositions are crucial in our proof of the theorem.

proposition 1.3 Let D be a bounded Lipschitz domain in RN with a Lipschitz defining function ρ. Let u be a harmonic function in D. Then the following conditions are equivalent:


427

(i) u ∈ H 1/2 (D), (ii)

u ∈ L2 (D)

and

u ∈ L2 (D)

and

D

(iii)

d(x)|∇u|2 dV < ∞,

sup

bD

|u|2 dS < ∞,

where d(x) is the distance from x to the boundary bD, D = {z ∈ D | ρ(z) < −} for > 0 and dS is the surface measure for bD . That (i) is equivalent to (ii) follows from Jerison and Kenig [20, Theorem 4.2]. It follows from Dahlberg [11] that (ii) and (iii) are equivalent. Proposition 1.3 also ¯ ∩ Dom(∂¯ ∗ ) since the system ∂¯ ⊕ ϑ is an elliptic system. In holds for u ∈ Dom(∂) fact, one can reduce the proof to the case when u has harmonic coefficients (see Straube [30]). proposition 1.4 Let D be a bounded Lipschitz domain in RN , and let u be a harmonic function in D. Then 1 u ∈ H −s (D), 0 ≤ s ≤ , 2 if and only if d(x)2s |u|2 dV < ∞, D

where d(x) is the distance from x to the boundary bD. The proof of this proposition uses complex interpolation arguments similar to those in Jerison and Kenig [20, Theorems 4.1 and 4.2], and we refer readers to the proof in that paper. 2. Forms with harmonic coefficients ¯ and ∂¯ ∗ N on the boundary of D, and In this section we study the traces of operators ∂N their duals. The dual operators have ranges in forms whose coefficients are harmonic functions. Let D be a bounded domain in Cn with C 1 -boundary. Let B(p,q) (bD) be the space of the restriction to bD of all (p, q)-forms with C 1 (D)-coefficients which are ¯ We use τ to denote the projection pointwise orthogonal to the ideal generated by ∂ρ. 1 1 (D) → τ : C(p,q) (D) → B(p,q) (bD), and ν to denote the projection ν : C(p,q) V(p,q) (bD), where V(p,q) (bD) denotes the space of the restriction to bD of all (p, q)-

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MICHEL AND SHAW

¯ In particular, τ ⊕ν = r, where r is simply the restriction forms that are multiples of ∂ρ. 1 map from C(p,q) (D) to the boundary. s (bD), 0 ≤ s ≤ 1, to denote the space of forms that are the We use H(p,q) completion of B(p,q) (bD)-forms with H s (bD)-norms. This is well defined also for Lipschitz domains since on bD, H 1 (bD) is well defined and the boundary value of any function in H 1 (D) to the boundary belongs to H 1/2 (bD) (see Jerison and Kenig [20]). lemma 2.1 Let D be a domain with C 1 -boundary bD, and let ρ be a C 1 –defining function ¯ and r(∂¯ ∗ N) can be for D. For any f ∈ L2(p,q) (D), the restriction maps r(∂N) −1/2

−1/2

extended as bounded operators from L2(p,q) (D) to H(p,q+1) (bD) and H(p,q−1) (bD), respectively. Let B = I − ∂¯ ∗ N ∂¯ denote the Bergman projection. Then τ B is bounded −1/2 from L2(p,q) (D) to H(p,q) (bD). The bounds depend only on the Lipschitz constant

of D. If D has only Lipschitz boundary bD, the operators τ and ν are also defined almost everywhere on bD. Proof 1 1 (D) and ψ ∈ C(p,q) (D), we have, using integration by parts, For any f ∈ C(p,q) 1 ¯ (ϑf, ψ) = f, ∂ψ + dS, (2.1) σ (ϑ, dρ)f, ψ |dρ| bD where dS is the surface measure of bD and σ (ϑ, dρ)f is the symbol of ϑ in the direction dρ. Let denote the adjoint of the exterior product. We have that ¯ σ (ϑ, dρ)f = ϑ(ρf ) = f ∂ρ. For any ψ ∈ H 1/2 (bD), we extend ψ from bD to ψ on D such that ψ H 1 (D) ≤ C ψ H 1/2 (bD) , where C depends only on the Lipschitz constant of D. Using (2.1), we have for any 1 (D), ψ ∈ C(p,q) 1 ¯ ¯ dS ≤ |(ϑf, ψ)| + f, ∂ψ f ∂ρ, ψ |dρ| bD (2.2) ≤ C(f + ϑf )ψ 1 H (D)

≤ C(f + ϑf )ψH 1/2 (bD) . From the density theorem and the Sobolev trace theorem for Lipschitz domains (see 1/2 1 (D) is dense in H(p,q) (bD). This implies that Grisvard [15]), the restriction of C(p,q) ¯ ∂ρ νf H −1/2 (bD) = f ≤ C f + ϑf . (2.3) |dρ| H −1/2 (bD)


429

1 ∞ Similarly, for any f ∈ C(p,q) (D) and φ ∈ C(p,q+1) (D), we have, using integration by parts, ¯ dρ f, φ 1 dS ¯∂f, φ = (f, ϑφ) + σ ∂, |dρ| bD (2.4) 1 ¯ = (f, ϑφ) + dS, ∂ρ ∧ f, φ |dρ| bD

¯ dρ)f is the symbol of ∂¯ in the direction dρ and σ (∂, ¯ dρ)f = ∂(ρf ¯ where σ (∂, )= ¯∂ρ ∧ f on bD. Using (2.4), we have for any φ ∈ C 1 (D), (p,q) ¯ φ + f, ∂φ ¯ ¯ φ 1 dS ≤ ∂f, f ∧ ∂ρ, |dρ| bD (2.5) ¯ φH 1 (D) ≤ f + ∂f ¯ φH 1/2 (bD) . ≤ C f + ∂f This implies that τf H −1/2 (bD)

¯ ∂ρ ¯ . = f ∧ ≤ C f + ∂f |dρ| H −1/2 (bD)

(2.6)

2 ¯ ∈ L2 We have that if f ∈ L2(p,q) (D), ∂f (p,q+1) (D), and ϑf ∈ L(p,q−1) (D), ¯ then the boundary value of f is in H −1/2 (bD) from (2.3) and (2.6). Since ∂Nf and ∗ ∗ ∗ ¯ ∩ Dom(∂¯ ), we have that ∂Nf ¯ and ∂¯ Nf have boundary data in ∂¯ Nf are in Dom(∂)

H −1/2 (bD). It is easily seen that the constants depend only on the Lipschitz constant and the diameter of D. ¯ = 0. Using (2.6), it follows that For any f ∈ L2(p,q) (D), we have that ∂Bf −1/2 ¯ bD ∈ H (bD). The lemma is proved. Bf ∧ ∂ρ| (p,q)

¯ to bD by R1 and R2 , respectively. We denote the restriction maps of ∂¯ ∗ N and ∂N 2 ∗ ¯ ). From Lemma 2.1 we For any f ∈ L(p,q) (D), R1 f = r(∂¯ Nf ) and R2 f = r(∂Nf have −1/2 R1 : L2(p,q) (D) −→ H(p,q−1) (bD) and

−1/2

R2 : L2(p,q) (D) −→ H(p,q+1) (bD). 1/2

Let T1 : H(p,q−1) (bD) → L2(p,q) (D) be the dual of R1 , and let it be defined as 1/2

follows. For a fixed v ∈ H(p,q−1) and any u ∈ L2(p,q) (D), we have, using Lemma 2.1, that ∗ ∂¯ Nu, v dS ≤ CR1 uH −1/2 (bD) vH 1/2 (bD) ≤ Cu, bD

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MICHEL AND SHAW

where C depends on v. Thus there exists an element g = T1 v ∈ L2(p,q) (D) such that ∗ (2.7) ∂¯ Nu, v dS = (u, T1 v) for any u ∈ L2(p,q) (D). bD

1/2

The dual of R2 is similarly defined. Let T2 : H(p,q+1) (bD) → L2(p,q) (D) such that 1/2

for any v ∈ H(p,q+1) (bD), ¯ u, v dS = u, T2 v ∂N bD

for any u ∈ L2(p,q) (D).

(2.8)

1/2

We also denote the adjoint operator of τ B by P1 . Then P1 : H(p,q) (bD) → L2(p,q) (D) 1/2

such that for any v ∈ H(p,q) (bD), (τ B)u, v dS = (u, P1 v) for any u ∈ L2(p,q) (D). bD

(2.9)

lemma 2.2 Let D be a bounded pseudoconvex domain with C 1 -boundary. Then for any v ∈ 1/2 H(p,q−1) (bD), the coefficients of T1 v are harmonic functions, and we have ¯ 1v = 0 ∂T

and

¯ 1v = 0 ∂ϑT

in D.

(2.10)

1/2

Also, for any v ∈ H(p,q+1) (bD), the coefficients of T2 v also are harmonic functions, and we have ¯ 2 v = 0 in D. and ϑ ∂T (2.11) ϑT2 v = 0 Proof Equation (2.10) is proved in Bonami and Charpentier [7]. For the convenience of the ∞ reader, we include a proof here. Let u = ϑφ for some φ ∈ C(p,q+1) (D) with compact support in D. From (2.7) we have ∗ ∂¯ Nϑφ, v dS = (u, T1 v) 0= bD

since ∂¯ ∗ Nϑφ = (∂¯ ∗ )2 Nφ = 0. Choosing φ with compact support in D, we have ¯ 1 v = 0. (ϑφ, T1 v) = φ, ∂T ¯ 1 v = 0 in the distribution sense. If we take u = ∂ϑg ¯ This proves that ∂T where g is a ∞ (p, q)-form with C0 (D)-coefficients, we have ¯ ¯ + ϑ ∂¯ g = ϑg. = ∂¯ ∗ N ∂ϑ ∂¯ ∗ N ∂ϑg


Thus 0=

bD

431

¯ ¯ ¯ 1v . v dS = ∂ϑg, T1 v = g, ∂ϑT ∂¯ ∗ N ∂ϑg,

¯ 1 v = 0 in the distribution Since g can be any test form, we have proved that ∂ϑT ¯ ¯ sense. From (2.10) we have that T1 v = (∂ϑ + ϑ ∂)v = 0. Thus the coefficients of T1 v are harmonic functions and (2.10) holds in the classical sense. Exchanging the role of ∂¯ and ϑ, (2.11) can be proved similarly. lemma 2.3 Let D be a bounded pseudoconvex domain with C 1 -boundary, and let P1 be the 1/2 operator defined by (2.9). Then for any v ∈ H(p,q) (bD), the coefficients of P1 v are harmonic functions and we have ¯ 1 v = 0, ∂P

ϑP1 v = 0

in D,

(2.12)

and P1 v = −ϑT1 v.

(2.13)

Proof ∞ (D) with compact support in D. We have (τ B)φ = −∂¯ ∗ N ∂φ ¯ on bD. Let φ ∈ C(p,q) Thus, using Lemma 2.2, we have ¯ v dS = ∂φ, ¯ −T1 v = φ, −ϑT1 v . (2.14) (τ B)φ, v dS = − ∂¯ ∗ N ∂φ, bD

bD

On the other hand, from the definition of P1 , we have (τ B)φ, v dS = (φ, P1 v). bD

(2.15)

Thus P1 v = −ϑT1 v and (2.13) is proved. Equation (2.12) follows easily from (2.13) and (2.10). lemma 2.4 Let D be a bounded pseudoconvex domain with C 1 -boundary, and let ρ be a C 1 – 1/2 ¯ on bD. Then for any f ∈ H(p,q) (bD), defining function for D. Let N = ∂ρ/|dρ| where 0 ≤ p ≤ n, 1 ≤ q ≤ n − 1, we have T1 f N = τf −N ∧ T2 f = νf

on bD, on bD.

(2.16) (2.17)

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MICHEL AND SHAW

Proof ∞ (D). From the definition of T and P , we have Let ψ ∈ C(p,q) 1 1 ∗ ¯ T1 f = ¯ f dS ∂ψ, ∂¯ N ∂ψ, bD ψ − (τ B)ψ, f dS = bD = τ ψ, f dS − (ψ, P1 f ).

(2.18)

bD

On the other hand, using integration by parts as in (2.1) and Lemma 2.3, we have ¯∂ψ, T1 f = ψ, ϑT1 f + ψ, T1 f N dS bD (2.19) = −(ψ, P1 f ) + ψ, T1 f N dS. bD

∞ (D), From (2.18) and (2.19), we have for any ψ ∈ C(p,q) τ ψ, f dS = ψ, T1 f N dS = bD

bD

bD

ψ, τf dS.

This implies (2.16). ∞ (D), that To prove (2.17), we have, for any ψ ∈ C(p,q−1) ¯ ϑ ∂(ρψ) ¯ ¯ ϑ ∂¯ + ∂ϑ ¯ (ρψ) = ∂(ρψ) ¯ ¯ ∧ψ ∂N = ∂N = ∂ρ

on bD.

Using the definition of T2 and (2.20), we have ¯ ¯ , T2 f ¯ ∧ ψ , T2 f = ϑ ∂(ρψ), T2 f − ϑ ρ ∂ψ ϑ ∂ρ ¯ ¯ ¯ ∂T ¯ 2f ∂Nϑ ∂(ρψ), f dS − ρ ∂ψ, = bD ¯ ∧ ψ, f dS − ρ ∂ψ, ¯ ∂T ¯ 2f . = ∂ρ

(2.20)

(2.21)

bD

On the other hand, using integration by parts as in (2.4), we have ¯ ∧ ψ , T2 f ϑ ∂ρ ¯ ∧ ψ, N ∧ T2 f dS ¯ ¯ ∂ρ = ∂ρ ∧ ψ, ∂T2 f − bD ¯ ¯ ∂T ¯ 2f − ¯ ∧ ψ, N ∧ T2 f dS = ∂(ρψ) − ρ ∧ ∂ψ, ∂ρ bD ¯ ∂T ¯ 2f − ¯ ∧ ψ, N ∧ T2 f dS, = − ρ ∧ ∂ψ, ∂ρ bD

(2.22)


433

¯ ¯ 2 f ) = (ρψ, ϑ ∂T ¯ 2 f ) = 0 from (2.11). Combinwhere we have used (∂(ρψ), ∂T ing (2.21) and (2.22), we have ¯ ∧ ψ, N ∧ T2 f dS = ¯ ∧ ψ, f dS. ∂ρ ∂ρ − bD

bD

Since ψ is arbitrary, we have that −N ∧ T2 f = νf

on bD.

This proves (2.17). 3. A priori estimates of Morrey-Kohn-Hörmander type In this section we prove some a priori estimates of the Morrey-Kohn-Hörmander type for smooth forms on domains with C 2 -boundary. We put a special weight function that includes the defining function. This idea was first used by Berndtsson [2], who derived the following identity (3.1) somewhat differently (see also Bonami and Charpentier [7], [8]). proposition 3.1 Let D be a domain such that the boundary bD has a C 2 –defining function ρ. Let ∞ (D), we have φ ∈ C 2 (D). For any f = |I |=p,|J |=q fI,J dzI ∧ d z¯ J ∈ C(p,q)

2 −φ 2 −φ ¯ ¯ (−ρ) ∂f e + (−ρ)|ϑφ f | e +2 ϑφ f, f ∂ρ φ +

D

=

f ∂ρ ¯ 2 e−φ 1 dS |dρ| bD

D

∂ 2φ fI,iK f¯I,j K e−φ ∂zi ∂ z¯ j D |I |=p,|K|=q−1 i,j ∂fI,J 2 −φ e + (−ρ) ∂ z¯ k D |I |=p,|J |=q k ∂ 2ρ + fI,iK f¯I,j K e−φ D ∂zi ∂ z¯ j

(−ρ)

|I |=p,|K|=q−1 i,j

and

2 −φ f ∂ρ ¯ e + ¯ 2 e−φ 1 dS (−ρ)∂f |dρ| D bD ¯ φ f, f = (−ρ)|ϑφ f |2 e−φ + 2 ρ ∂ϑ φ D

+

|I |=p,|K|=q−1 i,j

D

(−ρ)

∂ 2φ fI,iK f¯I,j K e−φ ∂zi ∂ z¯ j

(3.1)

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MICHEL AND SHAW

+

∂fI,J 2 −φ e (−ρ) ∂ z¯ k D k ∂ 2ρ fI,iK f¯I,j K e−φ . D ∂zi ∂ z¯ j

|I |=p,|J |=q

+

(3.2)

|I |=p,|K|=q−1 i,j

Proof Following Hörmander’s calculation, we have ∂f =

∂fI,J dzj ∧ dzI ∧ dzJ ∂zj I,J

and

j

ϑφ f = (−1)p−1

I,K

j

φ

δj fI,j K dzI ∧ dzK ,

(3.3)

φ

where δj u = eφ (∂/∂zj )(e−φ u). Thus, setting Lj = ∂/∂zj , we have 2 −φ ¯ (−ρ) ∂f e + (−ρ)|ϑφ f |2 e−φ D

=

I,J,L

+

j,9

D

jJ 9L − ρLj (fI,J ), L9 (fI,L ) φ

I,K j,k

φ

φ

− ρδj fI,j K , δk fI,kK

φ

(3.4)

,

jJ

jJ

where 9L = 0 unless j ∈ / J, 9 ∈ / L, and {j } ∪ J = {9} ∪ L, in which case 9L is the sign of permutation j9LJ . Rearranging the terms in (3.4), we have 2 −φ 2 ¯ (−ρ) ∂f e + (−ρ)ϑφ f e−φ D D = − ρLj fI,J , Lj fI,J φ I,J

−

j

I,K

+

j,k

I,K

− ρLk fI,j K , Lj fI,kK

j,k

φ

φ

− ρδj fI,j K , δk fI,kK

(3.5)

φ

φ

.

We now apply integration by parts to the last term in (3.4). We note that for each u, v ∈ C 2 (D), ∂ρ φ u, v − ρu, δj υ = ρLj u, υ φ + φ ∂zj φ


and

435

∂ 2φ φ φ φ δj , Lk u = δj Lk u − Lk δj u = u . ∂zj ∂zk

Thus we have ∂ρ φ φ φ φ δj u, υ − ρδj u, δk υ = ρLk δj u, υ + φ ∂zk φ ∂ρ φ φ φ = ρδj Lk u, υ + − ρ δj , Lk u, υ + δ u, υ φ φ ∂zk j φ (3.6) ∂ 2φ = − ρLk u, Lj υ φ + − ρ u, υ ∂zj ∂zk φ ∂ρ φ ∂ρ + δ u, υ − Lk u, υ . ∂zk j ∂z j φ φ Using (3.6) for each fixed I, K, we have φ φ − ρδj fI,j K , δk fI,kK j,k

∂ 2φ = fI,j K , fI,kK − ρLk fI,j K , Lj fI,kK φ + −ρ ∂zj ∂zk φ j,k j,k ∂ρ φ ∂ρ + δ fI,j K , fI,kK − Lk fI,j K , fI,kK . ∂zk j ∂zj φ φ

j,k

(3.7)

j,k

Substituting (3.7) into (3.5), we have 2 −φ ¯ (−ρ) ∂f e + (−ρ)|ϑφ f |2 e−φ D

D

∂ 2φ = − ρLj fI,J , Lj fI,J φ + fI,j K , fI,kK −ρ ∂zj ∂zk φ I,J j I,K j,k φ ∂ρ ∂ρ + fI,kK − Lk fI,j K , fI,kK δj fI,j K , ∂zk ∂zj φ φ I,K j,k I,K j,k ∂ 2φ −ρ = fI,j K , fI,kK − ρLj fI,J , Lj fI,J φ + ∂zj ∂zk φ I,J j I,K j,k 2 ∂ ρ ¯ + 2 − ϑφ f, f ∂ρ + fI,j K , fI,kK φ ∂zj zk φ I,K j,k −φ ∂ρ ∂ρ e dS. − fI,j K fI,kK |dρ| bD ∂zj ∂zk

I,K

j,k

(3.8)

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MICHEL AND SHAW

This proves (3.1). Equation (3.2) follows from ¯ ¯ ∧ ϑφ f, f = 2 ∂ρ 2 ϑφ f, f ∂ρ φ φ ¯ ¯ φ f, f = 2 ∂( ρ ∧ ϑφ f ), f φ − 2 ρ ∂ϑ φ ¯ φ f, f . = 2 ρϑφ f, ϑφ f φ − 2 ρ ∂ϑ φ This proves Proposition 3.1. proposition 3.2 Let D be a domain such that the boundary bD has a C 2 –defining function ρ. Let ∞ (D), we have φ ∈ C 2 (D). For any f = |I |=p,|J |=q fI,J dzI ∧ d z¯ J ∈ C(p,q)

¯ ∂f ¯ − ρ ∂f,

φ

+

I,J

k

φ

φ

− ρδk fI,J , δk fI,J

φ

¯ f + ρσ (ϑ, dφ)∂f, φ

2 ∂ 2 ρ ∂ ρ + fI,j K , fI,j K − fI,j K , fI,kK ∂zk ∂ z¯ k ∂zj ∂ z¯ k φ φ I,K k j ∂ 2φ ∂ 2φ + ρ fI,j K , fI,j K − ρ fI,j K , fI,kK ∂zk ∂ z¯ k ∂zj ∂ z¯ k φ φ ∂φ ∂φ φ φ + ρ fI,j K , δk fI,j K − ρ fI,j K , δk fI,kK ∂zk ∂z j φ φ e−φ ¯ ∧ f, ∂ρ ¯ ∧f ¯ = − ρϑφ f, ϑφ f φ + − ρf, 2ϑφ ∂f dS. + ∂ρ φ |dρ| bD Proof First we observe that ¯ ∂f ¯ ¯ ¯ ¯ ∧ f, ∂f ¯ − ρ ∂f, = − ∂(ρf ), ∂f + ∂ρ φ φ φ ¯ ¯ ∧ f ), f = − ρf, ϑφ ∂f + ϑφ (∂ρ φ φ e−φ ¯ ∧ f, ∂ρ ¯ ∧f + ∂ρ dS. |dρ| bD ¯ ∧ f ) = ϑφ (∂(ρf ¯ ¯ ), we have Using ϑφ (∂ρ ) − f ∂f ¯ ∧f ϑφ ∂ρ   ∂ρ fI,J dzk ∧ dzI ∧ dzJ = ϑφ  ∂zk I,J

k

(3.9)


437

∂ρ φ ∂ρ fI,J dzI ∧ d z¯ J − δ fI,J dzI ∧ d z¯ J ∂ z¯ k ∂ z¯ k k I,J k I,J k φ ∂ρ ¯ ∧ ϑφ f. + (−1)p δj fI,j K d z¯ k ∧ dzI ∧ d z¯ K − ∂ρ ∂ z¯ k

=−

φ

δk

I,K

k

j

(3.10) Using −

∂ ∂ρ φ ∂ φ φ δk fI,J = − δk fI,J ρδk fI,J + ρ ∂ z¯ k ∂ z¯ k ∂ z¯ k k

and

k

k

∂ φ ¯ φ + ϑφ ∂¯ f = − δ fI,J dzI ∧ d z¯ J , φ f = ∂ϑ ∂ z¯ k k I,J

k

we have ¯ ∧ f ,f ϑφ ∂ρ φ φ ∂ρ φ ∂ρ = − δk δj fI,j K , fI,kK fI,j K , fI,j K ∂ z¯ k ∂ z¯ k φ φ I,K k j φ φ ¯ ∧ ϑφ f, f + − ρ φ f, f + − ∂ρ ρδk fI,J , δk fI,J . φ φ I,J

k

φ

(3.11) For any smooth functions u, v, it follows that 2 ∂ ρ ∂φ ∂ρ φ ∂ρ u, v = u, v − u, v δj ∂ z¯ k ∂zj ∂ z¯ k ∂zj ∂ z¯ k φ φ φ 2 ∂ ρ ∂ 2φ = u, v + ρ u, v ∂zj ∂ z¯ k ∂zj ∂ z¯ k φ φ ∂φ ∂φ ∂u φ + ρ u, δk v + ρ ,v . ∂zj ∂zj ∂ z¯ k φ φ

(3.12)

Combining (3.11) and (3.12) with (3.9), we have φ φ ¯ ∂f ¯ + f , δ f − ρδ − ρ ∂f, I,J I,J k k φ I,J

k

φ

2 ∂ 2 ρ ∂ ρ + fI,j K , fI,j K − fI,j K , fI,kK ∂zk ∂ z¯ k ∂zj ∂ z¯ k φ φ I,K k j ∂ 2φ ∂ 2φ + ρ fI,j K , fI,j K − ρ fI,j K , fI,kK ∂zk ∂ z¯ k ∂zj ∂ z¯ k φ φ I,K

k

j

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MICHEL AND SHAW

∂φ ∂φ φ φ ρ fI,j K , δk fI,j K − ρ fI,j K , δk fI,kK ∂zk ∂zj φ φ I,K k j ∂φ ∂fI,j K ∂φ ∂fI,j K + ρ , fI,j K − ρ , fI,kK ∂zk ∂ z¯ k ∂zj ∂ z¯ k φ φ I,K k j ¯ ¯ φ f, f + − ρf, 2ϑφ ∂f ¯ ∧ ϑφ f, f + − ρ ∂ϑ = − ∂ρ φ φ φ e−φ ¯ ∧ f, ∂ρ ¯ ∧f dS + ∂ρ |dρ| bD −φ ¯ ∧ f, ∂ρ ¯ ∧ f e dS. ¯ = − ρϑφ f, ϑφ f φ + − ρf, 2ϑφ ∂f φ + ∂ρ |dρ| bD +

Since

∂φ ∂fI,j K ∂φ ∂fI,j K , fI,j K − ρ , fI,kK ρ ∂zk ∂ z¯ k ∂zj ∂ z¯ k φ φ I,K k j ∂f ∂φ ∂f ∂φ I,j K I,kK = , fI,kK − ρ , fI,kK ρ ∂zj ∂ z¯ j ∂zj ∂ z¯ k φ φ I,K k j ¯ f , = ρσ (ϑ, dφ)∂f, φ

Proposition 3.2 is proved. 4. Proof of Theorem 1 To prove Theorem 1 we first assume that D has C 2 -boundary and that ρ is a C 2 – defining function for D. We choose special weight functions in Proposition 3.1 and Proposition 3.2 to obtain the following estimates without weights. lemma 4.1 Let D be a bounded domain such that the boundary bD has a C 2 –defining function ρ. Let δ be the diameter of the domain D. For any smooth f = |I |=p,|J |=q fI,J dzI ∧ ∞ (D) such that ∂f ¯ = 0 and ∂ϑf ¯ = 0, we have d z¯ J ∈ C(p,q) ∂fI,J 2 2 2 2 δ (−ρ)|ϑf | + δ (−ρ) ∂ z¯ k D D |I |=p,|J |=q k ∂ 2ρ 2 2 (4.1) + (−ρ)|f | + δ fI,iK f¯I,j K ∂zi ∂ z¯ j D D |I |=p,|K|=q−1 i,j 2 1 f ∂ρ ¯ dS, ≤C |dρ| bD where C is a constant depending only on δ.


439

Proof We set the weight function φ = −t|z|2 where t is a positive number. Using (1.5), we have ¯ ¯ ¯ ¯ φ f = ∂¯ eφ ϑe−φ f = ∂ϑf − ∂¯ σ (ϑ, dφ)f = ∂ϑf − qtf − σ (ϑ, dφ)∂f. ∂ϑ It follows that ¯ ¯ 2 ρ ∂ϑφ f, f φ = 2 ρ ∂ϑf, f φ + 2qt (−ρ)|f |2 e−φ + E(f ), D

¯ f )φ . We can estimate E(f ) by where E(f ) = 2(−ρσ (ϑ, dφ)∂f, ∂fI,J 2 −φ C |E(f )| ≤ C (−ρ) e + (−ρ)t 2 |z|2 |fI,J |2 e−φ ∂ z¯ k D D where C is a constant depending only on q and n. Using (3.2) and the assumption, we obtain 2 −φ −φ f ∂ρ ¯ e − 2 ρ ∂ϑf, ¯ ¯ 2 e dS (−ρ)∂f f φ+ |dρ| D bD 2 e−φ f ∂ρ ¯ = dS |dρ| bD = (−ρ)|ϑφ f |2 e−φ + qt (−ρ)|f |2 e−φ D

+

∂fI,J 2 −φ e (−ρ) ∂ z¯ k D k ∂ 2ρ fI,iK f¯I,j K e−φ + E(f ). D ∂zi ∂ z¯ j

|I |=p,|J |=q

+

D

|I |=p,|K|=q−1 i,j

From (1.5) we have 2 |ϑφ f |2 = ϑf − σ (ϑ, dφ)f ≥ (1 − )|ϑf |2 − for any > 0. It follows from (4.2) that −φ f ∂ρ ¯ 2 e dS |dρ| bD ≥ (1 − )|ϑf |2 + qt +

D

(−ρ)|f |2 e−φ

|I |=p,|K|=q−1 i,j

1 2 2 2 t |z| |f |

D

∂ 2ρ fI,iK f¯I,j K e−φ ∂zi ∂ z¯ j

(4.2)

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MICHEL AND SHAW

+ (1 − C) −

|I |=p,|J |=q k

C+1

D

∂fI,J 2 −φ e (−ρ) ∂ z¯ k D

(−ρ)t 2 |z|2 |f |2 e−φ .

(4.3)

Without loss of generality we may assume that 0 ∈ D and supz∈D t|z|2 ≤ tδ 2 . Choose t > 0 such that o (4.4) t= 2 δ 2 for some 0 < o < 1/4 sufficiently small. Notice that 1 ≤ e−φ ≤ etδ = eo ≤ 2 and that t 2 |z|2 ≤ t 2 δ 2 = o t. Choosing first > 0 and then o > 0 sufficiently small, (4.1) follows from (4.3). lemma 4.2 Let D be a bounded domain such that the boundary bD has a C 2 –defining function ∞ I J ρ. For any f = |I |=p,|J |=q fI,J dz ∧ d z¯ ∈ C(p,q) (D) such that ϑf = 0 and ¯ = 0, we have ϑ ∂f ∂fI,J 2 2 2 ¯ ¯ (−ρ) + (−ρ)|f |2 δ − ρ ∂f, ∂f + δ ∂zk D D I,J k   2   ∂ 2 ρ ∂ ρ + δ2 fI,j K , fI,j K − fI,j K , fI,kK   ∂zk ∂ z¯ k ∂zj ∂ z¯ k I,K k j ¯ ∧ f, ∂ρ ¯ ∧ f 1 dS ≤C ∂ρ |dρ| bD (4.5) where C is a constant depending only on the diameter δ of the domain D. Proof We choose φ = t|z|2 for some positive constant t > 0 in Proposition 3.2. It is easy to see that ∂ 2φ ∂ 2φ ρ fI,j K , fI,j K − ρ fI,j K , fI,kK ∂zk ∂ z¯ k ∂zj ∂ z¯ k φ φ k j = (n − 1)tq (−ρ)|f |2 e−φ . D

Also, for any > 0, we have

φ φ − ρδk fI,J , δk fI,J φ

∂fI,J 2 −φ 1 e − ≥ (1 − ) (−ρ) (−ρ)t 2 |z|2 |fI,J |2 e−φ ∂zk D D


441

and 1 φ ∂φ φ 2 2 2 −φ δ fI,j K 2 e−φ . ρ ≤ f , δ f (−ρ)t |z| |f | e + (−ρ) I,j K I,j K I,j K k k ∂z k D D φ From (1.5), ¯ ¯ ¯ = 2 − ρf, ϑ ∂f + 2 − ρf, σ (ϑ, dφ)∂f . 2 − ρf, ϑφ ∂f φ φ φ Also, we have for any > 0,

− ρϑφ f, ϑφ f

φ

1 ≤ (1 + )(−ρϑf, ϑf )φ + 1 +

and

¯ − ρf, σ (ϑ, dφ)∂f

φ

¯ ∂f ¯ ≤ − ρ ∂f,

φ

1 +

D

D

(−ρ)t 2 |z|2 |f |2 e−φ

(−ρ)t 2 |z|2 |f |2 e−φ .

Using the above inequalities, it follows from Proposition 3.2 that ¯ ∂f ¯ + (1 − C) (1 − C) − ρ ∂f, φ C

I,J

k

D

∂fI,J 2 −φ e (−ρ) ∂z

k

(−ρ)t 2 |z|2 |f |2 e−φ + (n − 1)tq (−ρ)|f |2 e−φ D D   2   2 ∂ ∂ ρ ρ + fI,j K , fI,j K − fI,j K , fI,kK   ∂zk ∂ z¯ k ∂zj ∂ z¯ k I,K k j −φ + ¯ ¯ ∧ f, ∂ρ ¯ ∧ f e dS. ≤ (1 + ) − ρϑf, ϑf φ + 2 − ρf, ϑ ∂f ∂ρ φ |dρ| bD (4.6) −

¯ = 0, choosing > 0 sufficiently small and then choosing Since ϑf = ϑ ∂f t > 0 satisfying (4.4) with sufficiently small o , we can absorb all the terms with t 2 in (4.6). Equation (4.5) follows easily from (4.6), and the lemma is proved. lemma 4.3 Let D be a domain with C ∞ -boundary such that D has a C ∞ –plurisubharmonic 1/2 defining function ρ. For any α ∈ H(p,q−1) (bD), where 0 ≤ p ≤ n, 1 ≤ q ≤ n, we have the following estimates: 2 (−ρ)T1 α ≤ C |α|2 dS. (4.7) D

bD

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MICHEL AND SHAW 1/2

Also, for any α ∈ H(p,q+1) (bD), where 0 ≤ p ≤ n, 0 ≤ q ≤ n − 1,

(−ρ)|T2 α| ≤ C 2

D

bD

|α|2 dS,

(4.8)

where C is a constant depending only on the Lipschitz constant of D and the diameter of D. Proof ∞ (bD) is dense in L2(p,q−1) (bD), it suffices to prove (4.7) and (4.8) for Since C(p,q−1) ∞ ∞ (D). (bD). Let f = T1 α. We first prove (4.7), assuming that f ∈ C(p,q) α ∈ C(p,q−1) Since ρ is plurisubharmonic in D, we have ∂ 2ρ fI,iK f¯I,j K ≥ 0 ∂zi ∂ z¯ j

in D.

i,j

Thus, from (4.1) and (2.10) in Lemma 2.2, we have δ

2

(−ρ)|ϑf | + δ 2

D

2

|I |=p,|J |=q k

≤ Cδ

bD

∂fI,J 2 (−ρ) + (−ρ)|f |2 ∂ z¯ k D D

|f N |2 |dρ| dS,

(4.9)

where Cδ depends only on the diameter of D. Using (2.16), we have T1 αN = τ α Thus we have (−ρ)|ϑT1 α|2 + (−ρ)|T1 α|2 ≤ Cδ D

D

on bD.

bD

|τ α|2 |dρ| dS ≤ C

bD

|α|2 dS, (4.10)

where C depends only on the Lipschitz constant and the diameter of D. This proves ∞ (D). The proof for general f = T α follows from approxi(4.7) when T1 α ∈ C(p,q) 1 mating T1 α by forms smooth up to the boundary. Since T1 α has harmonic coefficients ∞ (D). If D is star shaped and 0 ∈ D, we can approximate f = T α by in D, f ∈ C(p,q) 1 ∞ dilation f , where f = f (z/(1 + )) for > 0. It is easy to see that f ∈ C(p,q) (D), ¯ = 0, and ∂ϑf ¯ = 0. Also, we have f → f in L2 (D) and ∂f f N −→ f N

in L2 (bD).

(4.11)

For general Lipschitz domains we use a partition of unity {ηi } corresponding to a finite open covering {Ui } of D such that each D∩Ui is star shaped. On each Ui ∩bD = ∅, we


443

use a dilation fi similar to the one described before and set f = ηi fi . It is easy to ∞ (D) with f → f , ϑf → ϑf , and ∂f ¯ → 0 in L2 (D, (−ρ) dV ), see that f ∈ C(p,q) where (−ρ) dV is the Lebesgue measure with weight (−ρ). Also, (4.11) holds. Since   ∂fI,J ∂ϑf ¯ ≤ C |ϑf | +  in D, ∂ z¯ + |f | k |I |=p,|J |=q k

¯ → 0 in L2 (D, (−ρ) dV ) from the using (3.2) applied to each f , we have ∂ϑf dominated convergence theorem. Thus (4.10) holds for any T1 α. This proves (4.7). To prove (4.8) we note that if ρ is plurisubharmonic in D, for every I, K, 2 ∂ 2 ρ ∂ ρ fI,j K , fI,j K − fI,j K , fI,kK ≥ 0 in D. ∂zk ∂ z¯ k ∂zj ∂ z¯ k k

j

This follows from diagonalizing the matrix (∂ 2 ρ/(∂zk ∂ z¯ k )) at each point z ∈ D. Let ¯ = 0. If f is smooth f = T2 α. From (2.11) in Lemma 2.2, we have ϑf = 0 and ϑ ∂f up to the boundary and δ is sufficiently small, we have from Lemma 4.2,

¯ ∂f ¯ + − ρ ∂f, ≤ Cδ

bD

I,J

k

∂fI,J 2 2 (−ρ) + (−ρ)|f | ∂z k D D

¯ ∧ f, ∂ρ ¯ ∧ f 1 dS, ∂ρ |dρ|

(4.12)

where Cδ is a constant depending only on the diameter. Using (2.17) in Lemma 2.4, we have N ∧ T2 α = να on bD, (4.13) and D

¯ 2 α 2 + (−ρ)∂T

D

(−ρ)|T2 α|2 ≤ Cd

bD

|να|2

1 dS ≤ C |dρ|

bD

|α|2 dS, (4.14)

where C depends only on the Lipschitz constant and the diameter of D. The general case also follows from the same dilation and approximating arguments as before, and the lemma is proved. Proof of Theorem 1 To finish the proof of Theorem 1, we first approximate D by a sequence of subdomains Dµ such that each Dµ is pseudoconvex and with a C ∞ –plurisubharmonic defining function ρµ on Dµ . This can be done easily by regularizing ρ + |z|2 for µ µ µ µ sufficiently small and letting → 0. We use T1 , T2 , R1 , R2 , and N µ to denote the

444

MICHEL AND SHAW µ

µ

corresponding operators on each Dµ . We apply Lemma 4.3 to T1 and T2 to obtain 1/2 for any α ∈ H(p,q−1) (bDµ ), where 0 ≤ p ≤ n, 1 ≤ q ≤ n, Dµ

µ 2 (−ρµ )T1 α ≤ C

bDµ

|α|2 dSµ ,

(4.15) 1/2

where C can be chosen independent of µ. Also, for any α ∈ H(p,q+1) (bDµ ), where 0 ≤ p ≤ n, 0 ≤ q ≤ n − 1, µ 2 (−ρµ ) T2 α ≤ C |α|2 dSµ . (4.16) Dµ

bDµ

−1/2

µ

Using Proposition 1.4, we have that T1 is bounded from L2(p,q−1)(bDµ ) to H(p,q)(Dµ ) −1/2

µ

and that T2 is bounded from L2(p,q+1) (bDµ ) to H(p,q) (Dµ ). Also, the bounds depend only on the Lipschitz constant and the diameter of the domain. Thus we have from 1/2 duality, for any f ∈ H(p,q) (Dµ ), µ

and

R1 f L2

(p,q−1) (bDµ )

≤ Cf H 1/2

µ R f

L2(p,q+1) (bDµ )

≤ Cf H 1/2

1

(4.17)

(p,q) (Dµ )

(p,q) (Dµ )

,

(4.18)

where C is a constant independent of µ. Using the trace theorem for elliptic equations (see Jerison and Kenig [20], Straube [30], and the remarks after Proposition 1.3), we 1/2

have for any f ∈ H(p,q) (Dµ ), ∗ µ ∂¯ N f and

µ ∂N ¯ f

1/2

H(p,q−1) (Dµ )

1/2

H(p,q+1) (Dµ )

≤ Cf H 1/2

(p,q) (Dµ )

≤ Cf H 1/2

(p,q) (Dµ )

,

(4.19)

(4.20)

where C is a constant independent of µ. Passing to the limit (cf. Michel and Shaw [25]), one obtains ∗ ∂¯ Nf 1/2 ≤ Cf H 1/2 (D) (4.21) H (D) (p,q−1)

and

∂Nf ¯

1/2

H(p,q+1) (D)

(p,q)

≤ Cf H 1/2

(p,q) (D)

.

Using Proposition 1.2(iv), we can write ∗ ¯ ¯ . ∂¯ N) + ∂¯ ∗ N ∂N N = ∂¯ ∂¯ ∗ + ∂¯ ∗ ∂¯ N 2 = ∂N

(4.22)


445

It follows from (4.21) and (4.22) that Nf H 1/2

(p,q) (D)

≤ Cf H 1/2

(p,q) (D)

.

Thus Theorem 1 is proved. Acknowledgments. We would like to thank professors Bo Berndtsson and Phillipe Charpentier for pointing out some mistakes in our original manuscript and calling our attention to a related paper by them (see Berndtsson and Charpentier [3]). Dr. Sophia Vassiliadou has also made helpful comments that have been incorporated into the paper. We would especially like to thank Professor Carlos Kenig for explaining his paper with David Jerison [20], which was used in the proof of Propositions 1.3 and 1.4.

References [1]

D. BARRETT, Behavior of the Bergman projection on the Diederich-Fornaess worm,

[2]

B. BERNDTSSON, “∂¯b and Carleson type inequalities” in Complex Analysis (College

Acta Math. 168 (1992), 1–10. MR 93c:32033

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Park, Md., 1985/86), II, Lecture Notes in Math. 1276, Springer, New York, 1987, 42–54. MR 89b:32004 B. BERNDTSSON and P. CHARPENTIER, A Sobolev mapping property of the Bergman kernel, Math. Z. 235 (2000), 1–10. MR CMP 1785069 H. P. BOAS and E. J. STRAUBE, Equivalence of regularity for the Bergman projection ¯ and the ∂-Neumann operator, Manuscripta Math. 67 (1990), 25–33. MR 90k:32057 , Sobolev estimates for the ∂-Neumann operator on domains in Cn admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), 81–88. MR 92b:32027 , “Global regularity of the ∂-Neumann problem: A survey of the L2 -Sobolev theory” in Several Complex Variables (Berkeley, 1995–1996), ed. M. Schneider and Y.-T. Siu, Math. Sci. Res. Inst. Publ. 37, Cambridge Univ. Press, Cambridge, 1999, 79–111. MR CMP 1748601 A. BONAMI and P. CHARPENTIER, Une estimation Sobolev 1/2 pour le projecteur de Bergman, C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), 173–176. MR 90b:32048 ¯ , Boundary values for the canonical solution to ∂-equation and W 1/2 estimates, preprint, Bordeaux, 1990. ¯ = f ” in P. CHARPENTIER, “Sur les valeurs au bord de solutions de l’équation ∂u Analyse complexe multivariable: Récents développements (Guadeloupe,1988), Sem. Conf. 5, EditEl, Rende, Italy, 1991, 51–81. MR 94f:32029 ¯ M. CHRIST, Global C ∞ irregularity of the ∂-Neumann problem for worm domains, J. Amer. Math. Soc. 9 (1996), 1171–1185. MR 96m:32014

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MICHEL AND SHAW

B. E. J. DAHLBERG, Weighted norm inequalities for the Lusin area integral and the

nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), 297–314. MR 82f:31003 K. DIEDERICH and J. E. FORNAESS, Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), 129–141. MR 55:10728 , Pseudoconvex domains: An example with nontrivial Nebenhülle, Math. Ann. 225 (1977), 275–292. MR 55:3320 L. E. EVANS and R. F. GARIEPY, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC, Boca Raton, 1992. MR 93f:28001 P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. MR 86m:35044 G. M. HENKIN and A. IORDAN, Compactness of the Neumann operator for hyperconvex domains with non-smooth B-regular boundary, Math. Ann. 307 (1997), 151–168. MR 98a:32018 G. M. HENKIN, A. IORDAN and J. J. KOHN, Estimations sous-elliptiques pour le ¯ problème ∂-Neumann dans un domaine strictement pseudoconvexe a` frontière lisse par morceaux, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 17–22. MR 97g:32014 ¨ L. HORMANDER , L2 estimates and existence theorems for the ∂¯ operator, Acta Math. 113 (1965), 89–152. MR 31:3691 , An Introduction to Complex Analysis in Several Variables, 3d ed., North-Holland Math. Library 7, North-Holland, Amsterdam, 1990. MR 91a:32001 D. JERISON and C. E. KENIG, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161–219. MR 96b:35042 J. J. KOHN, Harmonic integrals on strongly pseudoconvex manifolds, I, Ann. of Math. (2) 78 (1963), 112–148. MR 27:2999 ¯ , Subellipticity of the ∂-Neumann problem on pseudo-convex domains: Sufficient conditions, Acta Math. 142 (1979), 79–122. MR 80d:32020 , “Quantitative estimates for global regularity” in Analysis and Geometry in Several Complex Variables (Katata, Japan, 1997), Birkhäuser, Boston, 1999, 97–128. MR 2000f:32046 J.-L. LIONS and E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications, I, Grundlehren Math. Wiss. 181, Springer, New York, 1972. MR 50:2670 ¯ J. MICHEL and M.-C. SHAW, Subelliptic estimates for the ∂-Neumann operator on piecewise smooth strictly pseudoconvex domains, Duke Math. J. 93 (1998), 115–128. MR 99b:32019 R. M. RANGE, A remark on bounded strictly plurisubharmonic exhaustion functions, Proc. Amer. Math. Soc. 81 (1981), 220–222. MR 82e:32030 ¯ R. M. RANGE and Y. T. SIU, Uniform estimates for the ∂-equation on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann. 206 (1973), 325–354. MR 49:3214


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M.-C. SHAW, Local existence theorems with estimates for ∂ b on weakly pseudo-convex

[29]

E. M. STEIN, Singular Integrals and Differentiability Properties of Functions,

[30]

¯ E. J. STRAUBE, Plurisubharmonic functions and subellipticity of the ∂-Neumann

[31]

problem on nonsmooth domains, Math. Res. Lett. 4 (1997), 459–467. MR 98m:32024 ¯ S. VASSILIADOU, L2 existence and subelliptic estimates for the ∂-Neumann operator

CR manifolds, Math. Ann. 294 (1992), 677–700. MR 94b:32026

Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970. MR 44:7280

on certain piecewise smooth domains in Cn , to appear in Complex Variables Theory Appl.

Michel Université du Littoral, Centre Universitaire de la Mi-Voix, F-62228 Calais, France; [email protected] Shaw Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, USA; [email protected]

SEMICLASSICAL LIMITS FOR THE HYPERBOLIC PLANE SCOTT A. WOLPERT

Abstract We study the concentration properties of high-energy eigenfunctions for the LaplaceBeltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions. At the center of our investigation is the microlocal lift of an eigenfunction to SL(2; R) introduced by S. Zelditch. The microlocal lift is based on S. Helgason’s Fourier transform and has a straightforward description in terms of the Lie algebra sl(2; R). We begin with an elementary demonstration of Zelditch’s exact differential equation for the microlocal lift. Further, for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity, we show that the limit of microlocal lifts is a geodesic-flow-invariant positive measure. Our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the Macdonald-Bessel functions. We find that with a scaling of auxiliary parameters the corresponding high-energy limit converges to the positive Dirac measure for the lift to SL(2; R) of a single geodesic on the upper half-plane. In particular, at high energy the microlocal lift of a Macdonald-Bessel function is concentrated along the lift of a single geodesic. We further find that the convergence is uniform in parameters, and thus our considerations can be applied to study automorphic eigenfunctions, in particular, the Maass cusp forms for certain cofinite subgroups of SL(2; R). A formula is presented for the microlocal lift of a cusp form as an integral of a family of Dirac measures and a measure given directly by sums of products of the classical Fourier coefficients of the cusp form. The formula establishes that questions on “quantum chaos” for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients. In particular, we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums. We extend the considerations to the case of the nonholomorphic Eisenstein series for the modular group. We find that a result of W. Luo and P. Sarnak is equivalent to a limit-sum formula involving the DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 3, Received 14 January 2000. Revision received 25 August 2000. 2000 Mathematics Subject Classification. Primary 11F30, 11F72, 58G15, 58G25. Author’s work supported in part by National Science Foundation grant numbers DMS-9504176 and DMS9800701.

449

450

SCOTT A. WOLPERT

Riemann zeta-function and the elementary divisor function. The formula is suggestive of a formula of S. Ramanujan. 1. Introduction The correspondence principle of quantum mechanics provides that high-energy eigenfunctions of the Laplace-Beltrami operator concentrate along geodesics. The nature of the concentration for automorphic eigenfunctions and ergodic geodesic flow is of particular interest (see [2], [3], [4], [5], [6], [9], [18], [19], [20], [28], [33], [38], [39], [40], [41], [50], [51], [52], [53], [54]). For a cofinite, noncocompact subgroup of SL(2; R), a cuspidal automorphic eigenfunction for the hyperbolic Laplacian D has a Fourier series expansion an (y sinh πr)1/2 Kir (2π|n|y)e2π inx (1.1) ϕ(z) = n=0

for z = x + iy, y > 0, the variable for the upper half-plane H, eigenvalue −λ = −(1/4 + r 2 ) < −1/4, and Kir the Macdonald-Bessel function (see [43], [46]). Also associated to an eigenfunction ϕ are the coefficient sums 1/2 ˆ = πr −1 an e2π inxˆ ; Sϕ (t, x) 1≤|n|≤rt (2π )−1

the sums are of independent interest (see [8], [10], [14], [30], [42]). A theme of our investigations is that at high energy the concentration measure on the space of geodesics is approximately given in terms of |Sϕ |2 . Our results have applications for general cofinite, noncocompact subgroups, as well as for congruence subgroups of SL(2; Z). The first application concerns a congruence subgroup and unit-norm eigenfunctions; the spectral average of the sum ˆ 2 weak* converges in xˆ to 8t (Area(\H))−1 . In a second applisquares |Sϕ (t, x)| cation the results of Luo and Sarnak [33] and of D. Jakobson [28] for the uniform distribution for the limit of the modular Eisenstein series are interpreted as a limit-sum formula for the elementary summatory function. In a third application we present a lower bound for square coefficient sums and note that a mapping from eigenfunctions to coefficient sums is a uniform quasi-isometry. At the center of our investigation is the microlocal lift of an eigenfunction introduced by Zelditch [49], [50], [51], [52], [53]. Zelditch first observed that a DOcalculus can be based on Helgason’s Fourier transform in [49, Secs. 1, 2] and [51, Secs. 1, 2]. The basic construction is the microlocal lift, a finite-order distribution that encodes the oscillation for a pair of eigenfunctions. For u, v functions on the upper half-plane H, each satisfying the differential equation Df = −(1/4 + r 2 )f , we write

SEMICLASSICAL LIMITS FOR THE HYPERBOLIC PLANE

451

u, ˜ v˜ for the standard lifts to SL(2; R) and consider the sequence u0 = u, ˜ ˜ v0 = v, (2ir + 2m + 1)v2m+2 = E + v2m , (2ir − 2m + 1)v2m−2 = E − v2m for E + , respectively, E − , the SL(2; R) raising, respectively, lowering, operator; v2m is in the weight-2m representation for the compact subgroup of SL(2; R) (see [31]). The microlocal lift for the pair is defined by Q(u, v) = u0 m v2m . For σ ∈ C ∞ (SL(2; R) × R), a complete symbol for a DO properly supported on SL(2; R) (σ (A, τ ) is asymptotically a sum of homogeneous terms in the frequency τ with bounded left-invariant derivatives in A), the associated matrix element is σr Q(u, v) d V 2π Op(σ )v, u = SL(2;R)

for σr the symbol evaluated at τ = r and d V Haar measure (see [49], [51]). We generalize a result of Zelditch [51, Prop. 2.1] and show that the microlocal lift satisfies a partial differential equation; the equation is basic to our approach. A second microlocal lift is also considered. The Fejér sum M 2 −1 u2m QM (u) = (2M + 1) m=−M

provides a positive measure on SL(2; R); the assignment u to QM (u) is equivariant for the left action on SL(2; R). An integration by parts argument provides a bound for the difference of Q(u, u) and QM (u) for M large of order O(r −1 ) given uniform bounds for u2m . The Fejér sum construction provides an alternative to introducing Friedrichs symmetrization (see [9], [49]). Our initial interest is the microlocal lift of the Macdonald-Bessel functions. The Macdonald-Bessel functions were introduced one hundred years ago in a paper presented to the London Mathematical Society by H. M. Macdonald [34]. That paper includes a formula for the product of two Macdonald-Bessel functions in terms of the integral of a third Bessel function. We develop formulas for the microlocal lift from such an identity, ∞ 1/2 J0 y 2β cosh τ − 1 − β 2 sin rτ dτ 2Kir (βy)Kir (y) = π csch πr log β

(see [32], [34]). To present our main result, we first describe a family of measures on SL(2; R). The square root (the double-cover) of the unit cotangent bundle of H is

452

SCOTT A. WOLPERT

equivalent to SL(2; R). A geodesic has two unit cotangent fields and four square-root unit cotangent fields. The geodesic-indicator measure ' ( on SL(2; R) is the sum αβ

(

over the four lifts of the geodesic αβ of the lifted infinitesimal arc-length element. ( = We further write ' γ ∈∞ ' ( for the sum over ∞ , the discrete group ∞ (αβ)

γ (αβ)

of integer translations. We consider for t = 2πnr −1 the microlocal lift of K (z, t) = (ry sinh πr)1/2 Kir (2π|n|y)e2π inx ,

and we define the distribution Qsymm (t) =

K (z, t + k't)K (z, t)4m

k,m∈Z (

for 't = 2π r −1 . We show in Theorem 4.9 that, for αβ the geodesic on H with Euclidean center, the origin and radius |t|−1 , and d V Haar measure, in the sense of tempered distributions, Qsymm (t) d V is close to

π2 ( ' 8 ∞ (αβ)

uniformly for r large and |t| restricted to a compact subset of R+ . At high energy the microlocal lift of a Macdonald-Bessel function is concentrated along a single geodesic. Accordingly, at high energy the behavior of the microlocal lift of a sum of Macdonald-Bessel functions is explicitly a matter of the space of geodesics and sums of coefficients. We introduce in Sections 3.1–3.4 a coefficient summation scheme for studying quantities quadratic in the eigenfunction. The scheme provides a positive measure. In particular, for (x, ˆ t) ∈ R × R+ , define the distribution 2 ,ϕ,N = dt FN ∗ Sϕ (t, x) ˆ for dt denoting the Lebesgue-Stieljes derivative in t and for convolution in xˆ with the Fejér kernel FN . We use a slight improvement of the J. Deshouillers–H. Iwaniec coefficient sum bound in [11] and show for a unit-norm automorphic eigenfunction that ,ϕ,N is a uniformly bounded tempered distribution. In fact, for {ϕj } a sequence of unit-norm automorphic eigenfunctions, the sequences {Q(ϕj , ϕj )} and {,ϕj ,N } are relatively compact. We consider a weak∗ convergent sequence and write Qlimit = ˆ t) connoting the geodesic limj Q(ϕj , ϕj ) d V and µlimit = limN limj ,ϕj ,N . For (x, −1 on H with Euclidean center xˆ and radius t , the distributions ,ϕ,N and µlimit are given on G the space of nonvertical geodesics on H. The first application of our overall considerations is presented in Theorem 4.11: π ' ( µlimit (1.2) Qlimit = 8 G αβ

453


in the sense of tempered distributions on ∞ \ SL(2; R). The integral representation provides basic information, in particular, that Qlimit is a positive geodesic-flowG the space of invariant measure and that µlimit extends to a -invariant measure on geodesics on H. Vertical geodesics on H are found to be null for Qlimit . Recall that G is ergodic relative to the SL(2; R)-invariant measure. The the action of on invariance of µlimit is a significant hypothesis for the limits of the coefficient sums Sϕ . The -action has compact fundamental sets, and thus an invariant measure is compactly determined. We further find that the equality Qlimit = d V is equivalent to ˆ 2 to 4π −1 t. For the modular group the convergence the convergence of |Sϕj (t, x)| of Q(ϕj , ϕj ) to unity is found to agree with the residue formula at s = 1 of the

Rankin-Selberg convolution L-function Lϕ (s) = SL(2;Z)\H ϕ 2 (t)E (z; s) dA; E (z; s), the modular Eisenstein series. We show in Sections 5.1–5.4 that the above considerations can be extended to include the modular Eisenstein series. In particular, from the Maass-Selberg relation the microlocal lift 1 1 E z; + ir QE (r) = E z; + ir 2 2 2m m has magnitude comparable to log |r|. For limits QE ,limit = lim(log |rj |)−1 QE (rj ) d V j

and

2 ˆ , µE ,limit = lim lim(log |rj |)−1 dt FN ∗ SE (t, x) N

j

we find symm

QE ,limit =

π 8

G

' ( µE ,limit αβ

symm

in the sense of tempered distributions on ∞ \ SL(2; R) for QE ,limit the restriction 0 1 of the distribution to functions right-invariant by −1 0 . The limit of the microlocal lift QE (r) is analyzed in the joint work of Luo and Sarnak [33] and the work of Jakobson [28]. The authors find that (log |r|)−1 QE (r) weak∗ converges to 48π −1 d V relative to Cc (SL(2; Z)\ SL(2; R)). Their approach uses bounds for the Riemann zetafunction (see [44]), T. Meurman’s bounds for the L-function of a cusp form in [35], and the H. Petersson–N. Kuznetsov trace formula in [30], as well as the work of Iwaniec [25] and of J. Hoffstein and P. Lockhart [23]. We find that (log |r|)−1 QE (r) converging to 48π −1 d V is equivalent to 2 −1 2 −ir inν |ζ (1 + 2ir)| |r| log |r| σ2ir (n)n e converging to 48π −2 t 1≤n≤rt

454

SCOTT A. WOLPERT

weak∗ in ν for each positive t for the summatory function σα (n) = d|n d α and r tending to infinity. The convergence is suggestive of the Ramanujan formula (see [16]) x d(n)2 ∼ 2 (log x)3 π 1≤n≤x

and the Ingham formula in [24] for the divisor function (the additive divisor problem); the divisor-sum formulas provide that S(t, x) ˆ = 1≤n≤t d(n)e2π inxˆ satisfies |S(t, x)| ˆ 2∼

t (log t)3 π2

as a positive measure in x. ˆ The convergence is also suggestive of the residue formula at s = 1 for the Ramanujan identity ∞ |σ2ir (n)|2 n=1

ns

=

ζ 2 (s)ζ (s + 2ir)ζ (s − 2ir) ζ (2s)

(see [37]). The quantum unique ergodicity conjecture postulates that in general Qlimit is a constant multiple of d V (see [38], [39]; also see [5, Chap. 8, Sec. 3] and the discussion of the effect of the automorphic condition on the behavior of expansion coefficients). The work of Luo and Sarnak and of Jakobson establishes the conjecture for the modular Eisenstein series. In [48] we used a stationary-phase analysis of an SL(2; R)-translate of K (z, t), z ∈ H, to study the coefficient sums Sϕ (see [48, Th. 5.4]). In [1] A. Alvarez-Parrilla extended the stationary-phase analysis to include an SL(2; R)-translate of (Im z)s , z ∈ H, and studied the Eisenstein coefficient sums SE . Z. Rudnick and Sarnak [38] established the result that for an arithmetic surface a Hecke-basis semiclassical limit cannot have projection to H with nontrivial singular support contained in a finite union of closed geodesics. The renormalization for formula (1.2) is considered in Section 5.4. In the possible absence of quantum unique ergodicity, all L2 -mass could escape into the cusps for a semiclassical limit; Qlimit and µlimit could possibly be trivial. We normalize eigenfunctions to unit L2 -mass on a compact set and reconsider the semiclassical limit. The L2 -norms can now be tending to infinity, and the above considerations are not sufficient. In particular, it is an open question if the microlocal lifts form a uniformly bounded family of distributions. We find, though, in Proposition 5.11 that the L2 norms of renormalized eigenfunctions are bounded by log λ. In Proposition 5.12 we use Fejér summation to construct the microlocal lift and to establish the existence of a nontrivial semiclassical limit with the expected properties. In Theorem 5.14 we show that the resulting limit of square coefficient sums is the index-zero Fourier-Stieljes coefficient of a -invariant measure on the space of geodesics. The -invariance


455

has consequences for coefficient sums. The group action on the space of geodesics has a compact fundamental set as noted in Proposition 5.9. A positive lower bound for square coefficient sums is a consequence. Furthermore, for large eigenvalues the mapping from eigenfunctions to scaled index interval linear coefficient sums twisted by an additive character is a uniform quasi-isometry relative to the L2 -norms for a suitable compact set and the unit circle (parameterizing the character). In particular, the mapping from eigenfunctions to twisted linear coefficient sums is an injection. We begin our analysis in Sections 2.1–2.4 by considering the geodesic-indicator measures on H and the Radon transform; the adjoint is the weight-zero component of the integral transform (1.2). From a sequence of integral identities we show in Theorem 2.4 that suitable products of Macdonald-Bessel functions converge with rate r −1 to the Fourier-Stieljes coefficients of the geodesic-indicator. The focus of Sections 3.1–3.4 is the analysis of sums of Macdonald-Bessel functions and the interchange of summation and spectral limits. We begin by introducing measures on G constructed from the Fourier coefficients of an automorphic eigenfunction. In Theorem 3.5 we establish (1.2) in effect for translation-invariant test functions on H. In Theorem 3.6 we give a new bound for coefficient sums and establish (1.2) in effect for general test functions on H. The sum bound provides for ϕ2 = 1 that r −1 1≤n≤rt |an |2 is o(1) for t small, uniformly in r. The bound plays an essential role in our arguments and apparently is the only available short-range sum bound sharp in the r-aspect (see [26]). In Section 3.4 we show that the adjoint Radon transform is injective for translation-invariant measures. (The Radon transform is not surjective; translation invariance provides a special situation.) The analysis of Sections 4.1–4.5 concerns the microlocal lift on SL(2; R) and the necessary bounds for considering sums of microlocalized Macdonald-Bessel functions. We begin with bounds for the raisings and lowerings of the Macdonald-Bessel functions. Then we use Zelditch’s equation and an induction scheme on the SL(2; R)-weight to analyze the microlocal lift. Formula (1.2) is established in complete generality in Section 4.4. Initial consequences, including the connections to limits of coefficient sums Sϕ , are presented in Sections 4.4 and 4.5. Sections 5.1–5.4 are devoted to further applications. The analysis is extended to include the modular Eisenstein series; the analog of the relation (1.2) is presented in Theorem 5.6. Renormalization of the microlocal lift is considered in Section 5.4.

2. Products of Macdonald-Bessel functions and the Radon transform 2.1 The hyperbolic Radon transform is defined in terms of integration over geodesics in

456

SCOTT A. WOLPERT

the upper half-plane. We begin in Section 2.2 by introducing the Radon transform and the geodesic-indicator measure. The Fourier-Stieljes coefficients of the measure are presented in Proposition 2.1. In Section 2.3 we prescribe a test function and consider a sequence of integral identities to relate the product of the MacdonaldBessel functions to the Fourier-Stieljes coefficients of the geodesic-indicator. The relation complete with a remainder is given in Theorem 2.4. An alternate formula is presented in Section 2.4 for the square of the Macdonald-Bessel function. The bounds for the remainders are necessary in the consideration of sums in Sections 3.1–5.4. 2.2 G of complete geodesics on H the upper half-plane. G is We consider the space R = R ∪ {∞}, G {{α, β} | naturally parameterized by considering end points on R, α = β}. The Radon transform R : Cc (H) → Cc ( G) for compactly α, β ∈ supported functions is defined by R (f ) = f ds for f ∈ Cc (H), γ ∈ G, γ

G has for ds the hyperbolic arc-length element (see [22]). The space of geodesics −2 an SL(2; R)-invariant area element ω = (α − β) |dα ∧ dβ|. We wish to study the adjoint of the Radon transform. From the Riesz representation theorem, the spaces M (H) and M ( G) of regular Borel measures are the corresponding duals of the G). The pairing spaces of compactly supported continuous functions Cc (H) and Cc (

G) and ν ∈ M ( G) is (g, ν) = gν. The adjoint of the Radon transform for g ∈ Cc ( G A : M ( G) → M (H) is prescribed by f A (ν) = f (z)ν({α, β})δ ( (z) (2.1) αβ

G×H

H

for f ∈ Cc (H), z ∈ H, and δ ( the measure for arc-length integration along the αβ

(

geodesic αβ. In particular, the geodesic-indicator δ ( , a section of the trivial bundle αβ

G × M (H) → G, is the kernel for the integral representation of the adjoint A . G, the subspace of nonvertical geodesics, We study the kernel δ ( on G ⊂ αβ

G {{α, β} | α, β ∈ R, α = β}. First, we introduce alternate coordinates for G. A nonvertical geodesic is a Euclidean circle orthogonal to R. Set t = 2|β − α|−1 , the

reciprocal radius, and xˆ = (α + β)/2, the abscissa of the center. We have in terms of the (x, ˆ t)-coordinates that 2ω = |dt ∧ d x|. ˆ Define a function on R+ × R+ by

0 for ty ≥ 1, (2.2) S(t, y) = 1/2 −1 2 2 for ty < 1. y 1−t y

457


We are ready to consider the Fourier-Stieljes expansion for the kernel δ ( , a quantity αβ

on G valued in positive measures on H. For ∞ , the group of integer translations, denote the sum δ ( γ ∈∞

by δ

(

∞ (αβ)

γ (αβ)

.

proposition 2.1 Notation is as above. For z = x + iy ∈ H, (x, ˆ t)-coordinates for G, 1/2 ˆ ( = 2 e2π ik(x−x) S(t, y) cos 2πk t −2 − y 2 δ dx dy. ∞ (αβ)

k

Proof The matter is to describe the measure δ ( (z) in terms of the prescribed coordinates. For (

αβ

a point (x, y) on the geodesic αβ with coordinate (x, ˆ t), (x − x) ˆ 2 + y 2 = t −2 and the radius from the center (x, ˆ 0) to the point (x, y) has angle θ0 to the positive x-axis with ˆ − t 2 y 2 )1/2 . Thus in terms of the parameter y on H the element cos θ0 = sgn(x − x)(1 ( of hyperbolic arc-length along αβ is simply ds = | sec θ0 |y −1 |dy| = S(t, y)|dy|. Furthermore, the indicator measure of the geodesic is simply δ(x − xˆ + (t −2 − y 2 )1/2 ) + δ(x − xˆ − (t −2 − y 2 )1/2 ) in terms of the 1-dimensional Dirac delta. Thus from the Fourier-Stieljes expansion δ(x − a) = k e2π ik(x−a) dx we find that 1/2 ˆ δ ( (γ z) = 2e2π ik(x−x) S(t, y) cos 2πk t −2 − y 2 dx dy. γ ∈∞

αβ

k

The proof is complete. 2.3 We now show that certain products of Macdonald-Bessel functions converge to the ( . Throughout our considerations Fourier-Stieljes coefficients of the kernel δ ∞ (αβ) 2 2 −au for a 2au e

> 0. We write Kir (y) for the we use the test function h(u) = Macdonald-Bessel function (see [32, Sec. 5.7]) and start with an integral identity. lemma 2.2 Notation is as above. For a > 0, r > 0, α > 0, and β ≥ 1, ∞ Kir (βy)Kir (y)h(αy)y −1 dy 0 ∞ −1/2 dP P2 −Y 2 /(4a) 4 + P2 dY, e sin r arccosh 1 + = π csch π r 2 dY 0 where P 2 = α 2 Y 2 + (β − 1)2 β −1 .

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SCOTT A. WOLPERT

Proof We start with the standard formula (see [32, Chap. 5, Prob. 7]) ∞ π Kir (βy)Kir (y) = csch πr J0 yQ1/2 sin rτ dτ 2 log β for Q = 2β cosh τ − 1 − β 2 and J0 the order zero Bessel function. Next, multiply by h(αy)y −1 and integrate to obtain ∞ Kir (βy)Kir (y)h(αy)y −1 dy 0 ∞ ∞ (2.3) π = csch π r J0 yQ1/2 h(αy)y −1 dy sin rτ dτ ; 2 log β 0 the integrals are absolutely convergent since Q = βeτ + O(1) for τ large and J0 (x) is O(x −1/2 ) for x large positive. Now from the tables of Hankel transforms (see [13, p. 29, (10)]) we have that ∞ 2 J0 yQ1/2 h(αy)y −1 dy = e−Q/(4aα ) , 0

and thus the integral on the right-hand side of (2.3) is ∞ π 2 csch πr e−Q/(4aα ) sin rτ dτ. 2 log β

(2.4)

We next set α 2 Y 2 = Q and P 2 = (α 2 Y 2 + (β − 1)2 )β −1 and observe that 1+

P2 α2 Y 2 + β 2 + 1 Q + β2 + 1 = = = cosh τ. 2 2β 2β

We accordingly have that τ = arccosh(1 + P 2 /2) and that d arccosh(1 + u2 /2) = (4 + u2 )−1/2 2 du. The desired integral results from (2.4) after a change of variables since, for β ≥ 1, Q(τ ) is monotone on [log β, ∞). The proof is complete. The next matter is a simple identity for trigonometric integrals. lemma 2.3 For C positive,

1 0

2 1/2 ∞ 2 1/2 cos B 1 − Y 2 2 −CY 2 −X 2 /(4C) sin X + B dY = e 1/2 2CY e 1/2 X dX. 0 Y 1 − Y2 X2 + B 2


459

Proof We first consider the left-hand integral and substitute the series representation for the cosine and exponential to find the expansion for the integral 1 ∞ p−1/2 (−1)p+q 2p q+1 B C 2 Y 2q+1 1 − Y 2 dY, (2p)!q! 0 p,q=0

where

2

1

Y

2q+1

1−Y

2 p−1/2

0

1 dY = B q + 1, p + 2

is the Euler beta-function (see [15, Sec. 8.380]). Now for the right-hand side we use the series representation of the sine function to find 1/2 ∞ m sin X2 + B 2 (−1)m 2 X + B2 1/2 = (2m + 1)! X2 + B 2 m=0 ∞ m (−1)m m 2p 2m−2p = B X (2m + 1)! p = =

m=0 ∞ ∞

p=0 m=p ∞ p,q=0

We evaluate the integral ∞

e−X

2 /(4C)

0

p=0

m 2p 2m−2p (−1)m B X (2m + 1)! p

p + q 2p 2q (−1)p+q B X . (2p + 2q + 1)! p

X 2q+1 dX =

1 (4C)q+1 q!, 2

and on substituting the identity (2p+2q+1)!p!B(q+1, p+1/2) = (2p)!q!22q+1 (p+ q)! we have the desired equality of the two expansions. The proof is complete. We are now ready to present the integral formula relating products of MacdonaldBessel functions to the kernel δ ( . αβ

theorem 2.4 2 Notation is as above. For h(y) = 2ay 2 e−ay , given t0 , t1 , k0 positive, for n, r > 0 with 0 < t0 < t = 2π nr −1 < t1 , 0 ≤ k ≤ k0 , a > 0, and r large, ∞ r sinh π r Kir 2π(n + k)y Kir (2πny)h(y)y −1 dy 0

π = 2

t −1 0

1/2 cos 2π k t −2 − y 2 S(t, y)h(y) dy + O r −2 a 2 + r −1 a 1/2

460

SCOTT A. WOLPERT

for a remainder constant depending on t0 , t1 , and k0 . Proof We start with the left-hand side. The integral equals ∞ r sinh π r Kir (n + k)n−1 y Kir (y)h y(2πn)−1 y −1 dy, 0

which by Lemma 2.2 is ∞ −1/2 dP P2 −Y 2 /(4a) rπ 4 + P2 dY e sin r arccosh 1 + 2 dY 0 for P 2 = ((Y /(2π n))2 + (k/n)2 )(n/(n + k)). We consider the factors in the integrand. For t0 ≤ t = 2π nr −1 ≤ t1 , r 2 P 2 = t −2 (Y 2 + (2πk)2 )(1 + O(r −1 )), and similarly r(dP /dY ) = t −1 (Y 2 + (2πk)2 )−1/2 Y (1 + O(r −1 )). Furthermore, arccosh(1+P 2 /2) = P +O(P 3 ) for all positive P , and so sin(r arccosh(1+P 2 /2)) = sin rP + O(rP 3 ) for all positive r and P . Finally, we have that (4 + P 2 )−1/2 = 1/2 + O(P 2 ) for all P . Gathering the expansions, we have that the original integral of Macdonald-Bessel functions is 1/2 2 1/2 −1 π ∞ −Y 2 /(4a) e sin t −1 Y 2 + (2πk)2 Y dY t Y + (2πk)2 2 0 ∞ (2.5) dP −Y 2 /(4a) 2 3 2 dY . e r P + rP + 1 +O dY 0 To analyze the remainder, we use the coarse bounds that P is O((Y + 1)r −1 ) and dP /dY is O(r −1 ), valid for Y positive and t0 ≤ t ≤ t1 . The remainder simplifies to the quantity ∞ 2 e−Y /(4a) r −1 Y 3 + 1 r −1 dY , O 0

which is O(r −2 a 2 + r −1 a 1/2 ) by scaling considerations. Now we invoke the identity from Lemma 2.3 with X = Y t −1 , B = 2πkt −1 , and C = at −2 to find, for the principal term, 1/2 −1/2 −a(Y/t)2 π 1 cos 2π k t −2 − Y 2 t −2 e 2at −2 Y dY. 1 − Y2 2 0 The desired final integral results on substituting Y = yt. The proof is complete. 2.4 We are interested in a more detailed analysis of the large r behavior of the special integral ∞ −1 I 4aα 2 , r = π −1 r sinh πr Kir (y)2 h(αy)y −1 dy. 0

461


In Sections 3.1–3.4 the refined analysis is applied to the consideration of short-range coefficient sums. From Lemma 2.2, ∞ −1/2 P2 −AP 2 4 + P2 I (A, r) = r e sin r arccosh 1 + dP . 2 0 Set G(u) = uF (u) = u e−u −z2

z

2

u

ev dv, 2

0 v2

where F (z) = e 0 e dv is the probability integral (see [32, Sec. 2.3]). The function G(u) is positive for u positive, is O(u2 ) for u small, and satisfies G(u) = 1/2 + O(u−1 ) for u large positive. (The method of [32, p. 20] can be used to establish the last expansion.) lemma 2.5 Notation is as above. Given A0 > 0 for A ≥ A0 and given r ≥ 1 for A1/2 ≤ r, I (A, r) = G(r(2A1/2 )−1 ) + OA0 (A1/2 r −1 ); and given D0 > 0, furthermore, for A1/2 ≥ D0 r, I (A, r) = G(r(2A1/2 )−1 ) + OA0 ,D0 (A−1 ). Proof For the first expansion we start with the integral and integrate by parts twice—first with u = cos(r arccosh(1+P 2 /2)), second with u = r −1 sin(r arccosh(1+P 2 /2))— and then we change variables with A1/2 P = Q. The resulting formula is 1/2 1 A1/2 ∞ Q2 2 . d e−Q Q 4 + Q2 A−1 I (A, r) = + sin r arccosh 1 + 2 2r 0 2A We then note that, for A ≥ A0 , Q ≥ 0, it follows that 2 ≤ (4 + Q2 A−1 )1/2 ≤ −1/2 and that |(d/dQ)(4 + Q2 A−1 )1/2 | ≤ QA−1 2 + QA0 0 . It follows that the above

∞ −Q2 3 (1 + Q ) dQ, which is finite. The expression integral is dominated by 0 e for the integral I and the expansion for the function G now give I (A, r) = G(r(2A1/2 )−1 ) + O(A1/2 r −1 ) for the first specified range. For the second expansion we start with a change of variables for the integral: for Q2 = AP 2 , −1/2 ∞ Q2 r Q2 2 4+ I (A, r) = 1/2 e−Q sin r arccosh 1 + dQ. 2A A A 0 Now we have that arccosh(1 + P 2 /2) = P + O(P 3 ) for all positive P and thus for A1/2 ≥ D0 r ≥ D0 > 0 that Q2 = rA−1/2 Q + O rQ3 A−3/2 = rA−1/2 Q + O A−1 Q3 . r arccosh 1 + 2A

462

SCOTT A. WOLPERT

We also have the coarse bound that −1/2 2 4 + Q2 A−1 − 1 is O A−1 Q2 . Combining expansions, it follows that −1 ∞ −Q2 I (A, r) = r 2A1/2 e sin rQA−1/2 dQ 0 ∞ 2 e−Q Q2 + Q3 dQ . + O A−1 0

The explicit integral is tabulated as −1 −1 3 r 2 (4A)−1 1 F1 1, ; −r 2 (4A)−1 = r 2A1/2 F r 2A1/2 2 (see [12, p. 73, (18), and p. 373], [32, (9.9.1) and (9.13.3)]. The remainder integral is finite, and the proof is complete.

3. Measures from Fourier coefficients and sums of Macdonald-Bessel functions 3.1 Our plan is to express the square of an automorphic eigenfunction as an integral over the space of geodesics. We use Fejér summation to define a family of positive measures from the Fourier coefficients of an eigenfunction. Quadratic expressions in the eigenfunctions are then represented as integrals of the measures. In Section 3.2 we introduce the measures, establish their uniform boundedness, and consider their basic properties. In Section 3.3 we consider sums of the relation given in Theorem 2.4, that is, sums of products of Macdonald-Bessel functions. Our goal is to find the limits of sums of products. Only the basic sum-square bound is available for the Fourier coefficients; to be able to interchange the spectral and summation limit, a detailed analysis of the contribution of the Macdonald-Bessel functions is required. In Theorems 3.5 and 3.6 a high-energy limit of eigenfunction squares is presented as the integral of the geodesic-indicator and a limit of the constructed measures. In Corollary 3.7 we find that the high-energy limit is the adjoint Radon transform of the limit of the constructed measures. In Section 3.4 we use our formulation to show that the adjoint Radon transform is invertible for translation-invariant measures. 3.2 Our plan is to construct and analyze measures describing the concentration properties of automorphic eigenfunctions. Let ⊂ SL(2; R) be a cofinite group with a width-1


463

cusp at infinity. We consider -invariant eigenfunctions of the hyperbolic LaplaceBeltrami operator with finite L2 (\H)-norm (see [43], [46]). For Dϕ + λϕ = 0, λ = (1/4 + r 2 ) > 1/4, ϕ has the Fourier expansion an (y sinh πr)1/2 Kir (2π|n|y)e2π inx (3.1) ϕ(z) = n

for z = x + iy, a0 = 0, and the Macdonald-Bessel function. (Note the normalization of the Fourier coefficients.) We start with a bound for the Fourier coefficients M

|an |2 ≤ C ϕ22 (M + r)

(3.2)

n=1

(see [27], [48]). (The bound is a slight improvement of the Deshouillers-Iwaniec bound in [11].) Our constructions involve the Fejér kernel (see [29]). Recall that 2N 2 |k| 1 −N 1− Xk = X + · · · + XN , (3.3) 2N + 1 2N + 1 k=−2N

and, in particular, the Fejér kernel is simply 2N 1− FN = k=−2N

|k| e2π ik xˆ . 2N + 1

FN is positive and defines an operator by convolution on function spaces associated to R (see [29]). Convolution with FN converges to the identity as N tends to infinity for Cper (R) and L1per (R), the spaces of continuous and integrable 1-periodic functions. Convolution with FN is also a formally self-adjoint operator. We now give function space constructions of quantities from the Fourier coefficient sequence {an }n∈Z of an eigenfunction ϕ, normalized by ϕ2 = 1.

Definition 3.1 For an eigenfunction ϕ, with eigenvalue λ = 1/4 + r 2 , r positive, (x, ˆ t)-coordinates −1 on R × R, 't = 2π r , set ˆ t) = an e2π inxˆ , (n − 1)'t ≤ t < n't. χϕ (x, Furthermore, for (x, ˆ t)-coordinates on G, set σϕ,N = (2N + 1)

−1

π

N

χϕ x, ˆ Dt + j 't χϕ x, ˆ Dt + k't ,

j,k=−N D=±1

Fϕ,k (t) = π

|n|≤rt (2π )−1

an+k an r −1

464

SCOTT A. WOLPERT

(note that a0 = 0), and, for the Lebesgue-Stieljes derivative dFϕ,k , set µϕ = dFϕ,k (t)e2π ik xˆ . k

Comments and observations are in order. The quantity χϕ is akin to a probability amplitude associated to the state ϕ. For g ∈ Cc (R+ ), the 1-dimensional integral

a self-adjoint operator since σϕ,N R+ gσϕ,N in t already has the form Opg ϕ, ϕ for is a sum of Hermitian squares of (2N +1)−1/2 π 1/2 j χϕ (x, ˆ t +j 't). In Section 4.4 we find that the matrix elements Opg ϕ, ϕ give the probability for observing classical trajectories. The quantities χϕ and µϕ are (xˆ → xˆ + 1)-invariant; σϕ,N is a Fejér sum of χϕ and satisfies σϕ,N ≥ 0. From (3.2) the L2 -norm of χϕ on {0 ≤ xˆ ≤ 1, 0 ≤ t ≤ t0 } is bounded by C (t0 +1)1/2 , and similarly |Fk (t)| is bounded by C (t 2 + 2 (G), t (|k| + 2) + 1)1/2 . It follows that µϕ is at least a tempered distribution for Cper,c the space of 1-periodic, compactly supported, twice-differentiable functions on G 2 (G) has a Fourier expansion k fk (t)e2π ik xˆ with |fk (t)| ≤ Cf (|k| + f ∈ Cper,c 1)−2 . The distribution µϕ represents an elementary type of microlocalization of the eigenfunction square; µϕ encodes the concentration and oscillation properties of the eigenfunction. Quadratic expressions in ϕ are integrals of µϕ . Our analysis requires an appropriately convergent sequence of eigenfunctions. Definition 3.2 For dA the hyperbolic area element, a normalized sequence of eigenfunctions {ϕj } with eigenvalues tending to infinity is ∗-convergent, provided that ϕj2 dA converges weak* relative to Cc (\H) and provided that for each k the Lebesgue-Stieljes derivatives dFϕj,k converge weak* relative to the continuous functions for each closed subinterval of [0, ∞). Note that by weak* compactness of the unit ball of measures and diagonalization, a normalized sequence of eigenfunctions has a ∗-convergent subsequence. Weak* convergence of measures on G is considered relative to the system of spaces Cper,t0 (G), ˆ t) | all positive t0 , of continuous 1-periodic functions with support contained in {(x, 0 < t ≤ t0 }. proposition 3.3 Notation is as above. For {ϕj }, a ∗-convergent sequence, limj σϕj ,N = FN ∗limj µϕj relative to each Cper,t0 (G), and, in particular, each limj µϕj is a positive measure on G with limj µϕj bounded by C(t0 + 1).

465


Proof The basis of the considerations is the Fourier series expansion of a function in Cper (G). Functions with finite Fourier expansions in xˆ are dense in Cper (G). Accordingly, measures of uniformly bounded mass converge weak*, provided their sequences of Fourier-Stieljes coefficients weak* converge. Each measure σϕ,N is positive and thus has its mass given by its zeroth coefficient with an integral bounded by Fϕ,0 (t +N't), which by (3.2) is bounded by C(t + 1 + N't). Since, as j tends to infinity, r tends to infinity and 't tends to zero, we have for g ∈ Cc (R+ ) that lim χϕj x, ˆ Dt + (k + m)'t χϕj x, ˆ Dt + m't g(t)e−2π ik xˆ d xˆ dt j

D=±1

= lim j

=

χϕj x, ˆ Dt)g(t ˜ − m't)e−2π ik xˆ d xˆ dt ˆ Dt + k't χϕj (x,

D=±1

g(t) dFϕj,k (t)

since g(t ˜ − m't) tends to g(t) in Cc (R+ ), where g(τ ˜ ) is defined as g(0) for τ < 0 and g(τ ) for τ ≥ 0. The first conclusion follows, and, furthermore, it follows that FN ∗ limj µϕj is a positive measure with mass bounded in terms of C(t0 + 1). It also follows that limj µϕj is a positive measure with mass bounded in terms of C(t0 + 1). The proof is complete. 3.3 We continue our preparations, introduce test functions, and consider automorphic integrals. Let ∞ ⊂ be the stabilizer of the width-1 cusp at infinity. Definition 3.4 For z = x + iy ∈ H and k an integer, set hk (z) = 2ay 2 e−ay

2 −2π ikx

and

Hk (z) =

hk (γ z).

γ ∈∞ \

We now review certain basic bounds for incomplete theta series. Recall first that the ∞ \ translates of a point intersect the horoball B = {z | Im z ≥ 1} at most once. Recall also the truncation at height 1 of the Eisenstein series E 1 (z; 2) (see [7]); the function E 1 (z; 2) vanishes at each cusp. From the observation that h(y) ≤ 2ay 2 for 0 < a ≤ 1, it follows that H0 (z) ≤ 2a E 1 (z; 2), provided (∞ \)z ∩ B = ∅, and that H0 (z) ≤ 2a E 1 (z; 2) + h(Im z∞ ), provided (∞ \)z ∩ B = {z∞ }. Basic bounds for |Hk (z)| ≤ |H0 (z)| now follow: the restriction of Hk (z) to a compact set is bounded by a multiple of a, 0 < a ≤ 1; for a fixed, Hk (z) tends to zero as z tends to a cusp; also, Hk (z) is uniformly bounded for 0 < a ≤ 1.

466

SCOTT A. WOLPERT

We begin our consideration of the ∗-convergent limit by analyzing the zeroth coefficients. Let ν (a nonnegative tempered distribution on (0, ∞)) be the zeroth Fourier-Stieljes coefficient of the lift to H of the Cc (\H) weak∗ limit limj ϕj2 dA, and let σ (a nonnegative tempered distribution on [0, ∞) with possibly σ ({0}) > 0) be the weak* limit limj dFϕj,0 on each closed subset of [0, ∞). theorem 3.5 With the above notation, ∞

∞

h(y)ν(y) =

0

G a 1/2 t −1 σ (t).

0

Proof We must show that sequences of integrands have uniform majorants and that integrals converge. We begin with J (α) = 0≤x≤1 hϕ 2 dA, z = x + iy, 0 0 such that if |ξ/|ξ | − ξ 0 | < δ and |y − y 0 | < δ and |v − y 0 | < δ, then we have

y+v G = σ ∈ : ξ, σ − < −9|ξ | = ∅. 2 In particular, G has positive measure since G is open. Now we have e−2ξ,σ −(y+v)/2 dσ ≥ e−2ξ,σ −(y+v)/2 dσ ≥ e29|ξ | dσ ≥ C ξ 0 , y 0 e29|ξ | .

G

G

Now the theorem follows by the compactness of the unit ball in ξ space.

551

THE BERGMAN KERNEL FOR TUBES

Theorem 5 allows us to localize in the ξ variable. Now we discuss localizing on ∂. We define Sδ,y 0 = y ∈ ∂ : y − y 0 ≤ δ and Aδ,y 0 (ξ ) =

1 2|ξ |

Sδ,y 0

dr ξ , dσ (y). e−2ξ,y − |dr| |ξ |

We have the following. theorem 6 Let z0 , w0 ∈ Cn with z0 = w 0 = y 0 ∈ ∂. Let : ⊂ Rn be a small conic neighborhood of −dr(y 0 ). Then the function 1 1 ¯ dξ B (z, w) − eiz−w,ξ (2π)n : Aδ,y 0 (ξ ) can be extended as a real analytic function to a full neighborhood of (z0 , w0 ). Proof We begin by choosing convenient coordinates. Note that formula (15) is invariant under translations and real rotations. Hence we may assume that y 0 = 0. We may also assume that we have δ > 0 and ϕ real valued and real analytic near |y | ≤ δ such that r has the form r(y) = ϕ(y ) − yn with dϕ(0) = 0. Here y = (y1 , . . . , yn−1 ). Hence dr(0) = (0, . . . , 0, −1). So we may assume that Sδ,y 0 = y ∈ ∂ : yn = ϕ(y ), |y | ≤ δ . Given M > 0, we define : as follows: : = ξ ∈ Rn : ξn > M|ξ | .

(18)

Note that if M > 0 is large, : is a small conic neighborhood of −dr(0). We define V = Rn \ :, and for the rest of the proof we write Aδ in place of Aδ,y 0 . Using formula (15), we see that 1 ¯ 1 dξ B (z, w) − eiz−w,ξ (2π )n : Aδ (ξ ) 1 1 1 1 iz−w,ξ ¯ 1 iz−w,ξ ¯ = dξ + − dξ. e e (2π )n V A(ξ ) (2π)n : A(ξ ) Aδ (ξ ) By applying Theorem 5 and using the fact that (z − w) ¯ is near zero, we see that the theorem is proved if the quantity 1 1 Aδ − A − = A Aδ AAδ

552

FRANCSICS AND HANGES

is exponentially decreasing for ξ ∈ :, with M sufficiently large. This follows from the next two lemmas. lemma 7 Let δ > 0 be given. Then there exist M > 0 and 9 > 0 such that if ξn ≥ M|ξ |, then A(ξ ) − Aδ (ξ ) ≤ 1 e−9|ξ | . 9 Proof Since ∂ is convex, bounded, and analytic, it follows that there exists Cδ > 0 such that if y ∈ ∂ \ Sδ,y 0 , then yn ≥ Cδ . Now consider |y |ξn K y, ξ = y , ξ + yn ξn ≥ −|y ||ξ | + Cδ ξn ≥ − + Cδ ξ n ≥ C δ − ξn . M M Here K is the diameter of . The lemma follows once M is chosen large enough. Our next result uses stationary phase methods. Note that the critical point (which may be degenerate) for the function y , ξ + ϕ(y )ξn is given by the equation ϕ (y 0 and C > 0 such that if ξn ≥ M|ξ |, then Aδ (ξ ) ≥

C (n+1)/2 ξn

Proof We clearly have C > 0 such that C Aδ (ξ ) ≥ ξn

e−2ξn (ϕ(y< )−ϕ (y< ),y< ) ≥

|y |≤δ

C (n+1)/2 ξn

e−2(y ,ξ +ϕ(y )ξn ) dy .

The Taylor expansion about the critical point y 0 is large enough, there exists C > 0 such that if ξ ∈ :, we have 0 ≤ A(ξ ) − Aδ,p (ξ ) ≤ Ce−qξn . We also have Aδ,p (ξ ) = I (ξ ) + Iϕ (ξ ), 1 −2pξn I (ξ ) = − e e−2y ,ξ dy . 2ξn |y |≤δ Now if M > 0 is large enough, there exists C > 0 such that if ξ ∈ :, we have

where we define

(27)

558


|I (ξ )| ≤

C −pξn e . ξn

(28)

It follows from Lemma 8, or rather its proof (see inequality (19)), that there exist M > 0 and C > 0 such that if ξ ∈ :, we have |Iϕ (ξ )| ≥

C (n+1)/2 ξn

;

(29)

hence we also have, for ξn large, |Aδ,p (ξ )| ≥

C (n+1)/2 ξn

.

Since we have A ≥ Aδ,p , it follows that there exist M > 0 and 9 > 0 such that if ξ ∈ :, we have |Aδ,p (ξ ) − A(ξ )| + |I (ξ )| 1 1 1 ≤ e−9|ξ | . A(ξ ) − I (ξ ) ≤ |Aδ,p (ξ )Iϕ (ξ )| 9 ϕ Hence it follows that 1 (2π )n

:

¯ eiz−w,ξ

1 1 − dξ A(ξ ) Iϕ (ξ )

can be extended as a real analytic function near (z0 , w0 ), and the lemma is proved. 7. Gevrey regularity In this section we prove the first part of Theorem 2. The fact that m is the best Gevrey class possible is proved at the end of Section 9. Related interesting results on Gevrey regularity have also been obtained using 2 L -methods (see the work of Derridj and C. Zuily [9] and of Derridj [6]). theorem 9 Assume that the two distinct points z0 , w0 ∈ ∂T ⊂ C2 lie on the same characteristic line. Let m be the type of the point z0 . Then the Bergman kernel of the tube T , B (z, w), can be extended, as a smooth function of Gevrey class m, to a full neighborhood of the point (z0 , w0 ). Proof The assumption that z0 , w0 ∈ ∂T lie on the same characteristic line means that we have t ∈ R \ {0}, e ∈ R2 , |e| = 1 such that z0 = w 0 + te, e⊥∇r(y 0 ), e ∈ Ker r (y 0 ). After translating and rotating the domain , we may assume that y 0 = 0, ∇r(y 0 ) = (0, −1), and e = (1, 0). So we have x 0 ≡ z0 = w0 + t (1, 0). Near (0, 0) ∈ ∂ the boundary of is given by y2 = ϕ(y1 ). The function ϕ is

559


nonnegative, convex, real analytic, and ϕ(0) = ϕ (0) = · · · = ϕ (m−1) (0) = 0, ϕ (m) (0) = am! > 0, and m > 2 is even. We may also assume that ϕ satisfies the estimates 3a m m a m m ϕ (k) (y)y k y ≤ ≤ y (30) 2 k k! 2 k for k = 0, . . . , m and ϕ (4) (y) ≥ 0 (31) in the interval [−δ, δ] for some δ > 0. Let B90 (z0 , w0 ) ⊂ C4 be the open ball of radius 90 > 0 centered at (z0 , w0 ) ∈ C2 × C2 . We remark that it is sufficient to prove the Gevrey estimate in the set B90 z0 , w0 ∩ (z, w) ∈ C2 × C2 ; z ∈ , w ∈ , that is, for the points (z, w) ≡ (x + iy, u + iv) near (z0 , w0 ) satisfying the conditions y2 ≥ ϕ(y1 ),

v2 ≥ ϕ(v1 )

(32)

for all y1 , v1 ∈ (−δ, δ). After taking derivatives, there exists C > 0 such that 1 |α| |β| dξ. ξ1 ξ2 eiz−w,ξ C∂xα11 ∂yα12 ∂uα13 ∂vα14 ∂xβ21 ∂yβ22 ∂uβ23 ∂vβ24 B(z, w) = A(ξ ) R2 Let :M be the cone :M = {ξ ∈ R2 ; ξ2 > M|ξ1 |}. Then it follows from Lemma 9 that it is enough to prove the Gevrey estimate for the integral 1 |α| |β| dξ. (33) ξ1 ξ2 eiz−w,ξ Iϕ (ξ ) :M After the change of coordinates ξ1 = ρη, ξ2 = ρ m /a, we need to estimate the integral ρ |α|+m|β|+m η|α| m m dρ dη (34) ei(ρη(z1 −w1 )+(ρ /a)(z2 −w2 )) J ≡ |β|+1 Iϕ (ρη, ρ m /a) a :˜ M when (z, w) is near (z0 , w0 ) ≡ (x 0 , u0 ). The domain of integration is :˜ M = {(η, ρ) ∈ R2 ; ρ > (aM|η|)1/(m−1) }. Taking advantage of the fact that x10 = u01 , we deform the contour of the η-integration in J as η → η + 2iε(x10 − u01 ). The new exponential factor is eiE with E ≡ (η + 2i9(x10 − u01 ))ρ(z1 − w1 ) + (ρ m /a)(z2 − w2 ). We have the lower bound for ρm y 2 + v2 E = 2ε x10 − u01 (x1 − u1 )ρ + ρη y1 + v1 + a (35) 0 ρm 0 2 ≥ ε x1 − u1 ρ + ρη y1 + v1 + y 2 + v2 , a provided (x1 , u1 ) is close enough to (x10 , u01 ). Here we use assumption (32) and the convexity of the function ϕ to obtain

560


y2 + v2 ≥ ϕ(y1 ) + ϕ(v1 ) ≥ 2ϕ

y1 + v1 . 2

So the exponential factor in J satisfies the upper bound iE e ≤ e−ε(x10 −u01 )2 ρ−2(ρη((y1 +v1 )/2)+(ρ m /a)ϕ((y1 +v1 )/2)) . Consider the function (−δ, δ) ( y −→ ρηy +

ρm ϕ(y). a

(36)

It follows from the convexity and analyticity of ϕ that ϕ is strictly increasing, and therefore the function (36) has a unique critical point y∗ = y∗ (ρ, η). The critical point satisfies the equation ϕ (y∗ ) = −aη/ρ m−1 . If (ρ, η) ∈ :˜ M , then −aη/ρ m−1 lies in the set ϕ ((−δ/2, δ/2)); therefore y∗ ∈ (−δ/2, δ/2), provided that the constant M is large enough. Since the function (36) is convex, its minimum occurs at y∗ = y∗ (ρ, η). This yields the estimate iE e ≤ e−ε(x10 −u01 )2 ρ−2(ρηy∗ +(ρ m /a)ϕ(y∗ )) . (37) The next step in the proof is to obtain a lower bound for |Iϕ (ρ(η + 2iε(x10 − u01 )), ρ m /a)|. We have ρ m Iϕ ρ η + 2iε x10 − u01 , a δ 0 0 a m e−2(ρηy1 +(ρ /a)ϕ(y1 )) e−4iερ(x1 −u1 )y1 dy1 = (38) m 2ρ −δ δ a = e−f (y1 ) g(y1 ) dy1 , 2ρ m −δ where the functions f and g are defined as f (y) = 2(ρηy + (ρ m /a)ϕ(y)) and g(y) = e−4iερ(x1 −u1 )y . 0

0

To estimate the last integral, we use the stationary phase method as in Lemma 14. We start with decomposing the last integral into the sum of the four integrals δ e−(f (y)−f (y∗ )) g(y) − g(y∗ ) dy, (39) J1 = e−f (y∗ ) −δ

J2 = g(y∗ )e

−f (y∗ )

J3 = g(y∗ )e−f (y∗ )

δ

e−(f (y)−f (y∗ )) − e−(f

−δ ∞

−∞

e−(f

(y

∗ )/2)(y−y∗ )

2

(y

dy,

∗ )/2)(y−y∗ )

2

dy,

(40) (41)

561


J4 = −g(y∗ )e−f (y∗ )

|y|≥δ

e−(f

(y

∗ )/2)(y−y∗ )

2

dy.

(42)

The integral J3 can be evaluated explicitly: 2π g(y∗ )e−f (y∗ ) . J3 = f (y∗ )

(43)

The fourth integral, J4 = −g(y∗ )e

−f (y∗ )

|x+y∗ |≥δ

e−(f

(y

∗ )/2)x

2

dx,

can be estimated easily. If |x + y∗ | ≥ δ, then x 2 ≥ |y∗ |2 because y∗ ∈ [−δ/2, δ/2]. So we can use (f (y∗ )/2)x 2 ≥ (f (y∗ )/4)x 2 + (f (y∗ )/4)|y∗ |2 to obtain 4π 2 e−f (y∗ ) e−(1/4)f (y∗ )|y∗ | |J4 | ≤ f (y∗ ) since |g(y∗ )| = 1. In the exponent we havef (y∗ )|y∗ |2 = 2(ρ m /a)ϕ (y ∗ )|y ∗ |2 ≥ m(m−1)ρ m |y ∗ |m , using (30) with k = 2. The critical point can be estimated by using (30) with k = 1. Since |ϕ (y∗ )| ≤ (3am/2)|y∗ |m−1 , we get |y∗ | ≥

2 3m

1/(m−1)

|η|1/(m−1) . ρ

So the exponent f (y ∗ )|y∗|2 ≥ c1 (m)|η|m/(m−1) . Therefore we obtain 1 1 2π e−f (y∗ ) = |J3 | |J4 | ≤ 6 f (y∗ ) 6

(44)

(45)

for |η| ≥ η0 (a, m). To estimate the integrals J1 , J2 , we need the following lemma. lemma 10 The function f (y, ξ ) = 2(ξ1 y + ξ2 ϕ(y)) satisfies the inequality 2 f (y) ≥ f (y∗ ) + aξ2 y∗m−2 y − y∗ for all ξ ∈ :M and y ∈ [−δ, δ].

(46)

562


Proof Case I: ξ1 > 0 It follows from (30) that y∗ < 0 in this case. First we estimate f (y) in the interval [y∗ , δ]. We have f (y) = f (y∗ ) +

m−1 k=2

f (m) (y) f (k) (y∗ ) ˜ (y − y∗ )k + (y − y∗ )m k! m!

for some y˜ between y∗ and y. The inequality (30) implies that m ˜ f (m) (y) m m m y (y − y∗ ) ≥ aξ2 (y − y∗ ) = aξ2 y∗ −1 m! y∗ and

k k f (k) (y∗ )y∗k y m m y f (k) (y∗ ) − 1 ≥ aξ2 −1 . y∗ (y − y∗ )k = k k! k! y∗ y∗

So f (y) ≥ f (y ∗ ) + aξ2 y∗m

k m m y y − 1 = f (y∗ ) + aξ2 y∗m β −1 , k y∗ y∗ k=2

where the function β is defined as β(t) = (t + 1)m − mt − 1. It is elementary to see that β(t) ≥ t 2 for all t ∈ R. This proves the inequality (46) for y ∈ [y∗ , δ]. In the interval [−δ, y∗ ) it is enough to use a fourth-order Taylor expansion: f (y∗ ) f (3) (y∗ ) f (4) (y) ˜ (y − y∗ )2 + (y − y∗ )3 + (y − y∗ )4 . 2 3! 4! The third term is nonnegative when y ≤ y∗ because it follows from ξ1 < 0 and (30) that f (3) (y∗ ) < 0. The fourth term is always nonnegative because ϕ (4) ≥ 0. So using (30) again, we get f (y∗ ) m (y − y∗ )2 ≥ f (y∗ ) + a ξ2 y∗m−2 (y − y∗ )2 f (y) ≥ f (y∗ ) + 2 2 f (y) = f (y∗ ) +

for y ∈ [−δ, y∗ ]. This completes Case I. Case II: ξ1 < 0 We introduce the function ψ(x) = ϕ(−x) and define the function g as g(x) = 2(−ξ1 x + ξ2 ψ(x)) for x ∈ [−δ, δ]. Let the critical point of g be x∗ = x∗ (−ξ1 , ξ2 , ψ). If M > 0 is large enough, then x∗ ∈ (−δ/2, δ/2) for all ξ ∈ :M . Since the properties of ψ and g are similar to those of ϕ and f , Case I can be applied to g, so g(x) ≥ g(x∗ ) + aξ2 x∗m−2 (x − x∗ )2 for all x ∈ [−δ, δ]. Replacing x by −x, we get

563


g(−x) ≥ g(x∗ ) + aξ2 x∗m−2 (−x − x∗ )2 .

(47)

Notice that g(−x) = f (x). Since the critical point x∗ (−ξ1 , ξ2 , ψ) satisfies ϕ (x∗ ) = −ξ1 /ξ2 , we obtain that −x∗ (−ξ1 , ξ2 ) = y∗ (ξ1 , ξ2 ). Therefore the right-hand side of (47) is equal to f (y∗ ) + ay∗m−2 (x − y∗ )2 . This proves Case II. When ξ1 = 0, the inequality (46) is trivially satisfied, and the proof of Lemma 10 is complete. Applying Lemma 10 in estimating the integral J1 , we get δ m m−2 2 |J1 | ≤ max |g (y)|e−f (y∗ ) |y − y∗ |e−ρ y∗ (y−y∗ ) dy [−δ,δ] −δ ∞ m m−2 2 |s|e−ρ y∗ s ds ≤ max |g (y)|e−f (y∗ ) [−δ,δ]

=

c1

−∞

max |g (y)|e−f (y∗ ) .

ρ m y∗m−2 [−δ,δ]

We have |g (y)| = 4ερ|x10 − u01 | and |g(y∗ )| = 1. Moreover, we know from (30), (44) that −1/2 −1/2 f (y∗ ) ≤ c2 (a, m) ρ m−2 y∗m−2 ≤ c3 (a, m) |η|(m−2)/(m−1) m−2 ρ m−1 y∗ tends to zero if (η, ρ) ∈ :˜ M , and |η| → ∞. Therefore we get 1 1 2π e−f (y∗ ) = |J3 | |J1 | ≤ 6 f (y∗ ) 6

(48)

for |η| ≥ η0 (a, m, M, b). To estimate the second integral J2 , we introduce the function h(t) = h(t, y, η, ρ) = t (f (y) − f (y∗ )) + (1 − t)(1/2)f (y∗ )(y − y∗ )2 . Then δ δ 1 −f (y∗ ) −h(1) −h(0) −f (y∗ ) J2 = g(y∗ )e e −e dy = g(y∗ )e −h (t)e−h(t) dt dy. −δ

−δ

0

m−1

Here h (t) = f (y) − f (y∗ ) − (f (y∗ )/2)(y − y∗ )2 = k=3 (f (k) (y∗ )/k!)(y − k ˜ − y∗ )m for some y˜ between y and y∗ . It follows from (30) that y∗ ) + (f (m) (y)/m!)(y |f (k) (y∗ )|/k! = (2ρ m /a)(|ϕ (k) (y∗ )|/k!) ≤ 3ρ m mk |y∗ |m−k for k = 3, . . . , m − 1 and m ˜ = (2ρ m /a)(|ϕ (m) (y)|/m!) ˜ ≤ 3ρ m . So |h (t)| ≤ 3ρ m m that |f (m) (y)|/m! k=3 k |y∗ |m−k |y − y∗ |k . To estimate the exponent h(t), we use Lemma 10 and f (y∗ ) ≥ m(m−1)ρ m y∗m−2 to obtain h(t) ≥ tρ m y∗m−2 (y −y∗ )2 +(1−t)(m(m−1)/2)ρ m y∗m−2 (y − y∗ )2 ≥ ρ m y∗m−2 (y − y∗ )2 . These inequalities lead to

564


|J2 | ≤ |g(y∗ )|e−f (y∗ ) 3ρ m

m m

|y∗ |m−k

δ

m m−2

|y − y∗ |k e−ρ y∗ (y−y∗ ) dy k −δ k=3 m m ρ m |y∗ |m−k ≤ g y∗ e−f (y∗ ) 3 ck k ρ m |y∗ |m−2 (k+1)/2 k=3 m m m 2π −f (y∗ ) m 1−k/2 e ρ g y ≤ c4 (a, m) c |y | . ∗ k ∗ k f (y∗ ) k=3 In the last step we took advantage of 1/ ρ m |y∗ |m−2 ≤ c4 (a, m) 2π/(f (y∗ )). Since ρ m |y∗ |m ≥ c5 (a, m)|η|m/(m−1) , we see that 1 1 2π |J2 | ≤ e−f (y∗ ) = |J3 |, (49) 6 f (y∗ ) 6 2

provided |η| ≥ η0 (a, m, M, b). Now from (45), (48), and (49) we see that m 2ρ 0 ρ m 1 0 a Iϕ ρ η + 2iε x1 − u1 , a − J3 ≤ |J1 | + |J2 | + |J4 | ≤ 2 |J3 |. Therefore we can conclude that, for small ε, 0 ρ m 1 2ρ m 1 2π 0 ≥ |J3 | = Iϕ ρ η + 2iε x1 − u1 , e−f (y∗ ) a a 2 2 f (y∗ )

(50)

for all (η, ρ) ∈ :˜ M , |η| ≥ η0 (a, m, M, b). Our next step is to obtain a lower bound for |Iϕ (ρ(η + 2iε(x10 − u01 )), ρ m /a)| in the region {(η, ρ) ∈ R2 ; |η| ≤ η0 , ρ > ρ0 > 0}. We write ϕ(y) = ay m u(y), so that limy→0 u(y) = 1. Substituting y = −x/ρ into (38), we get ρ m 0 m 0 lim ρ Iϕ ρ η + 2iε x1 − u1 , ρ→∞ a δρ 0 0 a m e2((η+2iε(x1 −u1 ))x−x u(−x/ρ)) dx = lim ρ→∞ 2 −δρ a = N η + 2iε x10 − u01 , 2 where the function N is defined in (59). It easy to see that the last limit is uniform in η ∈ [−η0 , η0 ]. Since the roots of N are all imaginary, there is a constant C2 = C2 (ε, a, m, δ, b) > 0 such that |N (η + 2iε(x10 − u01 ))| ≥ (4/a)C2 for all |η| ≤ η0 , provided that ε is small enough. So there is a constant ρ0 = ρ0 (a, m, δ, m) > 1 such that m Iϕ ρ η + 2iε x 0 − u0 , ρ ≥ C2 1 1 a ρm

565


for all ρ ≥ ρ0 . Moreover, notice that the exponential factor e−f (y∗ ) is bounded when η is bounded. Indeed, using the Hölder inequality, we get −f (y∗ ) ≤ 2ρη|y∗ | − ρ m |y∗ |m ≤ ρ|y∗ |m + c1 (m)|η|m/(m−1) − ρ|y∗ |m ≤ c1 (m)|η0 |m/(m−1) = c2 (m, η0 ), that is, e−f (y∗ ) ≤ c3 (m, η0 ). So m −1 Iϕ ρ η + 2iε x 0 − u0 , ρ ≤ c4 (m, η0 )ρ m ef (y∗ ) 1 1 a

(51)

in the region {(η, ρ) ∈ R2 ; |η| ≤ η0 , ρ ≥ ρ0 > 0}. In the region {(η, ρ) ∈ R2 ; |η| ≤ η0 , 0 < ρ ≤ ρ0 }, it follows from (38) that Iϕ (ρη, ρ m /a) has a positive minimum. So there is a positive constant C = C(9, δ, m, M, b) > 0 such that |Iϕ (ρ(η + 2iε(x10 − u01 )), ρ m /a)| ≥ C, provided that 9 > 0 is small enough. Using the boundedness of e−f (y∗ ) again, we obtain the estimate (51) with a different constant c5 (m, η0 ) in the region |η| ≤ η0 , ρ ≤ ρ0 . Combining these estimates with (50), we obtain m −1 Iϕ ρ η + 2iε x 0 − u0 , ρ ≤ C 1 + ρ 3m/2 ef (y∗ ) (52) 1 1 a for all (η, ρ) ∈ :˜ M . In the last estimate we have used f (y∗ ) ≤ 3m(m − 1)ρ 2 |η|(m−2)/(m−1) . Returning to the integral J in (34), we get |α| m |J | ≤ |β|+1 ρ |α|+m|β|+m η + 2iε x10 − u01 a :˜ M ρ m −1 0 −ε(x10 −u01 )2 ρ−f (y∗ ) 0 ×e Iϕ ρ η + 2iε x1 − u1 , a dρ dη ∞ ρ m−1 /(aM) m ρ |α|+m|β|+m (1 + |η|)|α| 1 + ρ 3m/2 ≤ C |β|+1 a 0 −ρ m−1 /(aM) × e−ε(x1 −u1 ) ρ dη dρ ∞ |α| 0 0 2 ρ |α|+m|β|+2m−1 1 + ρ m−1 1 + ρ 3m/2 e−ε(x1 −u1 ) ρ dρ 0

≤

2mC a |β|+2 M

0 2

0

≤ C |α|+|β|+1 : m(|α| + |β|) + 1 . Here the constant C is independent of α and β. This completes the proof of the Gevrey estimate. 8. Analytic singularities away from the boundary diagonal Our goal in this section and in Section 9 is to prove the second halves of Theorems 1 and 2. To be precise, let z0 , w0 ∈ ∂T . Assume that z0 = w0 . We show that if z0

566


and w 0 lie on the same characteristic line, then B has no analytic extension past (z0 , w0 ). However, as we have already seen, a smooth extension of Gevrey class m exists if the type of z0 or w 0 is m. We show in Section 9 that this is the best Gevrey class possible. We begin by observing that, under the above assumption, there exists y 0 ∈ ∂, a weakly convex boundary point, such that z0 = y 0 = w0 . Our assumption also guarantees the existence of a vector a ∈ Rn , |a| = 1, and t0 ∈ R, t0 = 0, such that z0 = w 0 + t0 a. The vector a also satisfies a, dr(y 0 ) = 0

and

r (y 0 )a = 0.

All this follows from Proposition 1 and Remark 1. We now introduce the function U (t) as follows: (53) U (t) = B z0 , w0 + t dr(y 0 ) . We show that U has no analytic extension to any neighborhood of t = 0. It follows as a consequence of this that B has no analytic extension to any neighborhood of (z0 , w0 ). We begin by choosing convenient coordinates. Note that the formula (15) is invariant under translations and real rotations. Hence we may assume that y 0 = 0. We may also assume that we have δ > 0 and ϕ real valued and real analytic near |y | ≤ δ such that r has the form r(y) = ϕ(y ) − yn with dϕ(0) = 0. Here y = (y1 , . . . , yn−1 ). Hence dr(0) = (0, . . . , 0, −1). We may also assume that a = (a , 0), |a | = 1, a ∈ Rn−1 , with ϕ (0)a = 0. So we see that if ξ ∈ Rn , we have 0 0 z − w¯ + tdr y 0 , ξ = t0 a , ξ + tξn . Hence it follows that for t ∈ R, t near zero, we have dξ U (t) = ei(t0 a ,ξ +tξn ) A(ξ )−1 , n n (2π) R

(54)

where t0 ∈ R, t0 = 0, and a ∈ Rn−1 , |a | = 1. Now that we have chosen convenient coordinates, we apply Lemma 9. Note that : is defined in (25) and that Iϕ is defined in (26). We introduce dξ ei(t0 a ,ξ +tξn ) Iϕ (ξ )−1 . Uϕ (t) = n (2π) : We have the following immediate consequence of Lemma 9.

567


lemma 11 There exists a constant M > 0 such that U − Uϕ is an analytic function of t for t near zero. We now focus our attention on Uϕ . Our goal is to show that this function is not analytic near t = 0. 9. The two-dimensional case We now begin our study of the two-variable case. We assume that ⊂ R2 is open and convex with real analytic boundary. We may also assume that 0 ∈ ∂ and that we have δ > 0 and ϕ real valued and real analytic near |y1 | ≤ δ such that r has the form r(y) = ϕ(y1 ) − y2 with ϕ (0) = 0. Hence dr(0) = (0, −1). Our previous work allows us to focus on dξ ei(t0 ξ1 +tξ2 ) Iϕ (ξ )−1 , (55) Uϕ (t) = 2 (2π) : where

: = ξ ∈ R2 : M|ξ1 | ≤ ξ2 .

Since we assume weak convexity, we have ϕ (0) = 0. We may assume that we have, for |s| ≤ δ, a strictly positive analytic function u such that ϕ(s) = u(s)s m , where m ≥ 4 is an even integer. Rather than studying Uϕ , we introduce p i(t0 ξ1 +tξ2 ) ξ2 F (t) = Iϕ (ξ )−1 dξ. e 2m :

(56)

Note that F is the image of Uϕ under the elliptic pseudodifferential operator (2π 2 /m) |Dt |p . Hence it suffices to prove that F is not analytic at t = 0. We are free to choose the real number p. We choose p = −((2m + 1)/m). Clearly, we may assume without loss of generality that t0 = 1 and that u(0) = 1. We make the following change of variables in the formula for Dtk F : η=

ξ1 (ξ2 )1/m

and ρ = (ξ2 )1/m . We obtain

568


Dtk F (0) =

∞ M|η|≤ρ m−1

0

where we define

G(ρ, η) =

+|ρ|δ

−|ρ|δ

eiρη ρ mk G(ρ, η)−1 dη dρ,

e−2(ηt+t

m u(t/ρ))

dt.

(57)

(58)

We show that F is not analytic at zero by estimating (Dtk F )(0). Note that we have the identity G(ρ, η) = G(−ρ, −η) for all ρ = 0, η ∈ R. We now introduce I1 and I2 . We define ∞ +∞ I1 = eiρη ρ mk G(ρ, η)−1 dη dρ −∞

0

and

∞

I2 = 0

M|η|≥ρ m−1

Note that

eiρη ρ mk G(ρ, η)−1 dη dρ.

Dtk F (0) = I1 − I2 .

The estimate for I1 The argument is based on that presented in [13]. In that paper, the study is based on the function +∞ +∞ m 2(ηs−s m ) N (η) = e ds = e−2(ηs+s ) ds, (59) −∞

−∞

where m ≥ 2 is even. We know that N is an entire, even function of η. We also know that N has zeros only when m ≥ 4. In this case, all zeros are on the imaginary axis. This all follows from classical work of G. Pólya [28]. Since N (0) = 0, it follows that there exists R > 0 such that ±iR are the two zeros of N closest to the origin. We also know that ±iR are simple roots. This is a nontrivial fact that we prove in Section 11. We have the following. lemma 12 Let ±iR be the two zeros of N closest to the origin. Then there exists C > 0 such that for all k we have C(2k)! 2k 1 ∂ η N (0) ≥ R 2k .

569


Proof There exists f entire and even such that N (z) = (z2 + R 2 )f (z) for all z ∈ C. It follows that there exists an R1 > R such that f (z) = 0 for all z such that |z| ≤ R1 . Since 1/f is holomorphic near the closed disk |z| ≤ R1 , there exists a C > 0, depending on f , such that 2k 1 C(2k)! ∂ z f (0) ≤ R 2k d

1

z2j

for all k. Now, let P2d (z) = j =0 a2j be the Taylor polynomial of 1/f , centered at the origin, of order 2d. We choose d conveniently. Note that 1 (−1)k (2k)! (0) = . ∂z2k 2 2 z +R R 2k+2 Hence it follows that if k ≥ d, we have P2d (z) (−1)k (2k)!P2d (iR) 2k ∂z . (0) = z2 + R 2 R 2k+2 Now it follows that if k ≥ d, we have ∞ C(2k)! 2k 1 1 R 2j ∂ . z z2 + R 2 f − P2d (0) ≤ R 2k+2 R1 j =d+1

Combining the above, we see that if k ≥ d, we have 2j ∞ 2k 1 (2k)! R ∂ . z N (0) ≥ R 2k+2 P2d (iR) − C R1 j = d+1 Now the lemma follows since lim

d→∞

2j ∞ 1 R P2d (iR) − C > 0. = f (iR) R1 j =d+1

lemma 13 For every integer k, we have

I1 = πDηmk N

−1

(0).

Proof Since G is an even function of (ρ, η), we have, after integrating by parts, +∞ +∞ 1 eiρη Dηmk G−1 (ρ, η) dη dρ. I1 = 2 −∞ −∞ Let f be a smooth function with compact support such that f (0) = 1. Then we have

570


+∞ +∞ 1 I1 = lim f (9ρ) eiρη Dηmk G−1 (ρ, η) dη dρ 2 9→0 −∞ −∞ +∞ +∞ ρ 1 lim , 9η dη dρ f (ρ) eiρη Dηmk G−1 = 2 9→0 −∞ 9 −∞ +∞ +∞ −1 ρ 1 iρη mk lim = , 9η dρ dη f (ρ)e Dη G 2 9→0 −∞ −∞ 9 +∞ +∞ ρ 1 , 9η dρ dη f (ρ)eiρη lim Dηmk G−1 = 9→0 2 −∞ −∞ 9 +∞ +∞ 1 f (ρ)eiρη Dηmk N −1 (0) dρ dη = 2 −∞ −∞ = π Dηmk N −1 (0). To justify taking the limit inside the integral sign, we introduce F9 (η), which we define as follows: +∞ −1 ρ iρη mk , 9η dρ. f (ρ)e Dη G F9 (η) = 9 −∞ After integrating by parts twice, we have, for all η = 0, −1 +∞ iρη 2 ρ F9 (η) = 2 , 9η dρ. e ∂ρ f (ρ)Dηmk (G−1 ) 9 η −∞ It follows from Lemmas 14 and 15 that for each k = 1, 2 . . . there exists Ck > 0 such that we have j ∂ D mk G−1 ρ , 9η ≤ Ck ρ η 9 for 9 > 0, j = 0, 1, 2, and for all ρ, η. Hence we have Ck > 0, independent of 9, such that F9 (η) ≤ C 1 k 1 + η2 for all η ∈ R. Since f has compact support, independent of 9, we may take the limit inside the integral using the dominated convergence theorem. Combining Lemmas 12 and 13, we have the following estimate for I1 : There exists a constant C > 0 such that for all k we have (mk)! |I1 | ≥ C mk . (60) R The estimate for I2 We begin by estimating G(ρ, η) for η < 0, the arguments being similar for η > 0.

571


Define f (t) = −2(ηt + t m u(t/ρ)). Since we may assume u ≤ 3/2, we have f (t) ≥ 2|η|t − 3t m . Note that the function |η|t − 3t m has a critical point at t< = (|η|/(3m))1/(m−1) . We divide the argument into two cases. First, assume that t< ≤ ρδ. Assume that t< /2 ≤ t ≤ t< with t< ≥ 2. Since |η|t − 3t m is increasing on this interval, we have m 1 t< |η| m/(m−1) m − m . ≥ |η| + 3 f (t) ≥ |η| + |η|t< /2 − 3 2 3m 2 2 Hence we have G(ρ, η) ≥

t< |η|+C(|η|/(3m))m/(m−1) m e ≥ Cρe|η|+Cρ , 2

as long as M|η| ≥ ρ m−1 . Now assume that t< ≥ ρδ. This implies that the function |η|t − 3t m is increasing on the interval 0 ≤ t ≤ ρδ. If we assume that ρδ/2 ≤ t ≤ ρδ with t ≥ 1, we have m ρδ f (t) ≥ |η| + |η|ρδ/2 − 3 . 2 The fact that t< ≥ ρδ implies that |η| ≥ 3m(ρδ)m−1 . Hence it follows that 1 m m f (t) ≥ |η| + 3(ρδ) − m . 2 2 We obtain an estimate similar to that in the first case: ρδ |η|+Cρ m G(ρ, η) ≥ e . 2 We now can estimate I2 . We have ∞ +∞ mk −Cρ m ρ e dρ e−|η| dη. |I2 | ≤ C −∞

0

Hence there exists C > 0 such that for all k we have |I2 | ≤ C k k!.

(61)

Now if we combine estimates (60) and (61), it follows that for large k we have C(mk)! . R mk Hence F is not analytic near t = 0. Indeed, F is no better than Gevrey class m. |(Dtk F )(0)| ≥

10. The method of stationary phase In this section we discuss the precise version of the stationary phase formula that we need. We begin with an elementary, but very precise, result.

572


lemma 14 Let f and g be smooth, complex-valued functions defined near the closed interval [−p, p]. We define p e−f (t) g(t) dt. I (f, g, p) = −p

Assume that there exists a ∈ C, a > 0, and C1 > 0 such that |f (t) − at 2 | ≤ C1 |t|3 , and C1 p ≤ Let C2 > 0 satisfy

|t| ≤ p,

a . 2

|g(t) − g(0)| ≤ C2 |t|,

|t| ≤ p.

Then there exists a universal constant C > 0 such that I (f, g, p) − g(0) π ≤ CC2 + |g(0)|CC1 + |g(0)| 2π e− ap2 /2 . a a a ( a)2 Proof We write I (f, g, p) = I1 + I2 + I3 − I4 , where p I1 = e−f (t) g(t) − g(0) dt, −p p −f (t) 2 I2 = g(0) − e−at dt, e −p ∞

I3 = g(0) I4 = g(0)

−∞

e−at dt,

|t|≥p

2

e−at dt. 2

√ First, observe that when a > 0, we have I3 = g(0) π/a. When a > 0, this formula persists by analytic continuation. Note that if |t| ≥ p, it follows that (at 2 ) ≥ (a)p 2 /2 + (a)t 2 /2. Hence it follows that +∞ −ap 2 /2 −at 2 /2 −ap 2 /2 2π . e dt = |g(0)|e |I4 | ≤ |g(0)|e a −∞ Our hypothesis implies that if |t| ≤ p, then we have f (t) ≥ (a)t 2 /2. Hence p 2C2 ∞ −s 2 2 e−at /2 |t| dt ≤ e |s| ds. |I1 | ≤ C2 a −∞ −p It remains to estimate I2 . First, observe that

573


e

−f (t)

−e

−at 2

d −(λf (t)+(1−λ)at 2 ) e dλ 0 dλ 1 2 2 = at 2 − f (t) e−at e−λ(f (t)−at ) dλ.

=

1

0

Now, using the hypothesis, we obtain p 1 2 3 |t|3 e−at +C1 |t| dλ dt |I2 | ≤ |g(0)|C1 ≤ |g(0)|C1

−p p −p

0

|t|3 e−at

2 /2

dt ≤ |g(0)|C1

2 a

2

∞ −∞

e−s |s|3 ds. 2

We wish to estimate G(ρ, ζ ), which is defined in (58). We assume that ϕ(s) = u(s)s m , u(0) = 1, 1/2 ≤ u(s) ≤ 3/2 for |s| ≤ 2δ, s ∈ R. We also assume that u is holomorphic for s ∈ C, |s| ≤ 2δ. Throughout, ζ = η + iγ is a complex variable, ζ = η, ζ = γ . We assume that there exist M > 0 (large) and 9 > 0 (small) such that |η| ≤ ρ m−1 /M and |γ | ≤ 9 |η|. We define K as follows: t m K(t, ρ, ζ ) = 2 ζ t + t u , |t| ≤ 2ρδ. ρ We define t< (ρ, ζ ) to be the solution to the equation ∂K t< , ρ, ζ = 0. ∂t Note that ζ 1/(m−1) t< (ρ, ζ ) = − 1 + O(δ) . (62) m We always take the root closest to the real axis, so that t< (ρ, η) is real. Also note that ζ 1/(m−1) η 1/(m−1) − = − (63) 1 + O(9 ) . m m lemma 15 There exist M0 > 0 and 90 > 0 such that if M > M0 and 0 < 9 < 90 and if |η| ≤ ρ m−1 /M and |γ | ≤ 9|η|, then 1 2π −K(t< ,ρ,ζ ) G(ρ, ζ ) = e +O . Ktt (t< , ρ, ζ ) Ktt (t< , ρ, ζ ) Furthermore, we have

m/(m−1) η −K(t< , ρ, ζ ) = 2(m − 1) 1 + O(9) 1 + O(δ) m

and

574


2π = Ktt (t< , ρ, ζ )

(2−m)/(2(m−1)) η π 1 + O(9) 1 + O(δ) . m(m − 1) m

Proof Let q(t, ρ, ζ ) be holomorphic for |t| ≤ 2ρδ, |γ | ≤ 9 |η| for each ρ > 0. Assume that there exists C2 > 0 such that q(t, ρ, ζ ) − q t< (ρ, ζ ), ρ, ζ ≤ C2 t − t< (ρ, ζ ). Define

I=

ρδ

−ρδ

eK(t< (ρ,ζ ),ρ,ζ )−K(t,ρ,ζ ) q(t, ρ, ζ ) dt

and I1 =

|t−t< (ρ,η)|≤9|t< (ρ,η)|

eK(t< (ρ,ζ ),ρ,ζ )−K(t,ρ,ζ ) q(t, ρ, ζ ) dt.

Let I2 = I − I1 . Note that if M > 0 is large enough, we have |t< (ρ, ζ )| ≤ ρδ/2. Hence the major contribution comes from I1 . We begin by estimating I2 . Note that K(t, ρ, η) has an absolute minimum at t< (ρ, η). Hence, if |t − t< (ρ, η)| ≥ 9|t< (ρ, η)|, we have K(t, ρ, η) ≥ K t< (ρ, η) ± 9t< (ρ, η), ρ, η . Now, using (62) and (63), we have K(t< (ρ, ζ ), ρ, ζ ) − K(t, ρ, ζ ) m/(m−1) η ≤ 2 (1 + O(δ)) (1 + O(9 ))(1 − m) − (1 ± 9)m + m(1 ± 9) . m It follows that if 9 is small enough with 9 = O(9 3 ), there exists C > 0 such that we have m/(m−1) 2 η K t< (ρ, ζ ), ρ, ζ − K(t, ρ, ζ ) ≤ −9 C . m Hence there exists C > 0 such that |I2 | ≤ C e−9

2 C| η |m/(m−1) m

.

Hence the main contribution comes from I1 . To estimate I1 , we make the shift of contour t → t + t< (ρ, ζ ), where |t| ≤ 9|t< (ρ, η)|. We have eK(t< (ρ,ζ ),ρ,ζ )−K(t+t< (ρ,ζ ),ρ,ζ ) q t + t< (ρ, ζ ), ρ, ζ dt + R, I1 = |t|≤9|t< (ρ,η)|

where R is an integral over the paths t = ts± , 0 ≤ s ≤ 1, defined by

575


ts± = s t< (ρ, ζ ) ± 9|t< (ρ, η)| + (1 − s) t< (ρ, η) ± 9|t< (ρ, η)| . Observe that ts±

η 1/(m−1) = − (1 ± 9) 1 + O(9 ) 1 + O(δ) . m

It follows that if 9 , 9 are small enough, there exists C > 0 such that we have m/(m−1) ± 2 η . K t< (ρ, ζ ), ρ, ζ − K ts , ρ, ζ ≤ −9 C m Hence there exist C > 0, C > 0 such that |R| ≤ C e−9

2 C|η/m|m/(m−1)

.

Now it remains to study I1 − R, which can be estimated using Lemma 14.

11. An algorithm of Pólya lemma 16 The roots ±iR of N are simple. Proof To prove that N (iR) = 0, we use a method of Pólya [29]. Pólya developed an algorithm to compute the roots of a class of entire function assuming that the Taylor coefficients are known. Let F be an entire function of finite order, with genus p, satisfying the following conditions: (i) F has infinitely many roots, and all the roots of F are positive; (ii) F is real valued on the real line; (iii) F (0) = 1. Let 0 < α1 ≤ α2 ≤ · · · be the roots of F , and let (−1)k ak be the Taylor k k 2π i/n coefficients of F ; that is, F (z) = ∞ k=0 (−1) ak z . Taking a unit root ω = e with n > p, the power series of F (z)F (ωz) · · · F (ωn−1 z) has the following form: F (z)F (ωz) · · · F (ωn−1 z) =

∞ (−1)k an,k znk . k=0

olya’s algoThe coefficient an,k can be computed from coefficients {ak }∞ k=1 . Then P´ rithm provides the estimate an,1 1/n 1 1/n < α1 < . (64) an,1 a2n,1 We use this inequality to show that the first root of N is closer to the origin than the

576


first nonzero root of N . More precisely, we apply Pólya’s algorithm to the function m21/m−1 1/m−1 √ N 2 i z :(1/m) ∞ √ m m = e−t cos t z dt :(1/m) 0 ∞ :((2k + 1)/m) k z (−1)k = :(1/m):(2k + 1)

Fm (z) =

k=0

≡

∞

(−1)k ak zk .

k=0

It follows from Pólya [28] that Fm satisfies (i)–(iii). The genus of Fm is zero since the order of N is m/(m − 1) ∈ (1, 2). So we can apply (64) with n = 1 to obtain a1,1 α1 < , a2,1 where a1,1

:(3/m) , = 2:(1/m)

a2,1 =

Next we consider the function m Gm (z) = :(3/m)

:(3/m) 2:(1/m)

∞ 0

2 −2

√ sin(t z) 2 −t m t e dt. √ t z

The function Gm (z) is the derivative Fm (z) =

:(5/m) . :(1/m):(5)

√ m22/m−3 m √ i N 21/m−1 i z = − :(1/m) z 2:(1/m)

∞

t 2 e−t

0

m

√ sin(t z) dt √ t z

with the normalization Gm (0) = 1. Again, it follows from Pólya [28] that Gm satisfies (i)–(iii) and that the genus of Gm is zero. Let 0 < β1 ≤ β2 ≤ · · · be the roots of Gm , and let (−1)k bk be the Taylor coefficients of Gm ; that is, Gm (z) =

∞ ∞ :((2k + 3)/m) k z ≡ (−1)k (−1)k bk zk . :(3/m):(2k + 2) k=0

k=0

Applying (64) with n = 2, we get the lower bound 1 1/2 < β1 , b2,1 where

b2,1 =

:(5/m) :(3/m):(4)

2 −2

:(7/m) . :(3/m):(6)


To prove that α1 = β1 , it is sufficient to show that a1,1 1 1/2 < a2,1 b2,1

577

(65)

for all even integers m ≥ 4. After a short calculation one can see that the last inequality is equivalent to 3 2 1 3 5 3 1 7 10: : : < 15: . (66) +: : m m m m m m Using the logarithmic convexity of :, we have the estimate (1/x)e−Cx ≤ :(x) ≤ 1/x for all x ∈ (0, 1], with the Euler constant C. Therefore we have the upper bound (2/3)m3 for the left-hand side of (66), and we have the lower bound m3 (5/9 + 1/7)e−9C/m for the right-hand side of (66) for all m ≥ 7. Since the inequality 2 3 5 1 −9C/m m < m3 + e 3 9 7 is valid for m ≥ 114, we conclude that α1 (m) = β1 (m), provided that m ≥ 114. The remaining cases are verified by the computer program Mathematica. Taking n = 2 in (64), we have the estimate a2,1 1/2 1 1/2 , < β1 α1 < a4,1 b2,1 with "2 :(3/m) 2 :(5/m) −2 2:(1/m) :(1/m):(5) 2 :(9/m) :(5/m) :(3/m):(7/m) −4 . −2 +4 2 :(1/m):(5) :(1/m):(9) :(1/m) :(3):(7)

! a4,1 =

Acknowledgments. G. Francsics is grateful to the Schrödinger Institute, Vienna, and the Courant Institute of Mathematical Sciences, New York, where a portion of this work was completed.

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L. BOUTET DE MONVEL, “Singularity of the Bergman kernel” in Complex Geometry

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Francsics Department of Mathematics, Columbia University, New York, New York 10027, USA; [email protected] Hanges Department of Mathematics, Lehman College, City University of New York, Bronx, New York 10468, USA; [email protected]

THE WEIL-PETERSSON AND THURSTON SYMPLECTIC FORMS ¨ YAS¸AR SOZEN and FRANCIS BONAHON

Abstract We consider the Weil-Petersson form on the Teichmüller space T (S) of a surface S of genus at least 2, and we compute it in terms of the shearing coordinates for T (S) associated to a geodesic lamination λ on S. In the corresponding expression, the Weil-Petersson form coincides with Thurston’s intersection form on the space of transverse cocycles for λ. Let S be a closed oriented surface of genus at least 2. The Teichmüller space T (S) is the space of isotopy classes of complex structures on S. The space T (S) is a topological manifold, but, from its origins in complex geometry, it inherits two additional features, a structure as a complex manifold and a natural hermitian metric called the Weil-Petersson form. Recall that a hermitian form on a complex manifold is completely determined by its imaginary part, which is a real differential 2-form. A. Weil and L. Ahlfors showed that the Weil-Petersson 2-form is closed [Ah], [We3], namely, that the Weil-Petersson metric is a Kähler metric on the complex manifold T (S). W. Goldman discovered that the Weil-Petersson 2-form has a very topological interpretation and can be expressed as a cup product in a twisted cohomology group [Go]. In this paper we provide another topological expression for the Weil-Petersson form, in terms of the shearing coordinates for Teichmüller space associated to a maximal geodesic lamination on the surface. More precisely, if we fix a maximal geodesic lamination λ, the shearing coordinates for Teichmüller space identify T (S) with an open convex cone in the finite-dimensional vector space H (λ) of all “transverse cocycles” for λ. These shearing coordinates were introduced and developed in [Bo1], but they already appear in dual form in [Th]. This embedding of T (S) in H (λ) identifies each tangent space Tρ T (S) with the vector space H (λ). The main motivation for these shearing coordinates is that they are well adapted to the geodesic lamination λ and consequently provide useful tools for various problems in low-dimensional topology and geometry DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 3, Received 1 June 2000. Revision received 13 September 2000. 2000 Mathematics Subject Classification. Primary 32G15; Secondary 57R30.

581

¨ SOZEN AND BONAHON

582

(see, e.g., [Th] or [Bo3]). At each ρ ∈ T (S), the imaginary part of the Weil-Petersson form provides an antisymmetric bilinear form ω : Tρ T (S) × Tρ T (S) −→ R. On the other hand, H (λ) also has an antisymmetric bilinear form, namely, the Thurston intersection form τ : H (λ) × H (λ) −→ R, which essentially is an algebraic intersection number. The main result of this paper is the following theorem. theorem 1 Let S be a closed oriented surface of genus at least 2, and let λ be a maximal geodesic lamination on the surface. Then, for the identification Tρ T (S) ∼ = H (λ) provided by the shearing coordinates for T (S) associated to λ, the Weil-Petersson form ω is equal to the Thurston intersection form τ up to a multiplicative constant. The multiplicative constant in the statement of Theorem 1 depends on the conventions for the definition of the Weil-Petersson form. See Proposition 7 for a precise statement based on the conventions of §1. Since the Thurston form is constant in the shearing coordinates, it is closed as a differential form on H (λ). A corollary of Theorem 1 is that it provides another proof that the Weil-Petersson form is closed (cf. [Ah], [We3], [Go]). In the special case where the maximal geodesic lamination λ is obtained from a pair of pants decomposition of S by adding finitely many geodesics spiralling along these closed geodesics, the shearing coordinates are just linear combinations of the Fenchel-Nielsen coordinates associated to this pair of pants decomposition. In this context, Theorem 1 coincides with the Wolpert formula [Wo2] expressing the WeilPetersson form in these Fenchel-Nielsen coordinates. Another special case is that of transverse cocycles which are transverse measures for the geodesic lamination. Because the Thurston intersection form puts in duality shearing coordinates and length functions [Bo1], the restriction of Theorem 1 to this case is just Wolpert’s result [Wo2] that the Weil-Petersson form establishes a duality between earthquake vector fields and length functions. However, Theorem 1 is much more general since λ admits many fewer transverse measures than transverse cocycles. The fact that transverse cocycles are analytically much more complex than transverse measures [Bo2] also makes the proof technically more difficult.

THE WEIL-PETERSSON AND THURSTON SYMPLECTIC FORMS

583

There is a version of Theorem 1 for surfaces with punctures. Namely, if S is obtained by removing finitely many points from a compact surface, and if we are given a maximal geodesic lamination λ in S, the associated shearing coordinates define an open embedding of the Teichmüller space T (S) into the space H0 (λ) ⊂ H (λ) consisting of those transverse cocycles that satisfy a certain “cusp condition” at the punctures (see [Bo1, §12.3]). In this case the Thurston intersection form is a welldefined bilinear form on H0 (λ), and the proof of Theorem 1 automatically extends to this context to show that the Thurston and Weil-Petersson forms on Tρ T (S) ∼ = H0 (λ) coincide up to a multiplicative constant. In the special case where λ consists of finitely many leaves going from puncture to puncture, this was established in dual form in [Pe] and [PaP]. 1. The Weil-Petersson symplectic form Throughout the paper, S is a closed oriented surface of genus at least 2. Let S˜ denote the universal covering of S. A complex structure on S lifts to a complex structure on S˜ for which, by the uniformization theorem and because S has genus at least 2, S˜ becomes biholomorphic to the upper half-space H2 ⊂ C. Since every biholomorphic homeomorphism of H2 is of the form z → az + b/cz + d with a, b, c, d ∈ R, this defines a representation of the fundamental group π1 (S) into PSL2 (R) which is faithful, discrete, and well defined up to conjugation by an element of PSL2 (R). This identifies T (S) to the set of all conjugacy classes of discrete faithful representations of π1 (S) into PSL2 (R). If R denotes the space of conjugacy classes of all representations of π1 (S) into PSL2 (R), it is well known that the image of the above embedding T (S) → R is open (see [We1], [Ra]). Fix an element ρ ∈ T (S) ⊂ R . The standard theory of deformation of representations (see, e.g., [We2], [Ra], [Go]) identifies the tangent space Tρ T (S) = Tρ R to the first cohomology space H 1 (S, Adρ ) of the surface with coefficients in the Lie algebra sl2 (R) of PSL2 (R) twisted by the adjoint representation Adρ : π1 (S) → Aut(sl2 (R)), defined as follows. First of all, sl2 (R) consists of all real 2 × 2 matrices with trace zero, and, in this case, the adjoint representation is such that, for γ ∈ π1 (S), Adρ (γ ) is the conjugation automorphism u → ρ(γ )uρ(γ )−1 . The action of π1 (S) on the universal covering S˜ ˜ Z) into Z[π1 (S)]-modules. Similarly, turns the groups of the chain complex C∗ (S, the representation Adρ makes sl2 (R) a Z[π1 (S)]-module. The twisted cohomology modules H ∗ (S, Ad(ρ)) are defined as the homology modules of the complex ˜ Z , sl2 (R) . C ∗ (S, Adρ ) = HomZ[π1 (S)] C∗ S, Another way to rephrase this is that C n (S, Adρ ) consists of the group homomorphisms


584

˜ Z) → sl2 (R) that commute with the actions of π1 (S). Cn (S, The Cartan-Killing bilinear form B : sl2 (R)×sl2 (R) → R is such that B(u, v) = 4 Tr(uv). It is preserved by the adjoint representation and therefore enables us to define a cup product : C 1 (S, Adρ ) × C 1 (S, Adρ ) −→ C 2 (S, R), which induces an antisymmetric bilinear form ω : H 1 (S, Adρ ) × H 1 (S, Adρ ) −→ H 2 (S, R) ∼ = R, ∼ R is defined by evaluation on the fundamental where the isomorphism H 2 (S, R) = class of the oriented surface S. Goldman [Go] showed that, for the isomorphism Tρ T (S) ∼ = H 1 (S, Adρ ), this form coincides with the Weil-Petersson form of T (S) up to a multiplicative constant. 2. The Thurston intersection form Let the surface S be endowed with a hyperbolic metric m0 , namely, with a metric of constant curvature −1. A geodesic lamination is a closed subset λ of S which can be decomposed as a union of disjoint simple geodesics called its leaves. See, for instance, [PeH] and [CEG] for basic facts about geodesic laminations. An important property is that a geodesic lamination is actually a topological object, independent of the hyperbolic metric m0 which we put on the surface. If m1 is another hyperbolic metric on S, every m0 -geodesic lamination λ0 can be deformed to a unique m1 -geodesic lamination λ1 , and the converse is also true, so that there is a natural one-to-one correspondence between m0 -geodesic laminations and m1 -geodesic laminations. A geodesic lamination λ is maximal if it is maximal for inclusion among all geodesic laminations; this is equivalent to the property that the complement S − λ consists of finitely many disjoint ideal triangles, namely, triangles isometric to an infinite triangle in hyperbolic space H2 whose three vertices are on the circle at infinity. A transverse cocycle for the geodesic lamination λ is a function σ which assigns a number σ (k) ∈ R to each arc k ⊂ S which is transverse to (the leaves of) λ and which satisfies the following two conditions: (i) σ is finitely additive; namely, σ (k) = σ (k1 )+σ (k2 ) whenever the arc k transverse to λ is decomposed into two arcs k1 , k2 with disjoint interiors; (ii) σ (k) = σ (k ) whenever the arcs k and k are homotopic through a family of arcs that are all transverse to λ. The transverse cocycles for the geodesic lamination λ form a vector space H (λ) whose dimension is finite and can be explicitly computed from the topology of λ (see [Bo2]). In particular, when λ is maximal, which is the case of interest here, H (λ) is isomorphic to R3|χ (S)| , where χ(S) is the Euler characteristic of S.


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The computation of the space H (λ) uses a train track carrying the lamination λ. A ( fattened) train track on the surface S is a family of finitely many “long” rectangles e1 , . . . , en which are foliated by arcs parallel to the “short” sides and which meet only along arcs (possibly reduced to a point) contained in their short sides. In addition, a train track must satisfy the following: (i) each point of the “short” side of a rectangle also belongs to another rectangle, and each component of the union of the short sides of all rectangles is an arc, as opposed to a closed curve; (ii) note that the closure S − of the complement S − has a certain number of “spikes” corresponding to the points where at least 3 rectangles meet; we require that no component of S − be a disc with zero, 1, or 2 spikes or an annulus with no spike. The rectangles are called the edges of . The foliations of the edges of induce a foliation of , whose leaves are the ties of the train track. The finitely many ties where several edges meet are the switches of the train track . A tie that is not a switch is generic. A geodesic lamination λ is carried by the train track if it is contained in the interior of and its leaves are transverse to the ties of . For a given geodesic lamination λ, there are several constructions that provide a train track carrying λ (see, e.g., [PeH], [CEG]). For a train track , let W () be the vector space of all edge weight systems for , namely, maps a assigning a weight a(e) ∈ R to each edge e of and satisfying, for each switch s of , the switch relation p i=1

a(ei ) =

p+q

a(ej ),

j =p+1

where e1 , . . . , ep are the edges adjacent to one side of the switch s and ep+1 , . . . , ep+q are the edges adjacent to the other side. If the geodesic lamination λ is carried by the train track , a transverse cocycle σ ∈ H (λ) defines an edge weight system aσ ∈ W () by the property that aσ (e) = σ (ke ), where ke is an arbitrary tie of e. The switch relations are immediate consequences of the finite additivity of σ . This defines a map H (λ) → W () which can be shown to be injective [Bo2]. In addition, this map gives an isomorphism H (λ) ∼ = W (), provided that snugly carries λ, a technical condition that is always realized when the geodesic lamination λ is maximal. For convenience in the exposition, we can also arrange that the train track be generic in the sense that each switch is adjacent to exactly 3 edges. Hence, at each switch s of , we have one incoming edge esin touching s on one side and two right outgoing edges es and esleft touching s on the other side where, as seen from the


586

right

incoming edge esin and for the orientation of the surface S, es branches out to the left right and es branches out to the left. The Thurston intersection form on W () is the bilinear antisymmetric form τ : W () × W () → R defined by τ (a, b) =

right 1 right left a es b es − a esleft b es , 2 s

where the sum is over all switches s of and where a(e), b(e) are the weights associated by a, b ∈ W () to the edge e of . Using the isomorphism H (λ) ∼ = W (), this bilinear form induces the Thurston intersection form τ : H (λ) × H (λ) → R defined by τ (σ1 , σ2 ) =

right 1 right left σ1 e s σ2 es − σ1 esleft σ2 es , 2 s

where σ1 (e), σ2 (e) ∈ R are the weights associated by the transverse cocycles σ1 , σ2 ∈ H (λ) to the edge e of the train track . It has been shown (see [PeH, §3.2], [Bo1, §3]) that this intersection form on H (λ) is independent of the train track snugly carrying the geodesic lamination λ. In particular, see [Bo1, §3] for an interpretation of τ (σ1 , σ2 ) as an algebraic intersection number in a homology group with twisted coefficients. 3. Shearing coordinates Let λ be a maximal geodesic lamination in the surface S. The shearing coordinates for Teichmüller space, as developed in [Bo1], define a differentiable embedding ϕλ : T (S) → H (λ) from the Teichmüller space T (S) into the space H (λ) of all transverse cocycles for λ. For ρ ∈ T (S), the transverse cocycle ϕλ (ρ) associates to each transverse arc k a number ϕλ (ρ)(k) which, intuitively, measures the “shift to the left” between the two ideal triangles in S = H2 /ρ(π1 (S)) corresponding to the components of S − λ that contain the end points of k. The precise definition of ϕλ can be somewhat technical, but we only need to understand its tangent map. For this it is convenient to lift the situation to the universal covering S˜ of S. Fix an isometric identification between S˜ endowed with the hyperbolic metric corresponding to ρ ∈ T (S) and the hyperbolic space H2 , and choose the geodesic lamination λ as ˜ If k˜ is an arc transverse geodesic for this metric. Let λ˜ be the preimage of λ in S. ˜ = σ (k), where k is the projection of k. ˜ ˜ and if σ ∈ H (λ), we define σ (k) to λ, At ρ ∈ T (S), the tangent map of the local diffeomorphism ϕλ : T (S) → H (λ) gives an isomorphism Tρ ϕλ between Tρ T (S) ∼ = H 1 (S, Ad(ρ)) and Tϕλ (ρ) H (λ) = H (λ). If we differentiate the explicit formula for ϕλ−1 given in [Bo1, §5], we obtain

587


the following expression for Tρ ϕλ−1 (σ ) with σ ∈ H (λ) considered as a vector tangent to H (λ) at ϕλ (ρ). lemma 2 If σ ∈ H (λ) is a transverse cocycle for the maximal geodesic lamination λ, the element Tρ ϕλ−1 (σ ) ∈ Tρ T (S) ∼ = H 1 (S, Ad(ρ)) is represented by a cocycle uσ ∈ 1 C (S, Adρ ) such that, for every oriented arc k˜ transverse to λ˜ , ˜ = σ (k)t ˜ − + ˜d ) t − − t + , σ ( k uσ (k) g g g d+

d=d + ,d −

d

d

where the sum is over all components d of k˜ − λ˜ which are distinct from the components d + and d − containing, respectively, the positive and the negative end points of ˜ where k˜d is a subarc of k˜ joining the negative end point of k˜ to an arbitrary point k, in d, where gd+ and gd− are the leaves of λ˜ passing, respectively, through the positive ˜ and where tg ∈ sl2 (R) and negative end points of d and are oriented to the left of k, is the infinitesimal hyperbolic translation along the oriented geodesic g of S˜ ∼ = H2 . Note that the sum in the formula of Lemma 2 usually has infinitely many terms. The convergence is proved in [Bo1, §5]; compare also the gap formula in [Bo2, Theorem 10]. 4. A first expression for the Weil-Petersson form In this section we begin to compute the Weil-Petersson form ω : H 1 (S, Adρ ) × H 1 (S, Adρ ) −→ R ∼ H (λ) provided by ϕλ . The formula in terms of the identification H 1 (S, Ad(ρ)) = that we obtain here is reminiscent of that for the Thurston intersection form but not quite equal to it. We complete the proof of Theorem 1 by a limiting process in Section 5. Let be a generic train track carrying the complete geodesic lamination λ. Recall that being generic means that, at each switch s, we have one incoming edge esin right and two outgoing edges esleft and es diverging, respectively, to the left and to the ˜ denote right as seen from the incoming edge esin and for the orientation of S. Let ˜ the preimage of in S. Consider a transverse cocycle σ ∈ H (λ) and the cocycle uσ ∈ C 1 (S, Adρ ) ˜ where we have specified an associated to σ by Lemma 2. If e˜ is an edge of ˜ = uσ (k) ∈ sl2 (R), where k is any (oriented) tie orientation of its ties, define uσ (e) of e. ˜ This is independent of k but changes by multiplication by −1 if we reverse the orientation of the ties of e. ˜


588

If s is a switch of , choose an orientation for s and lift it to an oriented switch s˜ ˜ The orientation of s˜ gives an orientation for the ties of the incoming ˜ of in S. right in . Because the Cartan-Killing form B edge es˜ and of the outgoing edges esleft ˜ , es˜ right

)) ∈ R is invariant under the adjoint representation, the quantity B(uσ (esleft ˜ ), uσ (es˜ is independent of the lift s˜ of s, as well as of the choice of the orientation of s. We right right )). consequently write B(uσ (esleft ), uσ (es )) = B(uσ (esleft ˜ ), uσ (es˜ lemma 3 If σ1 , σ2 ∈ H (λ), then 1 right right ω uσ1 , uσ2 = , uσ2 esleft − B uσ2 es , uσ1 esleft , B u σ1 e s 2 s where s ranges over all switches of . Proof We first compute the cup product : C 1 (S, Adρ ) × C 1 (S, Adρ ) −→ C 2 (S, R) defined using the Cartan-Killing form B : sl2 (R) × sl2 (R) → R. By definition of the cup product, if u, v ∈ C 1 (S, Adρ ) and ( is a 2-simplex in S, 2 0 1 2 0 ˜ ,v ( ˜ ˜ ,u ( ˜ (u v)(() = B u ( −B v ( , 2 ˜ 2 is the front edge of a lift ( ˜ of ( and ( ˜ 0 is the back edge of (. ˜ If, in where (

addition, du = dv = 0, then (u v)(() depends only on the image of ( and on its orientation and not on the oriented identification between this image and the standard 2-simplex. To evaluate the 2-cocycle u v ∈ C 2 (S, R) on S, we triangulate the surface as follows. For each switch s of , we choose in the incoming edge esin an arc s transverse to λ with the same end points as s but with interior disjoint from s. Then s ∪ s right bounds in esin a triangle (s whose edges are s , s ∩ esleft , and s ∩ es (see Figure 1). The complement in of all these triangles (s is a disjoint union of rectangles. Splitting each rectangle into two triangles by a diagonal transverse to λ, we obtain a triangulation of whose edges are all transverse to λ. We arbitrarily extend this triangulation to a triangulation of S. By definition, uσ1 uσ2 ((), ω uσ1 , uσ2 = uσ1 uσ2 ([S]) = (

where ( ranges over all triangles of the triangulation of S and where [S] ∈ H2 (S; Z) is the fundamental class of the oriented surface S. If at least one of the edges of the triangle ( is disjoint from λ, we can choose


589

the lamination λ

the triangle (s

a switch

an edge subdivided into two triangles by a diagonal Figure 1. A triangulation adapted to the train track

the oriented identification of ( with the standard 2-simplex so that this edge is the front edge, and it follows that the contribution of ( to ω(uσ1 , uσ2 ) is equal to zero. Therefore, only the triangles (s associated to the switches s contribute to this sum. Similarly, we can choose the identification of (s with the standard simplex so right and s ∩ esleft , orienting s to that its front and back faces, respectively, are s ∩ es right the left as seen from esin . In this case, uσ ((0s ) = uσ (esleft ) and uσ ((2s ) = uσ (es ) for every transverse cocycle σ ∈ H (λ). It follows that the contribution of (s to ω(uσ1 , uσ2 ) is equal to right 1 right B uσ1 es , uσ2 esleft − B uσ2 es , uσ1 esleft . 2 We conclude that uσ1 uσ2 ((s ) ω uσ1 , uσ2 = s

1 right right = B uσ1 es , uσ2 esleft − B uσ2 es , uσ1 esleft , 2 s where s ranges over all switches of . This completes the proof of Lemma 3. 5. Approximating λ by thin train tracks The formula of Lemma 3 already looks fairly close to that of Theorem 1. We conclude the proof of Theorem 1 by passing to the limit as approximates the lamination λ. For this we use the following technical lemmas, whose proofs can be found in [Bo1, §1]. Let be a train track carrying the geodesic lamination λ, and let k be an oriented

590


generic tie of . If d is one of the components of k − λ, then we define the divergence radius or depth r(d) of d as follows. To avoid any ambiguity, let us lift the situation ˜ of and λ˜ of λ, and lift k to the universal covering S˜ of S. Consider the preimages ˜ By convention, r(d) is 1 if d contains one of the end points to an oriented tie k˜ of . of k. Otherwise, there are two leaves gd+ , gd− of λ˜ passing, respectively, through the positive and the negative end points of the component d˜ of k˜ − λ˜ associated to d by the projection k˜ → k. Then r(d) is the largest integer r ≥ 1 such that gd+ and gd− successively cross the same sequence of edges e−r+1 , e−r+2 , . . . , e0 , . . . , er−2 , er−1 ˜ Note that r(d) is independent of the ˜ where e0 is the edge of ˜ containing k. of , choice of the lift k˜ of k. For the following three lemmas, we fix a hyperbolic metric m on the closed surface S, and we consider an m-geodesic lamination λ carried by the fattened train track . Let k be a generic tie of . lemma 4 (Exponential decay for gap lengths) There exist constants A, B > 0 such that, for every component d of k − λ with depth r(d), ,m (d) ≤ Be−Ar(d) , where the constants A and B depend only on the metric m and on the fattened train track . lemma 5 (Bounded number of gaps of given depth) There is a universal constant C, depending only on the topology of S, such that the number of components d of k − λ with r(d) = r is bounded by C. If, in addition, we choose an orientation for the tie k, and if σ ∈ H (λ) is a transverse cocycle for λ, the σ -height hσ (d) of the component d of k − λ is defined as hσ (d) = σ (kd ), where the subarc kd of k joins the negative end point of k to an arbitrary point of d. lemma 6 (Linear growth for gap heights) For every component d of k − λ, hσ (d) ≤ σ r(d), where σ = maxe |σ (e)| as e ranges over all edges of . Again, we refer to [Bo1, §1] for proofs of Lemmas 4, 5, and 6. Using the definitions of the previous sections, Theorem 1 follows from the following statement. proposition 7 Let S be a closed oriented surface with genus at least 2, and let λ be a maximal

591


geodesic lamination on S. Then, for every σ1 , σ2 ∈ H (λ), the quantity 1 right right ω uσ1 , uσ2 = , uσ2 esleft − B uσ2 es , uσ1 esleft B u σ1 e s 2 s is equal to 2τ (σ1 , σ2 ) =

right right left σ1 e s σ2 esleft − σ2 es σ1 e s , s

where the sums are over all switches s of a generic train track carrying λ. Proof The strategy is the following: we prove that the train track can be chosen so that these quantities are arbitrarily close. This shows equality since we know that these terms are independent of . We start with a generic train track 0 carrying the lamination λ. By “unzipping zippers” (see [PeH, §§1.7, 2.4]), we define train tracks R which still carry the lamination λ and are thinner and thinner. In order to estimate the growth of the weights defined on the edges of R by σ1 and σ2 , we need to do this in a carefully controlled way. Let e1 , . . . , en be the edges of 0 . For each integer R ≥ 1, we delete from 0 the components of ei − λ, i = 1, . . . , n, with depth less than or equal to R (defining the depth of a component of ei − λ as the depth of the corresponding component of ki − λ, where ki is an arbitrary tie of ei ). The complement in 0 of these components of ei − λ can be slightly enlarged to a train track R containing λ. In addition, we can arrange by a perturbation that R is generic. By Lemma 3, for every R, ω uσ1 , uσ2 − 2τ (σ1 , σ2 ) right left 1 right B uσ1 fs , uσ2 fsleft − 2σ1 fs σ2 fs = 2 s right left 1 right , uσ1 fsleft − 2σ2 fs σ1 fs , B uσ2 fs − 2 s right

where the sums are over all switches s of R and where fs , fsleft are the edges of the train track R diverging, respectively, to the right and to the left at the switch s. We show that the contribution right right left B uσ1 fs , uσ2 fsleft − 2σ1 fs σ2 fs , of each switch s of R tends to zero as R tends to infinity. As usual, we say that the quantity X is an O(Y ) if the ratio |X|/|Y | is bounded above by a constant. In what follows the constants involved depend on the surface S,


592

∼ H2 , and on the original train track 0 , but not on the number on the identification S˜ = R or on the edges of the train track R considered. lemma 8 For every edge f of R and for every transverse cocycle σ ∈ H (λ), the weight σ (f ) is an O(σ 0 R), where σ 0 denotes the maximum of |σ (e)| as e ranges over all edges of 0 . Proof of Lemma 8 Let k be an oriented generic tie of 0 which meets f . Let kf be a component of k ∩ f , and let df+ and df− be the components of k − λ that contain the positive and negative end points of kf . Note that the depths r(df+ ) and r(df− ) are both less than or equal to R by construction of R . By definition of the σ -height hσ (d), σi (f ) = σi (kf ) = hσ (df+ ) − hσ (df− ). The result then follows from Lemma 6. ˜0 For each edge e of 0 , choose a preferred lift of e to an edge e˜ of the preimage ˜ of 0 in S, independently of R. Then, for each R, we can lift each edge f of R ˜ R of R such that f˜ intersects at least one of these to an edge f˜ of the preimage ˜ preferred edges e˜ of 0 . For each geodesic g of S˜ ∼ = H2 , let tg ∈ sl2 (R) denote the infinitesimal hyperbolic translation along g. lemma 9 With the above convention for edge lifts, for any two leaves g1 , g2 of λ˜ which cross the lift f˜ of the same edge f of R , the difference tg1 − tg2 is an O(e−AR ) for some constant A > 0. Proof of Lemma 9 Among the preferred lifts of edges of 0 , let e˜ be one that is crossed by f˜. Let ke˜ be a generic tie of e. ˜ Since the geodesics g1 , g2 pass through a fixed compact subset of H2 , namely, the union of the preferred edge lifts e˜ , they stay in a compact subset of the space of geodesics of H2 , and tg1 − tg2 = O d(g1 , g2 ) , where d(g1 , g2 ) denotes the distance from g1 to g2 in the space of geodesics of H2 . Because the geodesics g1 , g2 are disjoint, a classical hyperbolic geometry estimate (see [CEG, Lemma 5.2.6]) asserts that

593


d(g1 , g2 ) = O d g1 ∩ ke˜ , g2 ∩ ke˜ , where d(g1 ∩ ke˜ , g2 ∩ ke˜ ) denotes the distance in H2 between the intersection points of g1 and g2 with ke˜ . The distance d(g1 ∩ ke˜ , g2 ∩ ke˜ ) is bounded by the length of the subarc of ke˜ joining these two points. Because λ˜ ∩ ke˜ has measure zero (see, e.g., [PeH, §1.6]), this length is equal to d l(d), where the sum is over all components d of ke˜ − λ˜

which separate g1 ∩ ke˜ from g2 ∩ ke˜ and where l(d) denotes the length of d. Identifying d with its projection to the edge e of 0 , Lemma 4 shows that l(d) = O(e−Ar(d) ) for some constant A > 0, where r(d) is the depth of the component d in 0 . By construction of R , the depth r(d) is at least R + 1. It follows that ∞ −Ar(d) −Ar(d) tg1 − tg2 = O e e =O = O e−A R r(d)>R

r=R+1 r(d)=r

A

< A, where the last equality comes from the fact that the number of d for 0 < with r(d) = r is bounded independently of r by Lemma 5. This concludes the proof of Lemma 9. lemma 10 For geodesics g1 , g2 as in Lemma 9, B(tg1 , tg2 ) = 2 + O(e−AR ) for some constant A > 0. Proof of Lemma 10 By definition of the Cartan-Killing form, B(tg1 , tg2 ) = 4 Tr(tg1 tg2 ) where Tr denotes the trace. In particular, B(tg1 , tg1 ) = 2. Since g1 and g2 stay in a compact subset of the space of geodesics of H2 , the result immediately follows from Lemma 9. lemma 11 Let a lift f˜ of the edge f of R be chosen as specified above Lemma 9, let k˜ be an oriented generic tie of f˜, and let g be a leaf of λ˜ which crosses f˜, oriented to the left ˜ Then, for every transverse cocycle σ ∈ H (λ), of k. ˜ = σ (f )tg + O σ 0 e−AR uσ (k) in sl2 (R) for some constant A > 0, where σ 0 denotes the maximum of |σ (e)| as e ranges over all edges of 0 . Proof of Lemma 11 By Lemma 2, ˜ = σ (k)t ˜ − + uσ (k) g d+

d= d + ,d −

σ (k˜d ) tg + − tg − d

d


594

in sl2 (R), where the sum is over all components d of k˜ − λ˜ ⊂ S˜ other than the components d + and d − containing, respectively, the positive and negative end points ˜ where k˜d is an arc in k˜ joining the negative end point of k˜ to an arbitrary point of k, of d and where gd+ and gd− are the leaves of λ˜ passing, respectively, through the ˜ positive and negative end points of d, oriented to the left of k. Then ˜ − σ (f )tg = σ (k) ˜ t − − tg + ˜d ) t + − t − σ ( k uσ (k) g g g d+

d

d= d + ,d −

d

˜ since σ (f ) = σ (k). For a d contributing to the right-hand sum, Lemma 6 shows that σ (k˜d ) = hσ (d) − hσ (d − ) = O σ 0 r(d) , ˜ 0 containing k˜ and where we where the σ -heights hσ are measured in the tie of − use the fact that r(d) > R ≥ r(d ). Also, by Lemma 9, tg + − tg − = O(e−Ar(d) ) if d d A > 0 is chosen small enough. It follows that −Ar(d) σ (k˜d ) t + − t − = O σ 0 r(d)e d= d + ,d −

gd

gd

= O σ 0

= O σ 0 e

d ∞

r(d)e−Ar(d)

r=R+1 r(d)=r −A R

for 0 < A < A, using Lemma 5 to guarantee that for each r the number of d with r(d) = r is uniformly bounded. ˜ We have a similar bound for the first term σ (k)(t g −+ − tg ), from which we d conclude that ˜ − σ (f )tg = O σ 0 e−A R . uσ (k) This completes the proof of Lemma 11. We are now ready to prove Proposition 7. We have shown that ω uσ1 , uσ2 − 2τ (σ1 , σ2 ) right left 1 right , uσ2 fsleft − 2σ1 fs σ2 fs B uσ1 fs = 2 s right left 1 right − , uσ1 fsleft − 2σ2 fs σ1 fs , B uσ2 fs 2 s

595

THE WEIL-PETERSSON AND THURSTON SYMPLECTIC FORMS right

where the sums are over all switches s of R and where fs and fsleft are the edges of the train track R diverging, respectively, to the right and to the left at the switch s. ˜ R which is contained in one of Fix a switch s of R . Lift s to a switch s˜ of right the preferred lifts e˜ of the edges of 0 . The three edges fsin , fs , and fsleft of R right ˜ R which meet at s˜ . which meet at s lift to the three edges f˜sin , f˜s , and f˜sleft of right right left ˜ Let g and g be two leaves of λ which cross, respectively, the edges f˜s and f˜sleft . Pick an orientation for s˜ , and orient g right and g left to the left of s˜ . By Lemma 11, right right uσ1 fs = σ1 fs tg right + O e−AR , uσ2 fsleft = σ2 fsleft tg left + O e−AR , where we have incorporated the norms σ1 0 and σ2 0 in the constants of the symbols O. Since the geodesics g right and g left stay in a fixed compact subset of H2 , we conclude that right right left B uσ1 fs , uσ2 fsleft = σ1 fs σ2 fs B tg right , tg left + O Re−AR right left = 2σ1 fs σ2 fs + O R 2 e−AR , using Lemmas 8 and 10. We conclude that the contributions right right left B uσ1 fs , uσ2 fsleft − 2σ1 fs σ2 fs and

right right left B uσ2 fs , uσ1 fsleft − 2σ2 fs σ1 fs ,

of each switch s of R to ω(uσ1 , uσ2 ) − 2τ (σ1 , σ2 ) are both O(R 2 e−AR ). A counting argument shows that the number of switches of R is constant and equal to 6|χ (S)|. This proves that ω uσ1 , uσ2 − 2τ (σ1 , σ2 ) = O R 2 e−AR . Letting R tend to infinity now shows that ω(uσ1 , uσ2 ) = 2τ (σ1 , σ2 ). This completes the proof of Proposition 7 and, therefore, of Theorem 1.

References [Ah]

L. AHLFORS, Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of

Math. (2) 74 (1961), 171–191. MR 34:4480

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[Bo1]

[Bo2] [Bo3] [CEG]

[Go] [PaP]

[Pe] [PeH] [Ra] [Th] [We1] [We2] [We3]

[Wo1] [Wo2]


F. BONAHON, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s

symplectic form, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), 233–297. MR 97i:57011 , Transverse Hölder distributions for geodesic laminations, Topology 36 (1997), 103–122. MR 97j:57015 , Variations of the boundary geometry of 3-dimensional hyperbolic convex cores, J. Differential Geom. 50 (1998), 1–24. MR 2000j:57048 R. D. CANARY, D. B. A. EPSTEIN, and P. GREEN, “Notes on notes of Thurston” in Analytical and Geometric Aspects of Hyperbolic Space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, Cambridge, 1987, 3–92. MR 89e:57008 W. M. GOLDMAN, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200–225. MR 86i:32042 A. PAPADOPOULOS and R. C. PENNER, The Weil-Petersson symplectic structure at Thurston’s boundary, Trans. Amer. Math. Soc. 335 (1993), 891–904. MR 93d:57022 R. C. PENNER, Weil-Petersson volumes, J. Differential Geom. 35 (1992), 559–608. MR 93d:32029 R. C. PENNER and J. L. HARER, Combinatorics of Train Tracks, Ann. of Math. Stud. 125, Princeton Univ. Press, Princeton, 1992. MR 94b:57018 M. S. RAGHUNATHAN, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb. 68, Springer, New York, 1972. MR 58:22394a W. P. THURSTON, Minimal stretch maps between hyperbolic surfaces, preprint, 1986, arXiv:math.GT/9801039 A. WEIL, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369–384. MR 25:1241 , Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR 30:199 , “On the moduli of Riemann surfaces” (in French) in Oeuvres Scientifiques: Collected Papers, Vol. II (1951–1964) (in French), Springer, New York, 1979, 381–389. MR 80k:01067b S. A. WOLPERT, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), 207–234. MR 85e:32028 , On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), 969–997. MR 87b:32040


597

Sözen Department of Mathematics, University of Southern California, Los Angeles, California 900891113, USA; [email protected]; current: Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA; [email protected] Bonahon Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113, USA; [email protected]

LOCAL UNIQUENESS FOR THE DIRICHLETTO-NEUMANN MAP VIA THE TWO-PLANE TRANSFORM ALLAN GREENLEAF and GUNTHER UHLMANN

Abstract We consider the Cauchy data associated to the Schrödinger equation with a potential on a bounded domain ⊂ Rn , n ≥ 3. We show that the integral of the potential over a two-plane is determined by the Cauchy data of certain exponentially growing solutions on any open subset U ⊂ ∂ which contains ∩ ∂. 0. Introduction For , a bounded domain in Rn with Lipschitz boundary, ∂, and real-valued q(x) ∈ L∞ (), let (0.1)

q : H 1/2 (∂) −→ H −1/2 (∂) be the Dirichlet-to-Neumann map associated with the operator + q on , which is defined if λ = 0 is not a Dirichlet eigenvalue for + q on . More generally, one may consider the set of Cauchy data of solutions of ( + q(x))v = 0, which is defined even if λ = 0 is a Dirichlet eigenvalue. Set ∂v 1/2 −1/2 1 CDq = v , (∂) : v ∈ H (), ( + q)v = 0 , ∈ H (∂) × H ∂ ∂n ∂ (0.2) which is a subspace of H 1/2 × H −1/2 ; if q is defined, then CDq is simply the graph of q . This paper is concerned with the problem of obtaining partial knowledge of q(x) from partial knowledge of CDq , namely, its restriction to certain “small” open subsets of the boundary. The approach taken here is to use concentrated, exponentially growing, approximate solutions to relate CDq on an open set U ⊂ ∂ to the twoplane transform of the potential q(x) on two-planes whose intersections with ∂ are contained in U . DUKE MATHEMATICAL JOURNAL c 2001 Vol. 108, No. 3, Received 21 January 2000. Revision received 11 November 2000. 2000 Mathematics Subject Classification. Primary 35R30; Secondary 44A12. Greenleaf’s work partially supported by National Science Foundation grant number DMS-9877101. Uhlmann’s work partially supported by National Science Foundation grant number DMS-9705792 and the Royal Research Fund at the University of Washington.

599

600

GREENLEAF AND UHLMANN

Let M2,n denote the (3n − 6)-dimensional Grassmannian of all affine two-planes ⊂ Rn , and let f (y) dλ (y), f ∈ L2comp (Rn ), (0.3) R2,n f () =

denote the two-plane transform on Rn (see [H1], [H2]). Here, dλ is two-dimensional Lebesgue measure on ∈ M2,n , which can be defined by 1 f (x) dx. (0.4) f, dλ = lim n−2 →0 |B (0; )| {dist(x,) 0, CDq1 and CDq2 are equal on B for some sequence of exponentially growing solutions for all surface balls B = B n (x0 ; r) ∩ ∂ ⊂ ∂, then

dist supp(q1 − q2 ), ∂ ≥ Cr 2 ; that is, q1 = q2 on the tubular neighborhood {x ∈ : dist(x, ∂) ≤ Cr 2 } of ∂ in . Remark. The conclusions of Theorems 2 and 3 can be strengthened by combining them with a result in V. Isakov [Is]. Namely, if either C in Theorem 2, or if the support and uniqueness theorems are usually stated under the assumption that f (x) is continuous, of rapid decay in the case of the support theorem, but the proofs in [H2] are easily seen to extend to the case where f (x) = q(x)χ (x) with ⊂ Rn bounded, q ∈ C(). 1 The

602


assumption of Theorem 3 holds for some r > 0, we can conclude from Theorem 2 or Theorem 3 that supp(q1 − q2 ) . By [Is, Exercise 5.7.4], based on a technique of R. Kohn and M. Vogelius [K2], this, together with the condition that q1 = q2 on some open set U ⊂ ∂, implies that q1 ≡ q2 everywhere on . We are indebted to Adrian Nachman for pointing this out to us. 1. Approximate solutions To prove Theorem 1, we first construct exponentially growing approximate solutions for ( + q)v = 0. As considered in [C], [SU1], and [SU2], let Q = ρ ∈ Cn : ρ · ρ = 0 (1.1) be the (complex) √ characteristic variety of . Each ρ ∈ Q can be written as ρ = |ρ(|ρ/|ρ|) = (1/ 2)|ρ|(ωR + iωI ) ∈ R · (S n−1 + iS n−1 ), with ωR · ωI = 0. For ρ ∈ Q, let ρ = + 2ρ · ∇. Then

ρ + q(x) = e−ρ·x ( + q(x))eρ·x ,

(1.2)

so that, with v(x) = eρ·x u(x), ( ρ + q(x))u(x) = w(x) ⇐⇒ ( + q(x))v(x) = eρ·x w(x)

(1.3)

and, in particular, ( ρ + q(x))u(x) = 0 ⇐⇒ ( + q(x))v(x) = 0.

(1.4)

Now, given a potential q(x) and a two-plane ∈ M2,n , we construct an approximate solution uapp to ( ρ + q)u = 0, supported near . theorem 4 Let be Lipschitz, and let q(x) ∈ H s () for some s > n/2. Then, √ for any 0 < β < 1/4 fixed, the following holds: ∃ > 0 such that, for any ρ = (1/ 2)|ρ|(ωR +iωI ) ∈ Q and any two-plane parallel to 0 = span{ωR , ωI }, we can find an approximate solution uapp = uapp (x, ρ, ) to ( ρ + q(x))u = 0 satisfying uapp

L 2 (R n )

and

1/2 uapp 2 λ as |ρ| −→ ∞, ( ∩ ) L ()

2 , supp uapp ⊂ x ∈ Rn : dist(x, ) ≤ |ρ|β

≤ C,

( ρ + q)uapp 2 n ≤ C . L (R ) |ρ|

(1.5) (1.6)

(1.7)

603

LOCAL UNIQUENESS FOR THE DIRICHLET-TO-NEUMANN MAP (1)

(2)

Furthermore, for any two such solutions uapp , uapp associated with possibly different potentials q1 (x), q2 (x) and with ρ1 ∈ Q, ρ2 = eiθ ρ1 or ρ2 = eiθ ρ1 ∈ Q,

(2)

(1.8) u(1) app ·, ρ1 , uapp ·, ρ2 , −→ dλ weakly as |ρ1 | −→ ∞. In fact, as is seen below, uapp = u0 + u1 , with u0 depending only on and |ρ| and satisfying (1.5). Now, we may apply the results of [SU1] and [SU2] (see also [Ha]) to find a solution u2 of ( ρ + q)u2 = −( ρ + q)uapp ∈ L2comp (Rn ) uniformly in Ht1 and with a gain of |ρ|−1 in L2t as long as |ρ| ≥ C, with C depending only on q∞ and diam(). Here Hts and L2t are the weighted versions of these spaces, as in [SU2], for some fixed −1 < t < 0. By these results and (1.7), u2 L2 ≤ C|ρ|−1− . u2 H 1 (Rn ) ≤ c( ρ + q)uapp L2 (Rn ) ≤ c|ρ|− , t

t

t+1

(The statements in [SU1], [SU2] are for q ∈ C ∞ , but the proofs are easily seen to hold if q ∈ H s () with s > n/2. Also, the weights are irrelevant since we are working on .) Thus, u = uapp + u2 = u0 + u1 + u2 is an exact solution of ( ρ + q)u = 0 on Rn , satisfying u − u0 L2 ≤ c|ρ|−

and

u2 H s ≤ |ρ|s−1− ,

∀ 0 ≤ s ≤ 1.

Finally,

Fq = vz : |z| ≥ C = eρ·x u(x, , ρ) : ρ = Re(z)ωR + i Im(z)ωI , |z| ≥ C

is the associated family of exponentially growing solutions used in the statements of the theorems. Proof of Theorem 1 We assume that q1 , q2 and ∈ M2,n , U ⊂ ∂ are as in its statement. We make (j ) use of a variant of Alessandrini’s identity [A]. For j = 1, 2, let vρj be the exact (j )

solution to ( + qj )v = 0 constructed above, so that vρj (x) = eρj ·x u(j ) (x, , ρj ), (j )

(j )

with u(j ) = uapp + u2 . Taking ρ1 = ρ, ρ2 = −ρ, consider the quantity (2) (1) ∂v−ρ ∂vρ (2) · v−ρ − vρ(1) · dσ. I= ∂n ∂ ∂n (1)

(2)

Under the assumption that vρ and v−ρ have the same Cauchy data on U , I is equal to the integral of the same expression over ∂\U . Observing that (2) (1) ∂v−ρ ∂vρ ∂ ∂ ρ·x (1) −ρ·x =e +(ρ·n(x)) u =e −(ρ·n(x)) u(2) , and ∂n ∂n ∂n ∂n

604


we see that the exponentials cancel and that the integrand of I is equal to

∂u(1) (2) ∂u(2) · u − u(1) · + 2 ρ · n(x) u(1) u(2) . ∂n ∂n (j )

(j )

Since (1.6) implies that supp(uapp | ∂), supp((∂uapp /∂n)|∂ ) ⊂ U for |ρ| sufficiently large, we have that I=

(1)

∂\U

(2)

(1) (2) ∂u2 (2) (1) ∂u · u2 − u2 · 2 + 2 ρ · n(x) u2 u2 dσ. ∂n ∂n

We estimate (1) (1) (2) ∂u ∂u (2) 2 2 · u2 dσ ≤ · u2 H 1/2 (∂) ∂n −1/2 ∂\U ∂n H (∂) (1) (2) ≤ u2 H 1/2 (∂) · u2 H 1/2 (∂) (1) (2) ≤ C u2 H 1 () · u2 H 1 () by Sobolev restriction (1) (2) ≤ C u2 H 1 (Rn ) · u2 H 1 (Rn ) since is compact t

≤ C|ρ|−2 −→ 0

t

as |ρ| −→ ∞,

and we estimate similarly for the second term. Now, note that |ρ · n(x)| ≤ c|ρ| since ∂ is Lipschitz, and (j ) (j ) (j ) u σ u σ +(1/2)() ≤ c! |ρ|σ −1/2− u 2 ≤ ≤ c σ σ 2 2 2 L (∂) H (∂) H for any σ > 0, and thus the third term is dominated by (cσ! )2 |ρ| · |ρ|2σ −1−2 → 0 as |ρ| → 0 if we choose 0 < σ < . On the other hand, ∂v (2) ∂v (1) (2) · v − v (1) · dσ I= ∂n ∂n ∂

=

v (1) · v (2) − v (1) · v (2) dx by Green’s theorem

= − q1 v (1) · v (2) − v (1) · − q2 v (2) dx

= q2 − q1 v (1) v (2) dx

= q2 − q1 u(1) u(2) dx

605

LOCAL UNIQUENESS FOR THE DIRICHLET-TO-NEUMANN MAP (1)

(1)

(2)

(2)

since the exponentials cancel. As u(1) · u(2) = (uapp + u2 ) · (uapp + u2 ) and the (1) (2) leading term uapp uapp → dλ weakly as |ρ| → ∞ by (1.8), while the remaining (j ) (j ) terms → 0 since uapp L2 () ≤ C by (1.5) and u2 L2 () ≤ c|ρ|−1− , we conclude (q − q )() as |ρ| → ∞, finishing the proof of Theorem 1. that I → R2,n 2 1 Now, to start the proof of Theorem 4, we may use the rotation invariance of and the invariance of Q under S 1 = {eiθ }, and we note that it suffices to treat the case2 ρ = |ρ|(e"1 + i e"2 ), where {e"1 , . . . , e"n } is the standard orthonormal basis for Rn . Write x ∈ Rn as x = (x ! , x !! ) ∈ R2 × Rn−2 , and, similarly, write ξ = (ξ ! , ξ !! ). If ∈ M2,n is parallel to span{ωR , ωI } = span{e"1 , e"2 } = R2 × {0}, then = span{e"1 , e"2 } + (0, x0!! ) for some x0!! ∈ Rn−2 . Given that |ρ| > 1 and that x0!! ∈ Rn−2 , we define an approximate solution u(x, ρ, ) to ( ρ + q(x))u = 0 on Rn , of the form u(x, ρ, ) = u0 (x, ρ, ) + u1 (x, ρ, ). For notational convenience we usually suppress the dependence on ρ and and simply write u(x) = u0 (x) + u1 (x). We use various cutoff functions χj ; for j even or odd, χj always denotes a function of x ! or x !! , respectively. Also, B m (a; r) and S m−1 (a; r) denote the closed ball and sphere of radius r centered at a point a ∈ Rm . To define u0 , first fix χ0 ∈ C0∞ (R2 ) with χ0 ≡ 1 on B 2 (0; R) for any R > sup{|x ! | : (x ! , x !! ) ∈ for some x !! ∈ Rn−2 }; let C0 = χ0 L2 (R2 ) . Secondly, let ψ1 ∈ C0∞ (Rn−2 ) be radial, nonnegative, supported in the unit ball, and let it satisfy !!

2 !! dx = 1. ψ1 x Rn−2

Now, for β > 0 to be fixed later, we let δ be the small parameter δ = |ρ|−β , and we define ! x − x0!! !! −(n−2)/2 , ψ1 χ1 (x ) = δ δ so that χ1 L2 (Rn−2 ) = ψ1 L2 (Rn−2 ) = 1,

∀δ > 0.

(1.9)

Set u0 (x) = u0 (x ! , x !! ) = χ0 (x ! )χ1 (x !! ); then u0 is real, u0 L2 (Rn ) = C0 , and u0 L2 () → [λ ( ∩ )]1/2 as δ → 0+ , that is, as |ρ| → ∞. Note also that u0 H 1 ≤ cδ −1 = c|ρ|β , so that u0 H s ≤ c|ρ|sβ for 0 ≤ s ≤ 1. Since ρ =

+ 2ρ · ∇ = + 2|ρ|(e"1 + i e"2 ) · ∇ = + 4|ρ|∂¯x ! and ρ⊥Rn−2 , √ course, the length of this element of Q is 2|ρ|, but this is irrelevant for the proofs, and denoting the length of |ρ|(e"1 + i e"2 ) by |ρ| is notationally convenient. 2 Of

606


ρ + q(x) u0 = χ0 · χ1 + 2 ∇χ0 · ∇χ1 + χ0 χ1

+ 2(ρ · ∇)(χ0 )χ1 + 2χ0 (ρ · ∇)(χ1 ) + qχ0 χ1

= χ0 (x ! ) x !! + q (χ1 )(x !! ) on B 2 (0; R) × Rn−2 , the first and fourth terms after the first equality vanishing because (ρ · ∇)(χ0 ) = 2∂χ0 ≡ 0 on B 2 (0; R) and the second and fifth equalling zero because ∇χ1 ⊥R2 . To define the second term in the approximate solution, u1 (x), we make use of a truncated form of the Faddeev Green function, Gρ , and an associated projection operator. The operator ρ has, for ρ ∈ Q, (full) symbol

σ (ξ ) = − |ξ |2 − 2|ρ|ωI · ξ + i2|ρ|(ωR · ξ ) , (1.10) and so, for ρ/|ρ| = e1 + ie2 , we have

σ (ξ ) = − |ξ − |ρ|e"2 |2 − |ρ|2 + i(2|ρ|ξ1 ) , which has (full) characteristic variety 5ρ = ξ ∈ Rn : ξ1 = 0, |ξ − |ρ|e2 | = |ρ|

= {0} × S n−2 (|ρ|, 0, . . . , 0); |ρ| ⊂ Rξ1 × Rn−1 ξ2 ,ξ !! .

(1.11)

The Faddeev Green function is then defined by Gρ = (−σ (ξ )−1 )∨ ∈ S ! (Rn ). We now introduce, for an 0 > 0 to be fixed later, a tubular neighborhood of 5ρ , Tρ = ξ : dist(ξ, 5ρ ) < |ρ|−1/2−0 , (1.12) as well as its complement, TρC , and we let χTρ , χTρC be their characteristic functions. ρ , by Define a projection operator, Pρ , and a truncated Green function, G P and ρ f (ξ ) = χTρ (ξ ) · f(ξ )

∧ −1 Gρ f (ξ ) = χTρC (ξ ) · [−σ (ξ )] f (ξ )

(1.13) (1.14)

ρ = I − Pρ . for f ∈ S (Rn ). Note that ρ G ∞ n−2 Choose a ψ3 ∈ C0 (R ), supported in B n−2 (0; 2), radial, and with ψ3 ≡ 1 on supp(ψ1 ), and set χ3 (x !! ) = ψ3 ((x !! − x0!! )/δ). We now define the second term, u1 (x, ρ, ), in the approximate solution by

ρ ρ + q(x) u0 (x) , u1 (x) = −χ3 (x !! )G (1.15) and we set u(x) = u0 (x) + u1 (x). Then u1 (as well as u0 ) is supported in {x : dist(x, ) ≤ 2δ}, yielding (1.6). We see below that u1 L2 () ≤ C|ρ|− as |ρ| → ∞,

LOCAL UNIQUENESS FOR THE DIRICHLET-TO-NEUMANN MAP

607

so that (1.5) holds as well, so that the first part of (1.9) holds as well. To start the proof of (1.7), note that

ρ ( ρ + q)u0 ( ρ + q)(u0 + u1 ) = ( ρ + q)u0 − ( ρ + q)χ3 G

ρ ( ρ + q)u0 = ( ρ + q)u0 − χ3 ( ρ + q)G

ρ ( ρ + q)u0 − ρ + q, χ3 G ρ ( ρ + q)u0 = ( ρ + q)u0 − χ3 (I − Pρ )( ρ + q)u0 − χ3 q G

ρ ( ρ + q)u0 − x !! χ3 G ρ ( ρ + q)u0 − 2 ∇χ3 · ∇x !! G = χ3 Pρ ( ρ + q)u0

ρ ( ρ + q)u0 − qχ3 + 2 ∇χ3 · ∇x !! − χ !! χ3 G (1.16) on since χ3 ≡ 1 on supp(χ1 ). Now, since q1 χ3 ∈ L∞ , |∇χ3 | ≤ Cδ −1 = c|ρ|β , and | x !! χ3 | ≤ Cδ −2 = c|ρ|2β , (1.7) follows if we can show that for some > 0, Pρ ( ρ + q)u0 2 ≤ C|ρ|− , (1.17) L () !! −β− D , and (1.18) Gρ ( ρ + q)u0 L2 () ≤ C|ρ| (1.19) Gρ ( ρ + q)u0 L2 () ≤ C|ρ|−2β− , with C independent of |ρ| > 1. Before proceeding to prove these, we note that for any u(1) , u(2) constructed in this way for the same two-plane , !! !! (1) (2) 2 ! −(n−2) 2 x − x0 −→ dλ ψ1 u0 (x)u0 (x) = χ0 (x )δ in δ (1) (2)

(1) (2)

(1) (2)

as δ → 0 by (1.11), while u1 u0 + u0 u1 + u1 u1 → 0 in L2 (), yielding (1.8). Thus, we are reduced to establishing (1.17)–(1.19). 2. L2 -estimates We first prove (1.17)–(1.19) under the simplifying assumption that q1 , q2 ∈ C n−1+γ () for some γ > 0, turning to the Sobolev space case in Section 3. Start by noting that the desired estimates (1.17)–(1.19) cannot be simply obtained from operator norms; for example, Pρ L2 →L2 = 1 for all ρ. One needs to make use of the special structure of ( ρ + q)u0 ; we first deal with ρ u0 , leaving q(x) · u0 for the end. So we show that Pρ ρ u0 L2 ≤ C|ρ|− , and so on. Since ∇χ0 · ∇χ1 ≡ 0,

ρ u0 = χ0 x !! χ1 + x ! + 4|ρ|∂ x ! (χ0 ) · χ1 . (2.1) ρ are nonlocal operators, and The second term is supported on c , but Pρ and G we need to control the contribution from this term. However, because x ! (χ0 ) is a

608


fixed, δ-independent element of C0∞ (R2 ), this can be handled in the same way as the q(x) · u0 terms of (1.17)–(1.19), which are dealt with later. The contribution from 4|ρ|∂χ0 · χ1 is handled at the end. So, for the time being, we are interested in estimating Pρ (χ0 (x ! ) x !! χ1 (x !! ))L2 , and so on. Now, x !! χ1 (x !! ) = δ −2 χ5 (x !! ), where χ5 (x !! ) = δ −(n−2)/2 ψ5 ((x !! −x0!! )/δ) is associated with the radial function ψ5 = x !! ψ1 as χ1 is associated with ψ1 . Note 5 vanishes to second order at zero. Of course, χ0 ∈ C ∞ ⇒ χ 0 ∈ for future use that ψ 0 n S (R ), but, looking ahead to estimating the terms involving q(x) · u0 (x), we now ρ act on χ2 (x ! ) χ1 (x !! ), under prove the analogues of (1.17)–(1.19) where Pρ and G the weaker assumption that χ2 is radial and satisfies the uniform decay estimate, χ 2 (ξ ) ≤ C(1 + |ξ |)−α , (2.2)α for some α > 0. Now, by (1.14) and Plancherel,

∧ Pρ (χ2 χ1 ) 2 ≤ Pρ (χ2 χ1 ) L2 (Rn ) L () !! 5 δξ = δ −2 χ 2 (ξ ! )δ (n−2)/2 ψ

L2 (Tρ )

.

The characteristic variety 5ρ , of which Tρ is a tubular neighborhood, passes through the origin, and we may represent 5ρ near O as a graph over the ξ !! -plane: 5ρ = 5ρs ∪ 5ρn ∪ 5ρe , with 2 !! 2 1/2 !! |ρ| s 5ρ = ξ1 = 0, ξ2 = |ρ| − |ρ| − ξ , ξ ≤ 2 (2.3) ξ !! |ρ| ξ1 = 0, ξ2 = , ξ !! ≤ 2 2|ρ| a neighborhood of the south pole O, 2 !! 2 1/2 !! |ρ| n 5ρ = ξ1 = 0, ξ2 = |ρ| + |ρ| − ξ , ξ ≤ 2 !! 2 ξ |ρ| , ξ !! ≤ ξ1 = 0, ξ2 = 2|ρ| − 2|ρ| 2

(2.4)

a neighborhood of the north pole (0, 2|ρ|, 0, . . . , 0), and 5ρe a neighborhood of the equator {ξ ∈ 5ρ : ξ2 = |ρ|}. We have a corresponding decomposition Tρ = Tρs ∪ Tρn ∪ Tρe , where, for example, Tρs

!

ξ ,ξ

!!

!

:ξ ∈B

2

0,

!! 2

ξ 2|ρ|

−1/2−0

; |ρ|

!! |ρ| . , ξ ≤ 2

(2.5)

609


Recalling that χ2 and ψ3 are radial, so are χ 2 and χ 3 , and by abuse of notation, we consider these as functions of one variable satisfying (2.2)α and rapidly decreasing, respectively. Thus, using polar coordinates in ξ !! , |ρ|/2 2 2 χ χ 5 (δr)2 r n−3 dr 2 (ξ ! ) dξ ! δ n−6 ψ 2 χ1 L2 (T s ) ρ

0

√

B 2 ((0,r 2 /2|ρ|);|ρ|−1/2−0 )

dr 5 (δr)2 r n−2 | χ2 |2 dξ ! δ n−6 ψ r 0 B 2 ((0,0);|ρ|−1/2−0 ) |ρ|/2 2 2 2

r χ · B (0, 0); |ρ|−1/2 δ n−6 + √ 2 2|ρ| 2|ρ|1/4 dr 5 (δr)2 r n−2 . × ψ r 2|ρ|1/4

(2.6)

Since we are taking δ = |ρ|−β with β < 1/4, if we choose 0 < 0 < 2(1/4 − β), then the quantity |ρ|1/4 δ → ∞ as |ρ| → ∞, and so √ 1/4

2|ρ| δ −4 2 2 n−2 dr δ χ ψ 5 (r) r 2 χ1 L2 (T s ) ≤ c ρ r |ρ|1+20 0 2 2 r dr ψ χ 5 (r)2 r n−2 + √ 2 2 r 2δ |ρ| 2|ρ|1/4 δ

(|ρ|/2)δ

(2.7)

≤ c(δ 4 |ρ|)−1 , which is less than or equal to c|ρ|−2 with = (1/2)(1 − 4β) > 0. The other contributions to Pρ χ2 χ1 L2 , coming from Tρn and Tρe , are handled 5 . similarly and are even smaller, due to the decrease of χ 2 and ψ !! We next turn to estimating |D |Gρ ρ u0 L2 ; by the remark above we may concentrate on the χ2 χ1 term of ρ u0 . Then 2 2 !! D (2.8) Gρ (χ2 χ1 )L2 () ≤ ξ !! (σ (ξ ))−1 (χ2 χ1 )∧ (ξ )L2 (T C ) . ρ

We may cover TρC by TρC,s ∪ TρC,n ∪ TρC,e ∪ TρC,∞ , where  2

C  ξ !! C,s ! 2 −1/2−0 ; |ρ| 0, Tρ = ξ : ξ ∈ B  2|ρ|



C !! 2

ξ 1 |ρ|  ; |ρ| , ξ !! ≤ ; 0, 2|ρ| − 2|ρ| 4 2 

∩ B2

(2.9)

610


TρC,n is defined similarly: TρC,e

|ρ| 7|ρ| < ξ2 < , |ρ|−1/2 < dist ξ, 5ρ < |ρ|, ξ !! < 2|ρ| = ξ: 4 4

and TρC,∞

!! 3 = ξ : |ξ | ≥ 3|ρ|, ξ ≥ |ρ| . 2

(2.10)

(2.11)

One has the lower bounds on σ ,

C|ρ| dist ξ, 5ρ , |ξ | ≤ 3|ρ|, |σ (ξ )| ≥ |ξ | ≥ 3|ρ|, C|ξ |2 ,

(2.12)

with C (as always) uniform in |ρ|. The first inequality in (2.12) √ follows from noting that (1/2)∇σ (ξ ) = (ξ − |ρ|e"2 ) + i(|ρ|e"1 ), so that |∇σ (ξ )| = 2 2|ρ| on 5ρ , while the second follows from Re(σ (ξ )) = dist(ξ, |ρ|e"2 )2 − |ρ|2 . Using the first estimate in (2.12), we can then dominate the contribution to the right-hand side of (2.8) from the region TρC,s by δ n−6

|ξ !! |≤|ρ|/2 B 2 ((0,(|ξ !! |2 /2|ρ|));|ρ|−1/2−0 )C

|ρ|−2 ξ ! −

−2 e"2 2|ρ|

!! 2 ξ

(2.13)

2 2 !! 2 !! 5 δξ dξ . × χ 2 (ξ ! ) dξ ! ξ !! ψ The inner integral is the convolution ! ξ ≥ |ρ|−1/2−0 χ 2 |ρ|−2 χ 2 ∗R2 |ξ ! |2

.

ξ ! =(|ξ !! |2 /2|ρ|)e"2

An elementary calculation shows that, for χ 2 satisfying (2.2)α for some 0 < α < 1, and any 0 < a < 1, 

−1 ) , !  C |ξ ! | ≤ 1, 1 + log(a  1 2 χ |ξ | ≥ a ! χ 2 ∗R2 ≤ (2.14) |ξ |  |ξ ! |2 , |ξ ! | ≥ 1, C2 |ξ ! |−2 + C3 |ξ ! |−2α log a so that, taking a = |ρ|−(1/2)−0 and |ξ ! | = |ξ !! |2 /(2|ρ|), the inner integral in (2.13) is less than or equal to  √ C1 |ρ|−2 log |ρ|, 0 < |ξ !! | ≤ 2|ρ|1/2 C ξ !! −4 +C |ρ|2α−2 |ξ !! |−4α log ξ !! 2 / 2|ρ|1/2−0

, √2|ρ|1/2 ≤ |ξ !! | ≤ |ρ|/2. 2 3


611

Employing polar coordinates in ξ !! and rescaling by δ, we see that (2.13) is less than or equal to

√ 2|ρ|1/2 δ

dr ψ 5 (r)2 r n r 0 (|ρ|/2)δ dr ψ 5 (r)2 r n−4 + C2 δ −2 √ r 2|ρ|1/2 δ (|ρ|/2)δ dr ψ 5 (r)2 r n−2−4α + C3 δ 4α−4 |ρ|2α−2 log |ρ| √ . 1/2 r 2|ρ| δ

C1 δ

−6

|ρ|

−2

log |ρ|

With δ = |ρ|−β , β < 1/4, |ρ|1/2 δ → ∞ as |ρ| → ∞, and thus we estimate this for 5 ) by any N > 0 (using the rapid decay of ψ

−N

−N + C3 |ρ|(4−4α)β+2α−2 log |ρ| |ρ|1/2 δ , C1 |ρ|6β−2 log |ρ| + C2 δ −2 |ρ|1/2 δ the first term of which is less than the desired |ρ|−2β−2 , for any α > 0, if β < 1/4 and = (1/2)(1 − 4β); the second and third terms are rapidly decaying simply because β < 1/2. ρ χ2 χ1 2 2 Moving ahead for the moment to (1.19), the contribution to G L (which we want to be less than or equal to C|ρ|−4β−2 ) from TρC,s is handled in the same fashion, the only differences being the absence of the multiplier |D !! |∧ = |ξ !! | on the left and the improved gain we are demanding on the right. Taking these into account, we need to control C1 δ

√

dr ψ 5 (r)2 r n−2 r 0 (1/2)|ρ|δ dr ψ 5 (r)2 r n−6 + C2 √ r 2|ρ|1/2 δ (1/2)|ρ|δ dr ψ 5 (r)2 r n−4−4α + C3 δ 4α−2 |ρ|2α−2 log |ρ| √ 1/2 r 2|ρ| δ

−N ≤ C1 δ −4 |ρ|−2 log |ρ| + C2 |ρ|1/2 δ

−N + CN δ 4α−2 |ρ|2α−2 log |ρ| |ρ|1/2 δ ,

−4

|ρ|

−2

log |ρ|

2|ρ|1/2 δ

(2.15)

and this is less than or equal to C|ρ|−4β−2 , provided that β < 1/4, < (1/2)(1−4β), and N is sufficiently large. The contributions to (1.18) from TρC,n and TρC,e are handled similarly. To treat the contribution from TρC,∞ , we use the second estimate in (2.12) and calculate (for (1.18))

612


!! ξ (σ (ξ ))−1 (χ2 χ1 )∧ (ξ )2 2

L (TρC,∞ )

≤C ≤C

|ξ |≥3|ρ|

|ξ !! |≤|ρ|

δ

!! 2 ! !! ξ dξ dξ |ξ |4

!! 2 !! 2 !! 5 δξ ξ dξ δ n−6 |ρ|−2α−2 ψ

+

= C δ −6 |ρ|

2 !! 2 5 δξ χ 2 (ξ ! ) ψ

n−6

|ξ !! |≥|ρ| |ρ|δ −2α−2

5 δ n−6 ψ

2 −2α !! dξ δχ ξ !!

!!

(2.16)

dr ψ 5 (r)2 r n r 0 ∞ 2 n−2−2α dr 2α−4 +δ ψ5 (r) r r |ρ|δ −6 −2α−2

≤ C δ |ρ| + δ 2α−4 (|ρ|δ)−N , ∀N > 0,

which, for δ = |ρ|−β and N large, is less than or equal to C|ρ|−2β−2 , provided that β < 1/4 and < α + 1 − 4β. A similar analysis holds for the TρC,∞ contribution to (1.19). We now turn to controlling the q(x)u0 (x) terms in (1.17)–(1.19) as well as the contributions from the (χ0 ) · χ1 term in (2.1). Note that since q(x) is C n−1+γ (for some γ > 0), q(x) has an extension (see, e.g., [St, Ch. 6]) to a C n−1+γ function of compact support on Rn , which we also denote by q. The restriction of q to any ∈ M2,n is still C n−1+γ . Let {Dt : 0 < t < ∞} be the one-parameter group of partial dilations on ∗ ! S (Rn ),

(Dt f ) ξ ! , ξ !! = t n−2 f ξ ! , tξ !! , which, for f, g ∈ L1 , satisfy Rn Dt f dξ = Rn f dξ and Dt (f ∗ g) = Dt f ∗ Dt g. Then q ∗ u0 (ξ ) q u0 (ξ ) =

1 (ξ !! )eix0!! ·ξ !! = Dδ (Dδ −1 0 (ξ ! )ψ q ) ∗ δ −(n−2)/2 Dδ χ

1 eix0!! ·ξ !! . = Dδ Dδ −1 ( q ) ∗ δ −(n−2)/2 χ 0 ψ

(2.17)

Now, as δ = |ρ|−β → 0, Dδ −1 ( q ) = δ −(n−2) q (ξ ! , δξ !! ) converges weakly to the singular measure (2.18) Q(ξ ! ) ⊗ δ(ξ !! ) = Q(ξ ! ) dξ ! , where Q(ξ ! ) = Rn−2 q (ξ ! , ξ !! ) dξ !! ; note that q ∈ C n−1+γ implies that the integral defining Q converges, and Q satisfies (2.2)α , for α = 1 + γ . Letting F (ξ ) =

LOCAL UNIQUENESS FOR THE DIRICHLET-TO-NEUMANN MAP !!

613

!!

1 (ξ !! )eix0 ·ξ , it follows from (2.17) that χ 0 (ξ ! )ψ

q ) ∗ δ −(n−2)/2 F q u0 (ξ ) = Dδ Dδ −1 (

= Dδ (Qdξ ! ) ∗ δ −(n−2)/2 F

q − Qdξ ! ∗ δ −(n−2)/2 F . + Dδ Dδ −1

(2.19)

If we define χ 4 (ξ ! ) = Q ∗R2 χ 0 (ξ ! ), then χ 4 also satisfies condition (2.2)α (and thus ! (2.2)α for 0 < α < 1, so that (2.14) can be applied), and the first term in (2.19) is

1 δξ !! eiδx0!! ·ξ !! . 4 (ξ ! )δ (n−2)/2 ψ (2.20) Dδ Q dξ ! ∗ δ −(n−2)/2 F = χ ρ (qu0 )L2 , and G ρ (qu0 )L2 Thus, the contributions to Pδ (qu0 )L2 , |D !! |G from the first term in (2.19) may be handled as the main χ2 χ1 term was earlier, with the obvious absence of the factor δ −2 . To control the contributions from the second term in (2.19), we use the following elementary lemma. lemma 5 Let ϕ(x), f (x) be functions on Rm such that ϕ(x), |x|ϕ(x), f (x), and |∇f (x)| are in L1 (Rm ). Then, for all > 0, −m x ϕ − ϕ dy δ(x) ∗ f (x) Rm

≤ Cm ϕL1 + |x|ϕL1 · f L∞ (B(0;|x|−1)) + ∇f L∞ (B(x;1)) · . Applying this for = δ, ξ ! ∈ R2 fixed, and using F ∈ S , | q (ξ )| ≤ C(1 + −(n−1+γ ) , we find that, for all N > 0, |ξ |) −N

Dδ−1 ( q ) − Qdξ ! ∗ F (ξ ) ≤ CN (1 + |ξ ! |)−γ 1 + ξ !! δ. (2.21) Hence, the second term in (2.19) is less than or equal to CN δ n/2 (1 + |ξ ! |)−γ (1 + |δξ !! |)−N , and this allows the contributions to (1.17)–(1.19) to be dealt with as the χ2 x !! χ1 term was before. Finally, we need to establish estimates (1.17)–(1.19) for the 4|ρ|∂χ0 term in (2.1); thus, we need to show that

Pρ ∂χ0 · χ1 2 ≤ C|ρ|−1− , (2.22) L !!

−1−β− D , and (2.23) Gρ ∂χ0 · χ1 L2 ≤ C|ρ|

−1−2β− Gρ ∂χ0 · χ1 L2 ≤ C|ρ| , (2.24)

614


(ξ ! ) is rapidly decreasing and vanishes to first for some > 0. Using the fact that ∂χ 0 order at ξ ! = 0, we may replace (2.6) with 2 ∂χ0 χ1 2 s L (T )

ρ

|ρ|/2

! 2 ! n−2 ∂χ0 (ξ ) dξ δ

B 2 ((0,(r 2 /2|ρ|));|ρ|−(1/2)−0 )

0

1 (δr)2 r n−3 dr × ψ

√ 2|ρ|(1−20 )/4

dr 1 (δr)2 r n−2 |ρ|−2−40 δ n−2 ψ r 0 2 2 √2|ρ|1/2 r dr 1 (δr)2 r n−2 |ρ|−1−20 δ n−2 ψ + √ (1−2 )/4 2|ρ| r 0 2|ρ|

|ρ|/2 r 2 −N −1−20 n−2 dr 1 (δr)2 r n−2 ψ + √ |ρ| δ r 2|ρ|1/2 2|ρ|

≤ cN

≤ cN |ρ|−2−40 + |ρ|

√ 2|ρ|(1−20 )/4 δ

0

−3−20 −4

δ

√

2 n−2 dr ψ 1 r r

2|ρ|1/2

√ 2|ρ|(1−20 )/4 δ

+ |ρ|−1−20 +N δ 2N

(2.25)

2 n+2 dr ψ 1 r r

2 n−2−2N dr ψ 1 r √ r 2|ρ|1/2 δ |ρ|/2δ

−N ! ≤ cN |ρ|−2−40 + |ρ|−3−20 +4β |ρ|(1−20 )/4−β !

+ |ρ|−1−20 +N −2Nβ−N (1/2−β)

!

for any N, N ! ≥ 0. As before, the contributions from Tρn and Tρe are handled similarly. Since 0 < 1/2 − 2β, if N ! is chosen large enough, this yields (2.23) with ≤ 20 , which is weaker than the previously imposed < (1/2)(1−4β). The desired estimates (2.23) and (2.24) are even easier and hold for any β < 1/2. The contribution to (2.24) from TρC,s is controlled as in (2.13) but uses the factor δ n−2 and replaces the χ2 in ; this is then dominated in the same manner as in the discussion the integrand by ∂χ 0

following (2.14). The TρC,s contribution to (2.25) is estimated as in (2.15), but with the absence of the δ −4 . All other contributions are dealt with similarly. This concludes the proof of Theorem 4 for the case of potentials in the Hölder class C n−1+γ (), γ > 0. The restrictions on β and that we have needed are that β < 1/4 and < (1/2)(1 − 4β).

615


3. Remarks (i) The proof of Theorem 4 needs to be slightly modified if we assume that the potential q(x) belongs to the Sobolev space H n/2+σ () for some σ > 0. Since ∂ is Lipschitz, such a q(x) can, by the Calderón extension theorem, be extended to be in H n/2+σ (Rn ). Again denoting the extension by q, one has, by Cauchy-Schwarz,

R2

1 + ξ !! qˆ ξ ! , ξ !! dξ !!

Rn−2

2

σ

1 + |ξ ! |

2 dξ ! ≤ c qn/2+σ .

(3.1)

Thus, Q as in (2.18) belongs to L2 (R2 ; (1 + |ξ ! |)σ dξ ! ), so that χ 4 = Q ∗R2 χ 0 ∈ dξ ! ) ∩ L∞ . Replacing the uniform decay estimate (2.2)α with

σ

(3.2)σ χ 2 ∈ L2 R2 ; 1 + |ξ ! | dξ !

L2 (R2 ; (1 + |ξ ! |)σ

allows us to handle the first term in (2.19). Furthermore, if for ξ ! fixed we let φ(·) = q (ξ ! , ·) in Lemma 5, then φ(ξ !! ) and |ξ !! |φ(ξ !! ) are in L1 (Rn−2 ) with norms (as functions of ξ ! ) in L2 (R2 ; (1 + |ξ ! |)σ dξ ! ), and so the second term in (2.19) is less than or equal to cN χ6 (ξ ! )(1 + |δξ !! |)−N for all N, with χ6 satisfying condition (3.2)σ . So we are reduced to repeating the analysis of Section 2 with (2.2)α replaced by (3.2)σ . The decay of χ2 is used in only two places in the argument. In (2.14), under (3.2)σ , we have the same estimate except for the absence of |ξ ! |−2α ; however, this loss is absorbed into terms rapidly decreasing in |ρ|1/2 δ = |ρ|1/2−β where (2.14) is used. On the other hand, in (2.16) we may estimate the inner integral by dξ ! dξ ! χ2 (ξ ! )2 |χ2 |2 2 2 ≤ (1 + |ξ ! |)σ |ξ ! |4 |ξ ! |≥2|ρ| R2 |ξ ! |2 + ξ !! (3.3) !! −4−σ ≤ c|ρ| if ξ ≤ ρ and

R2

χ2 (ξ ! )2

!! −4 dξ ! 2 2 ≤ cξ |ξ ! |2 + ξ !!

if |ξ ! | ≥ ρ,

so that !! ξ (σ (ξ ))−1 (χ2 χ1 )∧ (ξ )2 2 C,∞ L (Tρ ) !! 2 !! 2 !! 5 δξ ξ dξ ≤C δ n−6 |ρ|−4−σ ψ |ξ !! |≤|ρ|

+

|ξ !! |≥|ρ|

5 δ n−6 ψ

2 −2 δχ ξ !! dξ !!

!!

(3.4)

616


−6 −4−σ = C δ |ρ| ≤ CN = CN

∞ dr 2 n−4 dr −2 ψ 5 (r)2 r n +δ ψ5 (r) r r r 0 |ρ|δ ! −6 −4−σ −2 −N + δ (|ρ|δ) δ |ρ|

N ! |ρ|6β−4−σ + |ρ|2β |ρ|β−1/2 , ∀N, (3.5) |ρ|δ

which is less than or equal to c|ρ|−2β− for N sufficiently large since β < 1/2. The restrictions on β and are as before. (ii) The construction of the approximate solutions given by Theorem 4 may be generalized by taking χ0 to be an arbitrary analytic function of z = x1 + ix2 defined on a domain ∩ ! ⊂ . Since ∂χ0 = x ! χ0 ≡ 0 on , the resulting u = u0 + u1 is still an approximate solution in the sense of Theorem 4, except that (1.8) no longer applies. Thus, Theorem 1 can be strengthened to conclude that (q1 − q2 )| is orthogonal in L2 ( ∩ , dλ ) to the Bergman space A2 ( ∩ ) of square-integrable holomorphic √ functions on ∩ . Furthermore, by repeating the construction using ρ = (1/ 2)|ρ|(ωR − iωI ), which induces the conjugate complex structure on , for which the ∂ operator equals the ∂ operator induced by ρ, we obtain 2 that (q1 − q2 )| is also orthogonal to the conjugate Bergman space A ( ∩ ) of antiholomorphic functions. (The analogue of this in two dimensions was obtained in [SU3].) It would be interesting to make further use of this information. (iii) To obtain variants of Theorem 1 establishing smaller sets of uniqueness in ∂, it might be useful to use approximate solutions associated to different twoplanes. For this it seems necessary to construct approximate solutions with much thinner supports, that is, to overcome the restriction β < 1/4 in Theorem 4. Such an improvement might also be useful in extending the results to qj ∈ L∞ . Acknowledgments. The authors would like to thank Alexander Bukhgeim and Masaru Ikehata for pointing out errors in an earlier version of this paper.

References [A]

G. ALESSANDRINI, Stable determination of conductivity by boundary measurements,

[C]

´ , “On an inverse boundary value problem” in Seminar on Numerical A. P. CALDERON

[F]

Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65–73. MR 81k:35160 L. D. FADDEEV, Growing solutions of the Schrödinger equation (in Russian), Dokl. Akad. Nauk SSSR 165 (1965), 514–517; English translation in Soviet Phys. Dokl. 10 (1966), 1033.

Appl. Anal. 27 (1988), 153–172. MR 89f:35195


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P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24,

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¨ P. HAHNER , A periodic Faddeev-type solution operator, J. Differential Equations 128

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S. HELGASON, The Radon transform on Euclidean spaces, compact two-point

Pitman, Boston, 1985. MR 86m:35044 (1996), 300–308. MR 97d:35021

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homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153–180. MR 30:2530 , The Radon Transform, Progr. Math. 5, Birkhäuser, Boston, 1980. MR 83f:43012 V. ISAKOV, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. 127, Springer, New York, 1998. MR 99b:35211 R. KOHN and M. VOGELIUS, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), 289–298. MR 85f:80008 , Determining conductivity by boundary measurements, II: Interior results, Comm. Pure Appl. Math. 38 (1985), 643–667. MR 86k:35155 A. NACHMAN, Reconstructions from boundary measurements, Ann. of Math. (2) 128 (1988), 531–576. MR 90i:35283 , Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), 71–96. MR 96k:35189 E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970. MR 44:7280 J. SYLVESTER and G. UHLMANN, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), 91–112. MR 87j:35377 , A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), 153–169. MR 88b:35205 , “Remarks on an inverse boundary value problem” in Pseudo-differential operators (Oberwolfach, Germany, 1986), Lecture Notes in Math. 1256, Springer, Berlin, 1987, 430–441. MR 88j:35156 , Inverse boundary value problems at the boundary-continuous dependence, Comm. Pure Appl. Math. 41 (1988), 197–219. MR 89f:35213

Greenleaf Department of Mathematics, University of Rochester, Rochester, New York 14627, USA; [email protected] Uhlmann Department of Mathematics, University of Washington, Seattle, Washington 98195, USA; [email protected]

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