1-Convex Manifolds are p-Kahler

Abh. Math. Sem. Univ. Hamburg 72 (2002), 255-268

1-Convex Manifolds are p-K~ihler By L. ALESSANDRINI, G. BASSANELLI, and M. LEONI Abstract. We study here K~ihler-type properties of 1-convex manifolds, using the duality between forms and compactly supported currents, and some properties of the Aeppli groups of q-convex manifolds. We prove that, when the exceptional set S of the 1-convex manifold X has dimension k, X is p-K~hler for every p > k, and is k - K ~ l e r if and only if "the fundamental class" of S does not vanish. There are classical examples where X is not k - K ~ l e r even with a smooth S, but we prove that this cannot happen if 2k >_ n = dim X, nor for suitable neighborhoods of S; in particular, X is always balanced (i.e., (n - 1)-K~aler).

Introduction A complex manifold which is K~ihler (i.e. 1-K~ihler) is also p-K~ihler for every p, 1 < p < dim X. Thus the notion of p-Kahlerianity (see Definition 1.5) is relevant only in the non-K~ihler case, and here a lot of interesting and natural examples belong to the class Ak of those complex manifolds which are K~thler outside an analytic subset S of dimension k; for instance, every modification of a K~ihler manifold is of this type (for other kinds of p-Kahler manifolds, see [2]). Let us firstly consider the compact case, in which the following characterization is known: If X is a compact complex manifold, the obstruction to p-Kiihlerianity is a positive current T ~ 0 ofbidimension (p, p) which is a component of a boundary (see [1 ]). In order to relate this characterization with the above classes Ak, let us remark that the obstruction T is positive and 00-closed, so that - as in the classical case of positive and d-closed currents - Federer's Support Theorem holds; this implies that T cannot be supported on any analytic subset Z with dim Z < p, neither can T be concentrated on Z, that is, x z T = 0. More than that (again as in the classical case) if Z is compact, irreducible, and dim Z = p, then x z T = c[Z] for a suitable c > 0 (see Theorem 1.4). For compact manifolds in ,Ak, one suspects that there is a relation between the obstruction T and the analytic subset S; from this idea and collecting the above remarks, we get the following conjecture (due to A. SILVA): Let X be a compact

complex manifold which is Kiihler outside an analytic subset S of dimension k. Then 2000 Mathematics Subject Classification. 32F10, 32Q15, 32U40, 32C35. Key words and phrases. 1-convex manifolds, Kfihler manifolds. Partially supported by MIUR research funds. 9 MathematischesSeminarder Umversit~itllamburg,2002

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L. Alessandrini, G. Bassanelli, and M. Leoni

X is p-Kgihlerfor p > k and X is k-Kgihler if and only if the irreducible components of S of dimension k satisfy some homological conditions in H2k(X, Z). We point out that the hard part of the conjecture is, of course, to prove the relation between T and S; precisely, to show that, if T is the obstruction to p-K~ihlerianity, then x s T cannot vanish. Nevertheless, there are many cases in which the conjecture certainly holds: i f k = 0, MIYAOKA shows that X is K~ihler (see [22]). I f k = 1 and S is smooth, then Theorem 5;.5 in [5] gives the conjecture for p = 1. In the class of n-dimensional Moishezon manifolds the conjecture is true for p = n - 1 (see [4]). There are also other cases of Moishezon manifolds for which the full conjecture holds (see [3]). In this paper we shall consider a non-compact case, namely we shall study 1convex manifolds. Let us recall that a 1-convex (i.e. strongly pseudoconvex) manifold is given by a desingularization of a Stein space Y which has just a finite number of isolated singularities; the map X f Y giving the desingularization is the Reininert reduction. The exceptional set S of f is the maximal compact analytic subset of X, and X E ~k for k = dim S. All 1-convex manifolds of dimension < 2 are K~ihler, but in higher dimensions this is not always true. In particular, if dim S = 1, then X is K~ihler if and only if S contains no effective curves homologous to zero in H2(X, Z) (see [6]). We shall show that for 1-convex manifolds the conjecture holds: T h e o r e m 4.2. Let X be a 1-convex manifold and let S be its exceptional set, dim S = k. Then X is p-Kgihler f o r every p > k, with a O-O-exact p-Kiihler form. T h e o r e m 4.8. Let X be a 1-convex manifold and let S be its exceptional set, and let S1 . . . . . Sr be the irreducible components of S of maximal dimension k. Then X is kKdhler if and only if ~ j n j S.j # 0 in H2k ( X, Z ) for every (nl . . . . . nr ) E 1~r -- {0}. For a little more general situation, related to the class ~4k, we get: Proposition 4.4. Let X be a complex manifold containing an exceptional subvariety S, such that X - S has a O0-exact Kiihler form. Then X is p-Kiihler f o r every p > dim S, with a O0-exact p-Kiihlerform. It may be of some interest to compare this kind of results with the q-complete case; see for instance the result proved in [10]: Let X be a q-complete manifold. Then X is q-Kgihler with a O~-exact q-Kiihler form. Finally we point out that all 1-convex manifolds are balanced (Theorem 4.12) and that a suitable neighborhood of S is always k-K~hler, when S is smooth: Proposition 4.10. Let X be a 1-convex manifold and let S be its exceptional set with dim S = k, and suppose S is smooth. Then every open 1-convex neighborhood U of S f o r which S is a deformation retract, is k-Kiihler. Let us remark that passing from the compact case to the 1-convex one exchanges the difficulties in proving the main conjecture: in fact, the problem is now to characterize the obstructions to p-K~ihlerianity in terms of positive OO-closed currents; after this, using the Remmert reduction, it is straightforward to see that such currents

1-Convex Manifolds are p-K~ihler

257

must be supported on S. The characterization follows the patterns of the original proof given by HARVEY and LAWSON in the compact case (see [19]): the idea is to apply Hahn-Banach Separation Theorem to the duality between forms and currents, in particular to the Aeppli cohomology groups (see Definition 2.3). This can be done also in the non-compact case (for a quite general statement see Theorem 3.2 and the following Remarks) when these groups are Hausdorff. About this last condition, recall that for a 1-convex manifold X and an analytic coherent sheaf Y', the groups HJ (X, Y') are finite-dimensional for j > 1, but (unfortunately there is a gap in [11]) this is not the case in general for the sheaf o~ of germs of real pluriharmonic functions, neither for the Aeppli cohomology groups, as the following example shows.

Example. Take the complex surface X := (C - Z) x (C - {0, 1}). X is a Stein, hence 1-convex, manifold, and by the Kiinneth formula H2(X, C) is not finite dimensional. But using the exact sequence (2.2), it is straightforward to see that H2(X, C) is isomorphic to H i ( x , de). Hence we need to prove that the Aeppli groups are Fr6chet topological vector spaces; this computation is done for q-convex manifolds (Theorem 2.4). 1 Forms and currents

Let X be a complex n-dimensional analytic manifold. For every p, q > 0, let ~P,q denote the sheaf of germs of (smooth) (p, q)-forms on X, and gP,q(x) the Fr6chet space of (p, q)-forms on X, endowed with the usual topology of uniform convergence on compact sets. Its dual space is K n - p ' n - q (X), the space of currents on X of degree (n - p, n - q) (or bidimension (p, q)) with compact support. An index ~ , like g~'P(x), denotes the set of the real vectors of the space, i.e. such that For the definitions of positive, strongly positive, weakly positive forms, see [3] or [20]; only for p = 1 or p = n - 1 these notions coincide. As regards currents, positivity is defined in terms of the duality:

Definition 1.1. A current T c Kn~p'n-P(x) is called weakly positive (positive, strongly positive) if T(co) > 0 for every strongly positive (positive, weakly positive) (p, p)-form w. Remark 1.2. A weakly positive current has measure coefficients, i.e. it is of order zero. Hence the following theorem can be applied to weakly positive currents. Theorem 1.3. (see [4], Theorem 2.1) Let f~ be an open set in C n, T a real current ofbidimension (p, p) and order zero, such that O-OT = 0. Ifsupp T c_ y, where Y

is an analytic subset of $2 of dimension k < p, then T = O. In the previous result, we have seen that the support of a 00-closed current of order zero cannot be too small; in the case of positive currents, we can handle also the case when the support of the current is an analytic subset of the same dimension as the current; more generally, we have the following result:

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Theorem 1.4. (see [8], Theorem 4.10) Let f2 be an open set in C,n, T a positive current of bidimension (p, t7) such that iO-OT > O. If Y is an analytic subset of ~2 of pure dimension p, then )o'T = f l Y ] f o r a suitable weakly plurisubharmonic function f > 0 on Y (i.e. x r T ( a ) = f r f a f o r every ot ~ 0 (or Wx > 0) denotes that w is strictly weakly positive, or strictly positive, or strictly strongly positive (in short, strictly positive) and, if T c Kn,~p'n-P(x), T > 0 means that T(w) _> 0 Vw > 0 (in short, T is a positive current): so we can handle the three cases simultaneously. On a complex manifold, the existence of a Kiihler metric is equivalent to the existence of a strictly weakly positive closed (1, 1)-form, which is the fundamental form of the Kiihler metric. When this is not the case, we may ask for the existence of a strictly weakly positive closed (p, p)-form, which is a sort of weak K~hler condition.

Definition 1.5. A complex manifold X is p-K~ihler if it carries a closed transverse (i.e. strictly weakly positive) (p, p)-form w, called a p-Kiihler form.

2 Fr~ehet sheaves Recall that the cohomology groups H J ( x , 3=') of a q-convex manifold are finitedimensional if j >_ q, for every coherent sheaf a~ on X (see Theorem 14 in [7]). But this is not enough for our purposes, because we need a result on Aeppli cohomology (see Definition 2.3), which is not finite dimensional, in general (there is a gap in [11], Theorem 3.1 of Section 3; see also [27], Sections 4 and 5). We prove in this section that the Aeppli cohomology groups (of high index) of a q-convex manifold are Hausdorff (hence Fr6chet) topological vector spaces; we shall use this result in Section 4, to apply the duality theorem 3.2. We need some general facts about the cohomology of a complex manifold X with values in a Fr6chet sheaf: the definitions of normal Fr6chet sheaf and Fr6chet homomorphism of sheaves are given in [13], Section 4. By Proposizione 2 and 4 ibidem, it is straightforward to prove:

Corollary 2.1. Let 0-+Y"-+

a~

a~ ' ' - + 0

be an exact sequence of normal Frdchet sheaves (with respect to the same countable Leray covering ~ of X) and Fr&het homomorphisms. If Hq('u, De') : H q + I ( ' u , ~ ) = O, then the coboundarymap 3q. 9 H q ( x , 5~') --> H q + I ( X , a~ ' ) is a topological isomorphism. Moreover, it is not hard to prove:

Proposition 2.2. Let 0 - + ~ ' ~ D r " ~ 0

be an exact sequence of normal Frdchet sheaves (with respect to the same Leray covering of X), and Frdchet homomorphisms. (i) I f dim H q ( x , J~) < oo, then H q ( x , 3=') is Hausdorff

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259

(ii) If d i m H q ( X , jr) < c~ and Hq+I(x, jr') is Hausdorff then Hq(x, jrt') is Hausdorff

As usual, we give ~J (X) the topology of uniform convergence on compact sets. and the quotient topology to the de Rham groups H j (X); these topological spaces are Hausdorff. Let us consider now the following resolution of the sheaf o~ of germs of real pluriharmonic functions (f2J denotes, as usual, the sheaf of germs of holomorphic j-forms) (see [1] or [11] and [12]). 0 ~

~ ......+ ~ p - 1 o'_p...~l~ p --+ Op . . . ~ ~ Cr-1 ~ ~0 ...+

sp,p ~

sp+l,p+l a~l

~2p--1 O2p-1

(2.1)

(sp+2,p+ 1 1~ sp+l,p+2)i R a~p..-2 ~2p+4 ~N -'+ "'"

where: J

d ~ J = (t~ J~+ l 'f ~ 2( ~" sJk ' j+- k )l ) " " \

\-~, k=O

"

/

2p-j ,~J = ( ( ~ 8J-P+k,p-k)

for

/R

for

0<j

< p-I;

p < j 5 2p-1;

k=0

O-l(f) = (--O f, L -Of); 6rj(c0J+l , tiotkj-klj .j+l~) ' Jk=0' ~t)

=

(_0qgj+l q)j+l _t_00/0,J,

{Oolk'j-k q- -OOlk+l'j-k-11J-a Jk=0, OolJ'O -t- g)j+l, _0q)j+a) for

O < j < p - 2 (if p > l);

- - folk ~Tp-l(q)P, / ' p-l-k~p-1 Jk=0' q)p.) = (qgP + aOIO'p-1, {ootk,P-1-k 4- -ootk+l,P-Z-kl p-2 0~P -1,0 + q)P); --

Jk=0

'

O'j ({OlJ-P+k'p-k}~PoJ) = ({OOlj-p+k'p-k -~--OOlj-p+k+l'p-k-112p-j-I'~ Jk=0 1 for

p < j < 2p-1;

Cr2p = i O0;

~r2p+l = O'2p+2 = d = O + 0. If p = 0, the sequence (2.1) becomes 0 ~

a~ "+ S O'0 ~

~1,1 L (~2,1 f~ sl,2)N d

~4 ._._~ . . .

The sequence (2.1) is an exact sequence of Fr6chet sheaves and Fr6chet homomorphisms. Definition 2.3. The Aeppli groups of a complex manifold X are defined as: A~+I,p+I(x ) =

~P+I'p+I(x) / d a = 0} = (Kera2p+l)(X) {or 9 ~IR {Or 9 ~ P + I ' p + I ( x ) / ly = i 0 0 / 3 } (Imcr2p)(X)

Yp > 0 -

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L. Alessandrini, G. Bassanelli, and M. Leoni

v~,P(x ) =

{o/ 9 g~['P(x) / i 000t = 0} = (Kera2p)(X) {or 9 gP't'(x) / a = -Off + O-fi} (Imcr2p-l)(X)

'r

> 1

Let us consider also

W;+2,p+I(x) = (Kera2p+2)(X) (Im a2p+l)(X)

Vp >_O.

It is well-known that such groups can be defined also in terms of currents; the groups involving currents (or forms) with compact support are denoted by a subscript c, as for instance VcP~P(x) (see also [11], [12] and [27], Sections 4 and 5). Let us consider now the main result of this section: T h e o r e m 2.4. Let X be a complex manifold such that." dim H j(X, ~k) < oo

u > O, u > jo.

Then A P + I ' P + I ( x ) and wP+z'P+I (X) are Hausdorfffor p + 1 > jo, and vP'P (x) is Hausdorff for p > jo. Proof Consider the exact sequence ,O ~ R -~ O -~ oW ~ 0

(2.2)

where i(c) = ic, c 9 R , and 2Ref(z) = f ( z ) + f ( z ) . It gives

9.. ~ H J ( x , O) ~ H J ( x , oze) --+ HJ+I(x, lI~) -+ HJ+I(x, (9) ~ .... Adapting a construction given by CASSA in [14], we can build a countable covering A of X which is a Leray covering for all sheaves involved in (2.1) and (2.2). Moreover, all these sheaves are normal with respect to A. Notice that H j+l (X, lI~) is a 12ech cohomology group, which is isomorphic to the de Rham cohomology group H j +1 (X). This isomorphism is given by a composition of coboundary maps, coming, out from the short exact sequences associated to the sequence 0--+ N--+ g0 ___~gl ~ . . . By Corollary 2.1, HJ+I(x, If{) "~ HJ+I(x) is a topological isomorphism, so HJ+I(x, R) is Hausdorff. Now suppose j > j0: by Proposition 2.2, also HJ(x, eTt) is Hausdorff. If p = 0, from (2.1) we get 0 --+ o~f --+ go,0 _.+ ~.era2p+a --+ O, which gives

0 --~ HO(x, Jr) --~ HO(x, gO,O) --~ HO(x, JCer~2p+l) ~-~ HI(X, Jr) --~ O. By [ 13] (Section 4, Proposizioni 2 and 4) 8 0 is a topological homomorphism, and it gives a topological isomorphism AR1,1(X) .-= (Kera2p+l)(X) _~ H I ( x ' o~(), (Im aZp)(X) so that, if p = 0, A p+I'p+I (X) is Hausdorff.

1-Convex Manifolds are p-K~ihler

261

I f p > 0, from (2.1) we get 0--+ 3(ercr2p ---> 8 p'p --~ 3s

0 --+ HO(x, ,Tfer~2p) --+ H~

NP'P) ~ H~

-~ 0, hence

s0 3(ercrzp+l) --+

H I ( x , 3(er~2p) --+ 0 and as above 6o becomes a topological isomorphism

AP+a'p+I(x) ~ Ha(x, Xerffzp). Using 0 --+ ,Tferff2p-1 --> ~2p-1 __+ ~er~2p --+ O, and the previous tecniques, we prove that 8,1 9 H i ( x , a%erff2p) --+ H2(X, 3(er~zp-1) is a topological isomorphism. In this way, we get a topological isomorphism A p+I'p+I(X) '~

HP+I(x, 3(er~rp). Now consider 0 --+ 3r --+ s

__.> 3s

O'1 --'> 0, which gives

...._._> H2P(X, ~0) ._..>H2P(X, 3feral) ~ H2p+I(x, a~) H2p+I(x ' ~0) ~ . . . . Since by hypothesis, dimHJ(X,~2 k) < cx~ u Vj > p + 1 > jo, and H 2p+l (X, J-e) is Hausdorff, by Proposition 2.2 H 2p (X, 3(er al) is Hausdorff. Using the other short exact sequences coming from (2.1) we get that Hp+I(x, o~.erap) is Hausdorff, hence we conclude that AP+I'p+I(x) is Hausdorff. In a rather similar way we get the result for vP'P(x) and W~+2'p+1 (X).

[]

Corollary 2.5. Let us consider the following sequence ~ P - I ' p - I ( x ) -~ ~P'P(x) ~ (SP+I'P(X) @ 8P'P+a(x))~ -% 8~P+2(X)

(2.3)

wherev := iO-O, u := d, w := d. I f d i m H J ( X , ~ 2 k) < c~ Vk >_ O, Vj > p, then v and u are topological homomorphisms, AP'P(x) is a Fr6chet t.v.s, whose (topological) dual space is Vcn,~p'n-p (X). Proof Recall that AP'P(x)

Keru and wP+I'P(x) = Kerw. then use Theo-

rem 2.4. The result on A ~ ' P ( x ) comes from Theorem 2.4 in [12] (see also the proof of Proposition (5.5) in [27]). []

3 Some duality results Let us consider sequence (2.3) (p > O)

~ - I ' p - I ( x ) & sP'P(x) --~ (~P+I,P(X) ~ SP'P+I(x))N and the dual sequence

Ke,n-p+1 R 'n-p+l (X) ~v~ Kn,~p'n-P (x) ~ur (Kn-p-I'n-P (x) ~ Kn-p'n-p-I(x))N where v := iO-O, u :-- d, v' := iOg, u t := 0 + 5, in the sense ut(S + S) =

OS + OS VS E K n-p'n-p-1.

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L. Alessandrini, G. Bassanelli, and M. Leoni

Definition 3.1. We shall denote Ker v' by P Hcp (X), and call it the space of pluriharmonic currents of bidimension (p, p) and compact support, and Im u' by BeP(x), and call it the space of currents of bidimension (p, p) and compact support which are the (p, p)-component of a boundary with compact support. Moreover, the closed convex cone of positive currents of bidimension (p, p) and compact support is denoted by Pcp (X). T h e o r e m 3.2. Let X be a complex n-dimensional manifold, K a compact subset of X. (i) There exists a current T E PeP(x) f3 PHcP(X), T # O, with supp T C K, if andonly ifthere is no o~ E 8~'P(x) with co ~ I m v ando)x > 0 Yx c K. (ii) If u := d : 8P'P(X) --* (~P+a'p(x) (1~ ~P'P+a(x))~ is a topological homomorphism, then." there exists a current T ~ P f (X) M BPc(X), T # O, with s u p p T c_ K, if and only if there is no o) c 8P'P ( x ) with do) = 0 and Wx > 0 Yx ~ K.

Proof In both cases, one side is straightforward, and needs no hypothesis on u. As for the other side, let us choose a hermitian metric on X with associated fundamental form ~p, and consider a := {~0 ~ gP'P(x) [ 3e > 0suchthat~px > e~Pxp

Yx ~ K}.

By assumption (i): A 71 Im v = t3. Notice that A is a non empty open convex set in 8~'P(x), hence by Hahn Banach Theorem (see e.g. [29] II.3.1) there exists T ~ Kn~p'n-P(x) such that T(co)=0

u

T(cp)>0

'r

Since K e r v ' = ( I m v ) • we get T 6 K e r r ' , that is, T E P H P ( X ) . Moreover, for every positive (p, p)-form rl and Ye > 0, 7/+ 2e~ p E A, hence T(r/) = lim T(~I + 2egrP) > O. E

Notice that supp T _ K: in fact pick ot c 8P'P(x) with suppot C X \ K; u 1~, cot q- 2 ~ p E A, so that

E

0 < T(cot q- 2 ~ p) = cT(ot) + T(21p p) : this implies T(ot) = 0, because c is arbitrary. Thus, (i) is achieved. To prove (ii), notice that A M Ker u = 13, so that there exists T ~ Kn,~p'n-p (X) such that T(w)=0

u

,

T(~0)>0

'r162

Since u is a topological homomorphism Im u' = Im u t, hence (Keru) • = I m u t =

B~(X).

[]

Remark 3.3. If X is compact, its Dolbeault cohomology is finite-dimensional, hence by Corollary 2.15, u is always a topological homomorphism. Hence, if X is compact, we can choose K = X, and we get from Theorem 3.2 (ii) a proof of Theorem 14 in [19], of Theorem 4.7 in [21], of Theorem 1.17 in [1].

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263

Remark 3.4. Using the same techniques as above, we can prove also the following result: Let X be a complex manifold, K a compact subset of X. If v : = i 0 0 : ~-I'p-I(x) --+ ~ ' P ( x ) is a topological homomorphism, there exists a current T 6 PcP-I(x) N I m v ' , T ~: 0, with supp T _ K, if and only if there is no co 8 ~ - I ' P - I ( X ) with 00co = 0 and cox > 0 Yx ~ K. This gives in particular the existence of a Gauduehon (or standard) metric (a hermitian metric with K~ihler form ~0 such t h a t iO-O~on-1 : 0, n = dim X) on every compact manifold X. Indeed, if X is compact, v is a topological homomorphism, and moreover there are no nonconstant plurisubharmonic functions on X, hence no positive currents in Im v', for p = n. The above result gives thus the existence of a Gauduchon metric, because every strictly positive (n - 1, n - 1)-form is the (n - 1)-power of a strictly positive (1, 1)-form.

4 1-convex manifolds The references for this argument are the classical paper [7] and the survey [16]. Here, we shall discuss the existence of strictly positive closed forms on 1-convex manifolds, and on manifolds which have an exceptional subvariety.

Definition 4.1. (see [24]) A complex space X is said 1-convex, if there exist a proper surjective holomorphic map f : X ~ Y onto a Stein space Y, and a finite set B C Y such that, i f S : = f - l ( B ), the induced map X \ S --+ Y \ B is a biholomorphism and Oy = f , O x . The map f is called the Remmert reduction, and S, which is the maximal compact analytic set of positive dimension of X, is called the exceptional set of X. T h e o r e m 4.2. Let X be a 1-convex manifold, dim X = n, and let S be its exceptional set, dim S = k. Then X is p-Ktihler f o r every p > k (with a O-O-exact p-Kahler form).

Proof Let f : X ---> Y be the Remmert reduction; Y is embeddable in C N , hence it carries a K~ihler form Y = iO-Jg. Let fl : = f * F and T E Pc('(X) n PHcP.(x) (all kinds of positivity are available); since/~P is 00-exact, T(fl p) ---- 0, so that supp T c S, because/3 > 0 and/3 > 0 on X \ S. By Theorem 1.6, this implies T = 0, since p > dim S. By Theorem 3.2 (i) we get a form w ~ 8 ~ ' P ( x ) , co = iOOa, cox > 0 Yx E K = S. Take now a cut-off function X such that )f ~ (2~(X), 0 < X _< 1, )f = 1 on S. For M > > 0, Mfl p + iO0(Xot ) is a 00-exact real form, which is strictly "positive" (in particular, strictly weakly positive) on X. [] Remark 4.3. a) As the referee pointed out, Theorem 3.2 (i) is not essential in the above proof. In fact S has a (k + 1)-complete open neighbourhood U (see [9]), thus from [10] it follows that there exists co ~ 8~'P ( u ) , co = iO-Ool, cox > 0 Yx E U. b) Obviously, X is not k-K~ibler with a OO-exact form: indeed, if co = iO-Oot is such a form, we would have that 0 < S(co) = iO-gS(oO = O. More generally, we get:

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L. Alessandrini, G. Bassanelli, and M. Leoni

Proposition 4.4. Let X be a complex manifold, dim X = n, and let S be an exceptional subvariety, such that X \ S has a O-O-exact Kiihler form. Then X is p-Kiihler for every p > dim S (with a O0-exact p-Kiihler form). Proof By the hypothesis, there is a smooth strictly plurisubharmonic function u on X \ S. Since S is exceptional, there exist an analytic space Y, a proper holomorphic map f : X --+ Y and a discrete set D such that f ( S ) = D and X \ S is biholomorphic to Y \ D; and Sing Y consists of isolated normal singularities, belonging to D. Consider v := u o f - 1 on Y \ D: by means of L e m m a 1 in [17], there exists a smooth strictly plurisubharmonic function g on Y such that v = g on Y \ V, for a suitable small neighborhood V of D. We conclude as in Theorem 4.2, using y := iO-Jg. [] Now we shall study the case p = k. L e m m a 4.5. Let X be a 1-convex manifold with exceptional set S, dim S = k. Then the map F induced by the identity, F : A~k (X) --+ H2k(X, 1~) , is surjective.

Proof Since dim S = k, hence H j (X, ~r) = 0 u >_ O, u > k (see f.i. L e m m a 1 in [15]). Thus the hypothesis of Proposition 5 in [26] is satisfied (in order to consider real Aeppli groups a slight modification is enough), and we get the result. [] Remark 4.6. By Corollary 2.5, Hence the dual map:

g c,~ n-k'n-k

k'k(x). (X) is the topological dual of A R

F* : (H2k(x, ~))* = H2c(n-k)(x, •) ~ Vcn~k'n-k(X) given by F*((qg)) = {~on-k'n-k } is injective (here (.) denotes the class in H~2(n-k)(X, R) and {.} the class in the Aeppli group). This fact and the Kronecker duality give the proof of the following lemma. L e m m a 4.7. Let X be a 1-convex manifold of dimension n, with exceptional set S of dimension k; let S1 . . . . . Sr be the irreducible components of S of dimension k. Let Cl . . . . . Cr be real numbers. Then the following conditions are equivalent: r k 1) Y~j=lcj[Sj] E Be(X). 2) (Z~=I Cj[Sj]) ~-O E H , 2 ( n - k ) ( x , ~ ) . 3) )-~=1 cj(Sj) = 0 E Hzk(X, R). T h e o r e m 4.8. Let X be a 1-convex manifold with exceptional set S, and let $1 . . . . . Sr be the irreducible 3 components of S of maximal dimension k. Then X is k-Kgihler if and only if )-~=1 n j ( S j ) # 0 ~ H2k(X, Z ) f o r all choices of (nl .....

n~) ~ ~

- {0}.

Proof Let f : X ~ C u be the proper map given by the composition of the Reininert reduction f0 : X --+ Y and the embedding of Y in c N ; we can suppose fls : S ~ 0 ~ C N . Let z = (z l . . . . . ZN ) be coordinates on C u , and ), = i O-OIz[2 the standard K~ihler form. As said in Theorem 4.2, fl := f * ( y ) is a positive closed real (1, l)-form on X, which is strictly positive on X - S.

1-Convex Manifolds are p-Kahler

265

Since X is 1-convex, dim HJ (X, ~"2r) < O0 Yj >_ 1; this implies, via Corollary 2.5, that u : g~'k(x) -+ (r @ gk'k+l(X))• is a topological homomorphism, so that we can use Theorem 3.2.(ii). Let T ~ P~(X) fq Bkc(X) as before, T(/3 k) = 0, so that suppT _ S. Since T, being positive and pluriharmonic, "ignores" the irreducible components of S having dimension less than k (Theorem 1.3), by Theorem 1.4 we get T = ~ j =F l Xsj T = Y~=I cj[Sj] with cj >_ O. Therefore T = ~ j =F l cj[Sj] ~ Bkc(X) and by the previous lemma this means:

~

cj(Sj) = 0 E

H2k(X, ]~).

j=l

Therefore (see Lemma 6 in [6]) there exists some natural numbers nl . . . . . nr such that Y~=I nj(Sj) = 0 in the abelian group H2k(X, Z). This is contrary to the hypothesis, if T # 0. Hence Pck(X) N Bkc(X) = {0} so that by Theorem 3.2 ii) we get a closed real (k, k)-form o9, which is strictly positive on S (therefore also in a neighborhood U of S). Now we shall modify this form step by step to get a k-K~ihler form; so we can take e > 0 such that f - l ( B ( 0 , e)) CC U, take K0 := B(0, e), K1 := B(0, 1) B(O, e) . . . . . Kn+l := B(0, n + 1 ) - B(0, n), and call Aj := f - 1 (K j), j > 1. Take a sequence {~rj} of real numbers, ~rj > 1, such that W+crjflk > o

onAj.

Take a smooth function g : [0, + ~ ) -~ R such that g(t) > 1, g'(t) > 0 Yt, and g(Izl 2) > aj on Kj Yj > 1. Let h : [0, +c~) -+ ~ be given by h(x) = f~ g(t)dt: then for every z we get

i O-Oh(IZ 12) = h ' ( Iz 12)i 0 Iz 12 A OIZ 12 + h'(Iz 12)i O-OIz12. The first addendum is a strictly strongly positive (1, 1)-form, hence on Kj we have

(y = iO-OlZl2): iOOh(Iz[ 2) > g(lzl2)y > ~ j y . Every term is a positive (1, 1)-form on K j, hence

(iO-~h(Izl2)) k > cr~y k > ~r~yk on K j, which gives

f.(iO-~h(izl2))k > ~jflk > --oa on Aj. Call qJ := f*(iO-Oh([zl2)) k ; qJ is a real closed (k, k)-form, and qs + o9 > 0; so, by means of qJ § w, X is k-K~ihler. [] Corollary 4.9. If X is a l-convex manifold whose exceptional set S is irreducible of dimension k, then X is k-Kiihler if and only if the fundamental class of S does not vanish in H2k(X, Z).

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L. Alessandrini, G. Bassanelli, and M. Leoni

Proposition 4.10.

Let X be a 1-convex manifold with exceptional set S of dimension k and suppose S is smooth. Then every open 1-convex neighborhood U of S for which S is a deformation retract, is k-Kiihler.

Proof Since U is l-convex, the map: u = d 9 E~'k(U) ~ (gk+l'k(u) ~ gk'k+l(u))~ is a topological homomorphism and Vc,]~ n-k'n-k (U) is the topological dual of A~ k (U) (Corollary 2.5). By the duality theorem 3.2(ii), if there exists no current T satisfying T c P~(U) ~ Bkc(U), T # O, Supp(T) _ S, then there is some oJ 6 g~'k(U) such that d~o = 0 and OJx > 0 Vx c S. In this case, one proceeds as in the final step of the proof of Theorem 4.8 to get the expected k-K~hler form on U. Suppose the current T exists. By Theorem 1.4, T = c[S] with c > 0. On the other hand, Remark 4.6 assures that the map:

F*" tlc2,(n-k)(U, R) --+ V~,n R- k 'n - k (U) Since T = c[S], T is closed and therefore represents a class in H~2.(n-k)(U, ~) whose image is {T}, which is zero in Vc,n ~- k ' n - k (U) because T

is injective.

Bkc(U). Therefore T = dQ. Consider now the retraction map r : U --+ S, which gives an isomorphism: r* : H . i ( s , ~ ) -+ H J ( u , ~ )

Vj c N

Let ~7be a volume form for S. Since T is supported on S:

0 = Q(dr*~7) -- dQ(r*o) = T(r*o) = c / i ~ which is the required contradiction.

> 0 []

Remark 4.11. a) Given a couple (X, S) as in the last Proposition, neighborhoods U as required always exist: more than that, there are neighborhoods U such that they are also open in suitable Moishezon manifolds (see for example [28], Proposition 1.4). b) In [23], page 174, there is an example of a (compact) Moishezon threefold M which contains a hypersurface D such that X : = M - D is a 1-convex threefold which is not Kfihler and has a smooth rational curve S as exceptional set (see also [30], page 241). Thus Theorem 4.2 cannot be improved to include the case p = k, even when the form is not required to be exact. This emphasizes the "global" meaning of the result: X - S has a 00-exact K~,hler form, S has a K~ihler neighborhood (Proposition 4.10), but there is no global K~hler form. Nervertheless, in this example the obstruction is precisely the curve S, which is homologous to zero, compatibly with our Corollary 4.9. T h e o r e m 4.12. Let X be a 1-convex manifold and S be its exceptional set, dim S =

k. If k > ~-~, then X is k-Kgihler. In particular, X is always balanced (that is." (n - 1)-Kiihler).

1-Convex Manifolds are p-K~ihler

Proof Using the Remmert reduction f : X --+ Y and the inclusion i : S ~ can construct as in [18] an exact sequence:

267 X, we

9'' -"> Op+l(Y, Z ) --> Hp(S, Z ) J~ m p ( X , Z ) --~ Op(Y, Z ) -->" . . . Since Y is a Stein space, Hp(Y, Z ) = 0 'r

> n = dim Y ([25], Theorem 3). Fix

k > ~-~ and call p : = 2k: then H p + I ( Y , Z ) = 0 so that the map i . : H p ( S , Z ) ---> r n j ( jS) E H2k(S, Z), where (nl . . . . nr) E N r H p ( X , ~ ) is injective. Let Y~j.=I {0}: this class cannot vanish (in H2k(S, Z)), because $1 . . . . . Sr are irreducible components of maximal dimension k, hence the class does not vanish also in H2k(X, Z). By Theorem 4.8, X is k-K~ihler. []

References [1] L. ALESSANDRINI and M. ANDREATTA,Closed transverse (p, p)-forms on compact complex manifolds. Compositio Math. 61 (1987), 181-200. Erratum ibidem 63 (1987), 143. [2] L. ALESSANDRINI and G. BASSANELLI, Compact p-Kahler manifolds. Geometriae Dedicata 38 (1991), 199-210. [3] _ _ , Positive 00-closed currents and non-Kahler geometry. J. of Geometrical Analysis 2 (1992), 291-316. [4] , Metric properties of manifolds bimeromorphic to compact K~ihler spaces. J. Differential Geom. 37 (1993), 95-121. [5] , Compact complex threefolds which are K~ihler outside a smooth rational curve. Math. Nachr. 207 (1999), 21-59. [6] _ _ , On the embedding of 1-convex manifolds with 1-dimensional exceptional set. Ann. Inst. Fourier 51 (2001), 99-108. [7] A. ANDREOTTI and H. GRAUERT, Th6or~mes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr. 90 (1962), 193-259. [8] G. BASSANELLI, A cut-off theorem for plurisubharmonic currents. Forum Math. 6 (1994), 567-595. [9] D. BARLET, Convexit6 au voisinage d'un cycle. Functions of several complex variables, IV (Sem. Francois Norguet, 1977-1979). Lecture Notes in Math. 807, Springer, 1980, pp. 102-121. [10] , Convexit6 de l'espace des cycles. Bull. S.M.F. 106 (1978), 373-397. [11] B. BIGOLIN, Gruppi di Aeppli. Ann. SNS Pisa 23 (1969), 259-287. [12] , Osservazioni sulla coomologia del 00. Ann. SNS Pisa 24 (1970), 571-583. [13] A. CASSA Coomologia separata sulle varietal analitiche complesse. Ann. SNS Pisa 25 (1971), 291-323. [14] , New spectral sequences for duality theorems on compact analytic spaces. Boll. UMI 18-B (1981), 429-453. [15] M. COLTOIU, On the embedding of l-convex manifolds with l-dimensional exceptional set. Comment. Math. Helvetici 60 (1985), 458-465. [16] , Q-convexity. A survey. In: Complex analysis and geometry, Pitman Research Notes in Math. Series 366 (1997), 83-93. [17] A. FUJIKI, K~ihlerian normal complex surfaces. T~hoku Math. J. 35 (1983), 101-117. [18] H. GRAUERT and O. RIEMENSCHNEIDER, Verschwindungss~itze for analytische Kohomologiegruppen auf Komplexen Raumen. Invent. Math. 11 (1970), 263-292.

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[19] R. HARVEY and J. R. LAWSON, An intrinsec characterization of Kahler manifolds. Invent. Math. 74 (1983), 169-198. [20] R. HARVEYand A. W. KNAPP, Positive (p, p)-forms, Wirtinger's inequality and currents. In: Complex Analysis and Related Topics in Differential Geometry 1972-73. Dekker, 1974, pp. 43-62. [21] M. L. MICHELSON, On the existence of special metrics in complex geometry. Acta Math. 143 (1983), 261-295. [22] Y. MIYAOKA, Extension theorems for K/ihler metrics. Proc. Japan Acad. 50 (1974), 407-410. [23] B. G. MOISHEZON,On n-dimensional compact varieties with n algebraically independent meromorphic functions. A.M.S. Transl. 63 (1967), 51- 177. [24] R. NARASIMHAN,The Levi problem for complex spaces II. Math. Ann. 146 (1962), 195-216. [25] , On the homology groups of Stein spaces. Inv. Math. 2 (1967), 377-385. [26] F. NORGUET, Remarques sur la cohomologie des varirtrs analytiques complexes. Bull S.M.F. 100 (1972), 435-447. [27] F. NORGUET and Y. T. SIU, Holomorphic convexity of spaces of analytic cycles. Bull. S.M.F. 105 (1977), 191-223. [28] T. PETERNELL,On strongly pseudo-convex K/ahler manifolds, Invent. Math. 70 (1982), 157-168. [29] H. H. SCHA.FER, Topological Vector Spaces. Graduate Texts in Mathematics 3, Springer, 1970. [30] VO VAN TAN, On certain non-K/ihlerian Strongly Pseudoconvex Manifolds. J. of Geometric Analysis 4 (1994), 233-245. Received: 18 December 2001 Communicated by: O. Riemenschneider Authors' addresses: Lucia Alessandrini, Universith di Parma, Dipartimento di Matematica, Via D'Azeglio, 85/A, 1-43100 Parma, Italy E-mail: lucia, a l e s s a n d r i n i @ u n i p r ,

it.

Giovanni Bassanelli, Universifft di Parma, Dipartimento di Matematica, Via D'Azeglio, 85/A, 1-43100 Parma, Italy E-mail: g i o v a n n i , b a s s a n e J l i @ u n i p r ,

it.

Marco Leoni, Universith di Parma, Dipartimento di Matematica, Via D'Azeglio, 85/A, 143100 Parma, Italy E-mail: marco, l e o n i @ u n i p r , it .