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l with the three properties given above. Set X = X U {boundary points of X} Let P be a boundary point of X, defined by a sequence {Xv}v:2;l' We definc neighbourhoods of P in X as follows. Let E > 0 be small and De = {z Ilz - al < E} (a E q or De = {z Ilzi > ~} U {oo} (a = 00), where a = limp(xv). Let Oe be the~()nnected component of p-1 (De) containing all but finitely many of the XII' and let ne be the union of ne with those boundary points Q with the following property: if {Yv}v:2;l defines Q, then {l/ YII f/: Oe} is finite (this is independent of the sequence {Yv} d:finillg Q). The {Ie (t: > 0 small) form a fundamental syste1n of neighbourhood of P EX - X.
I
This topology is Hausdorff if P, Q are boundary points defined by {Tv}, {Y,/} respectively, and P =/= Q, then, by the definition of the equivalence relation, there is c > 0 such that the components 0",10",2 of p-1(De) containing all but finitely many of the Tv, Yv respectively are distinct, and tle,l n tle,2 = 0. Moreover, p clearly extends to a continuous map p : JY -+ jp'1: i5(P) = a = limp(xv). A boundary disc around but finitely p : 0 -+ De
point P of X is said to the algebmic if the following holds: let De be a small = p(P) ancllet 0 be the connected component of p-1(De) containing all many points of a sequence defining P; then p(O) c Dc - {a} and the map - {a} is a finite covering.
a
If we set 6.R = {z E C Ilzl < R} and 6. R = 6.R - {O}, then there is n ~ 1 such that the map p : -+ Dc - {a} is isomorphic to the map Pn : 6.;,/n -+ DE - {a} given by TJn(Z) = a + zn if a E «::, Pn(z) = z-n if a = 00 (see Example 1 after Definition (1.10)).
n
This construction can be used to obtain the Riemann surface of a holomorphic function as conceived by Riemann. To do this, we first introduce the sheaf of germs of holomorphic functions on a Riemann surface.
2) a = b. Let U be a connected open set, a E U, and f, g, holomorphic functions on U so that the pairs (U, f), (U, g) define f , 9 respectively. We claim that N(U, f)nNW, g) = in fact, if flx (x E U) is a germ in:\h"Etintersection, then both f and 9 induce tl~e germ flx at x, hence coincide in some neighbourhood of x. Since U is connected, the principle of analytic continuation implies that f == g, so that -a f = -a 9 , a contradiction.
o
Let X be a Riemann surface, and let a E X. We consider pairs (U, I), where U is an open neighbourhood of a and f is a holomorphic flillction on U. Two such pairs (U, f) and (V, g) are said to be equivalent, and define the same germ of holomorphic function at a, if there exists an open neighbourhood W of a, W C Un V, such that flW = glW. An equivalence class is called a germ of holomorphic function at a; the class of a pair (U, f) is called the germ of f at a and denoted by f . The value at a of f is defined by f (a) = f(a) for any pair (U, f) defining f. "-0. -a -a
-a
If we choose a chart (U, U {co} if a = co), then 7r'r7r-l(D,,"{a}) ---7 DE - {a} is a finite covering (of n-sheets). In particular, 7r-l (DE - {a}) has only finitely many connected components. Moreover, if W is a connected component of V', then 7r'[W is again a covering, and so maps W onto pI - S. Hence V' has only finitely many connected components. (We shall see below that it is, in fact, connected.) Let WI, ... , Wr be the components of V'. Then 7rj = 7r[lIVj ---7 pI - S is a finite covering, hence every boundary point of Wj is algebraic. Let 1rj : Wj ---7 pI be the algebraic completion of lfj : Wj ---7 pI - S. If P E vVj - Wj and a = frj(P), there is a neighbourhood U of P and E > such that 1rjlU ---7 Dc is isomorphic to the map z f--+ a + zm (or z f--+ z-m) for some m > 0, so that, in particular, 7rjlU ---7 DE is proper. It follows that for any a E S, there is E > so that 1rjl7rjl(Dc) ---7 D( is proper. Since
°
D
°
°
irj[Wj
-->
jp'l -
S is proper, 1rj : lIVj
pI is proper, so that
---7
TVj
is compact.
Let pz": V ---7 C be the second projection (x, y) f--+ y. Then the function holomorphic on VI, so that T)j =I)[ Wj is holomorphic.
1)
=
PzlV' is
vVe claim that I)j extends to a meromorphic function on Wj. To see this, let a E S. Let P E vVj, 7rj(P) = a, and choose local coordinates z at P and tv at a so that 7rj becomes the map Z f--+ zm = w. If U is a small neighbourhood of P, by the definition of V and '1], vve have, if z f. 0,
al(W) n-l( z). + ... + an(,w,) ~+ --'/, ao( w)
Moreover, the av/ao 0, N 2 maxv
>
° so that
, =-
= u'n+1
m'2:n+l
Iwl:>E
,. 01
and boundedly
> O.
f(z -+-w)dW) d ( ---'W
](' e, the map --+
EBCn ® (Oajm7) i=l
x. vV; xC'''
-+
..,N of holomorphic
maps
is injective, because the kernel consists exactly of sections s with ordai (s) :c:: k Vi. This proves the theorem. The next finiteness theorem we shall need is somewhat
more difficult to prove.
Let X be a compact Riemann surface and 7T : E -+ X a holomorphic vector bundle on X. The sheaf of germs of sections lE of E is the sheaf U >--+ lEt U) = {space of holomorphic sections of E on U}. We shall denote by HI (X, E) the first cohomology space HI(X,lE).
u
Theorem 2. If 7T : E -+ X is a holomorphic vector b'undle on a compact R-iemann surface X, the fir.st cohomology HI (X, E) ~s a finite dimens'ional C-vector space.
IIsll = max sup ISi(x)1 I·
xEUi
sup ISi(X)1 xEV;
First remark the following: There exists a constant we have
e > 0 such
that,
"18
E
HO(X, E),
In fact, let Xo E [ri be such that Is;(xo)1 = 11.slluChoose j such that Xo E Vj. ::;
e ISj(xo)1
::;
e 11.s111/,
where = maXi,j SUPXEUinUjIlg;j(x)ll, Ilgll denoting the operator llorm of g E GL(n, (considerd as a linear map of into itself).
Proof. Let U c X be open, and suppose that there is a (holomorphic)trivialisation hu : 7T-I(U) -+ U X en Then, if V is open and V cc U (relatively compact in U), we shall denote by Eb('V) the space of bounded holomorphic sections of E on V, viz, the space of sections s: Il -+ E such that if hu 0 s(x) = (x, f(x)), x E Il, f(x) E then sUPxE1/ If(x)1 < 00; we set 11.o11v = SUPxEV If(x)l· With this norm, Eb(Y) is a Banach space; a different trivialisation h~T : 'Jr--I (U) -+ U X en gives rise to an equivalent norm on Eb(j>').
cn,
If U, hu are as above, and if, in addition, U is analytic~lly isomorphic to an open set in e, then HI(U, E) = O. This follows from the Ivlittag-Leffier theorem in §5 and the fact that if U is isomorphic to 0 C and hu : iT-I (U) -+ U x C" is ",Ii isomorphism,
We have
cn
::;
.s f-----> EB(Si mod zf)
by a family {S;}i=l,
Is;(xo)1 = Igij(~O)Sj(xo)1
ISi(x)1 ::; sUPziE1/i(lzfll#l)
II.sf ::; e 11.s111/ ::; Tke Iisf
1} is an analytic isomorphism.
< !}, then U V;
Vi,
I ::;
and ordai(.s):C:: k, we have
HO(X,E)
(c) There exist neighbourhoods Wi of [Ti and trivialisations hi : 7T-I(Wi) with corresponding transition functions gij : Wi n 1iVj -+ GL(n, C).
e
Hence, if x E
in Ui, so that SUP1/i1#
e
q
Let ai E Ui be the point with zi(a;) = O. We now prove the following (Schwarz's Lemma): Let s E HO(X, E) and suppose that ordai (.s) :c:: k (k :c:: 0 a given integer), i =
then HJ
,
E) c:::EEln copies HI(C!, 0).
e
Let 6.(1') be the disc {z E Ilzl < r}, r > O. We choose a finite of coordinate neighbourhoods on X and holomorphic trivialisations Wi x cr' of E on lVi with the following properties:
Z'i}·i=ll
hi
'Jr-I
(T'Vi)
..,N
-+
1) Zi is an isomorphism of Wi onto .6.(2). 2) Setting Ui(r) = z;I(.6.(r)), For!
:s; r:S; 2, we denote by U(r) the covering {Ui(r)}._ t_l,
and v E 7f-l(X)
...,N
= Ex, we write Ihi(v)1 for 1101if hi(v) = (x,w),
Zl(r)={~EZl(U(r),E)1
if
~=(fij),
b
Let N be an integer 2: 1. Let! :s; p < l' < 1 as above, and let CO(r,N) = = (Ci) E Cg(r) ordai(ci) 2: N}, where ai is the point in Wi with zi(ai) = O. By Schwarz's lemma (see proof of Theorem 1), we have
I
we have Ui Ui(!) = X. of X.
10
E
Also, if x E Wi
en
Set
jijEEb(Ui(T)nUk))Vi,j},
then
Thus, if we choose N such that C(~)N
:s; ~, we obtain:
For 1 E CO(r, N), we have
IhllT :s; IIhilT + C(~f IhIIT' i.e. IhIIT:S; 2110111T·In particular, oCO(r, N) C Zl(r) is a closed subspace and the quotient H = ZlCr)/oCO(r, N) is a Banach space. Moreover, Cg(r)/CO(r, N) is finite dimensional, so that the image of oCg(r) in H has finite dimension and so is closed in H (see the proof of the functional analysis theorem in §8). It follows that oCg(r) is closed in Zt(r), and Hl(r) = Zl(r)/oCg(r) is a (Hausdorff) Banach space.
IhllT=max
sup
,
XEUi(T)
Ihi(Ci(X))I
if
1=(Ci)ECg(r).
t
With these norms, Zl(r) and Cg(r) are Banach spaces. Let :s; p < l' < 1. We have: If 1 E CO(U(r), E) and 01 E Zl(r), then 1 E Cg(r); moreove~, there exists a constant C > 0 depending only on {Wi, Z'i, hi} such that
In fact, if 1 = (c;) and Xo E Ui(r); choose j such that xo E Uj(p).
(ei -Cj)(:2:o) +Cj(xo), and hi(cj(xo))
= hi ohjl(hj(Cj(xo))),
Ihi(Cj(:l:o))I :s; Clhj(Cj(xo))I:s;
Let Hl(r)
= Zl(r)/oCg(r).
(x) over x
Note. The proof of Theorem 2 in an earlier version of these notes used a non-trivial theorem of 1. Schwartz on perturbations of surjective linear maps between Banach spaces by compact ones. The arrangement of that proof avoiding Schwartz's theorem as given above was suggested by Madhav Nori. Theorem 2 is quite powerful. As an immediate application, we shall prove the following theorem.
Clh'llp ,
where C is the supremum of the norms ofthe matrices hiohjl Hence
[hi
We have Ci(XO) =
so that
Now, by Montel's theorem (which asserts that a uniformly bounded sequence of holomorphic functions on an open set 0 in iC has a subsequence converging uniformly on compact subsets of 0), the restriction map Zt(l) -; Zl(r) (1' < 1) is compact [since Ui(r) n Uj(r) is relatively compact in Ui(l) n Uj(I)]. Thus, the induced map Zl(1) -; Hl(r) is both compact and surjective. By the open mapping theorem, Hl(r) has a relatively compact neighbourhood of 0 (e.g. the image of the open unit ball in zt(l)) Thus, Hl(r)::: H1(X,E) is finite dimensional.
E
Ui(l) nUj(l).
(Ci(.I'O))I:s; IlhllT+Clhlip
Then, the natural map
Theorem 3. Let X be a compact Riemann
surface and 7f : L -; X a holornorphic line blmdle. Then L has a meTamorphic section which is not holomorphic. In pal··tu;UI(lr: (a) Any line bundle L on X 2S isomorphic to L(D) for some cl'ivisOTD on X and (b) ther'e exists a non·constant function on X.
Proof. Let a E X and let (U, z) be a coordinate neighbourhood assume also that there is a hoIomOIphic trivialisation hu :'7f-1(U) is an isomorphism for! :s;s :s; 1; 'In fact, the remark above shows that the map is injective. Surjectivity follows from the Leray theorem since the isomorphism Hl (U(2), E) -; HI (U (s), E) . fact~rs through Hl (s). Also, the restriction map Ht (1) -; Ht (s) is an lsomorplusm; 111 particular, the map Zl( 1) -; H"C (1') induced by the restriction Zl(1) ~,Zl(r) is surjective.
of a with -; U X C.
= 0;
Let /;; 2: 1 be an integer and Sk be the meromorphic section of Lover U for which hu '~(:»)k ),/ :c E U - {a}, Consider the covering U = {U, X - {a.} ~) and set \~ ,x . .t(k) U· f} f(k) j(k) I j(k) 0 h . ('" r" 1 fiJ12 ='Bk -La ;seo 21 =- 12 ane ij =1 ot erWlse t,]EjI,L;. c \) rph' 1. lsce_llles 1 an element f(k) E Zl(U,L). Since H (X,L) is finite dimensional and H1(U,L)-;
I
L
H1(X, L) is injective, if d = dime H1(X, L), there exist constants zero) such that
Cd+l
Cl,""
(not all
at (Xl, ... ,Xd) = f-l(z) But av(z)
=
(z E W - {a})
(w~Vz\)~N' so av
are bounded,
Since any meromorphic function on pl is rational . algebraic over C(f) of degree::; d.
The section s = s = - I:~+l CVS"
of L on X - {a} is meromorphic U - {a}.
U2
(and not holomorphic)
on X since
+ Ulan
Remark. This argument shows that if 9 = dimH1(X, 0) and a E X, there is a (nonconstant meromorphic function on X) holomorphic on X - {a}, with a pole of order ::;9 + 1 at a. Theorem :3(b) can be used to prove the following. Theorem 4. Let X be a compact Riemann surface and let M(X) be the field of merom orphic functions on X. Then /vi(X) is an algebraic function field in one variable. More pT'ecisely, if f is a non-constant merom orphic function on X, A1(X) is a finite algebraic extens'ion of the field C(f) of rational functions in .1. Proof. Let .f be a non-constant meromorphic function on X. We consider .1 : X --+ pl as a holomorphic map into pl (the poles map to 00 in pI = C Ij {oo}). Let C c X be the critical points of this map (points where f is not a local homeomorphism) and B C pI the image of C : B = .f(C). B,C are finite and let it = .1-1(C). Then f : X - it --+ pl - B is a finite covering, say of d sheets. Let 9 that
E
JV1(X).
V\le claim that there exist meTamorphic functions (g(:T))d
+ aL(f(:c))(g(x))d-l
+ ... + UdU(T))
ell, ...
r-
on pI such
= O.
To see this, if S is the set of poles of g, we define forz E pI - B i/h elementary symmetric function in ), ... , g(Xd), where {:2'1,. Clearly, we have (by definition of elementary symmetric functions),
1 ](8). for x E X - Acally to all of 1P'1.
,ad
.f (S)
by =
uvC z)
=
f-l
Thus, we have only to show that the a,/ extend mer omorphi-
Let a E B Ij .f(5') and let U be a neighbourhood of a such that the only poles of 9 on .f-l(U) lie in f-l(a), and such that there is a holomorphic fUllction w on U with w( a) = 0, w 't O. Then, there is an integer N > 0 such thaJ. (w 0 .f)N 9 is holomorphic on .1-l(U). If now VV is an open set with a E W cc V, then (w o.f)N 9 is bounded on f-1 (W), so that the l/th elementary symmetric function bv(z) of the values of (w 0 f)Ng
so extend holomorphically
to a.
has at most a pole at a. this shows that any 9 E M(X)
is
Choose go such that the degree [C(f,go): C(f)] is maximal. We claim that C(f,go) = M(X); in fact, if h E M(X), h rf- C(f, go), then, since C(f) has characteristic 0, the field C(f)(gO, h) = C(f)(g) for some 9 E M(X). But then, the degree of 9 over C(f) = [C(f) (go , h) : C(f)] is greater than [C(f)(go) : C(f)], a contradiction. This proves the theorem.
Set A~l(TiV) = CE'(W) 0c=(w) AO,l(liV) A~I(W) = HO(W, E) 0o(w) AO,I(W).
In proving Mittag-Leffler's theorem (H1(U,0) = 0, U c ic), we reduced the result to solving the equation ~~ = f. The method given there, when formalised, leads to an important interpretation of HI(X, E) [E being a holomorphic vector bundle on the Riemann surface X] called the Dolbeault isomorphism.
, We
X open.
If EIW
is trivial, we have
If W is such that EIW is trivial, there is a unique O(W)-linear map 8E,W : CE'(W) --+ A~I(W) induced by the map 1 W x iCn), then the map f{O(W,E) Ig,O(W) C00(W) -> CE'(W) [where HO(W, E) is the space of holomorphic sections of E over W] given by s ® f f-+ f . s is an isomorphism.
THE DOLBEAULT ISOMORPHISM. Let 11' : E --+ X be a holomorphic on the Riemann surfac8 X, and consider the map
We have: ker(8) coker(8) is naturally
HO(X,E), isomorphic
the space of holomorphic to HI(X, E).
sections
vector bundle
of E over X and
Proof. The statement that ker(8) = HO(X,E) is local. If U c X is open, 1I'-1(U)--+ UxiCn is an isomorphism and S E CE'(X), then 8sIU = af = 0 where (x, f(x)) = hu(s(x)), x E U. This is the case if and only if f is holomorphic.
°~
To prove the second part, we first prove the following lemma.
Proof. Let U = {U;}iEI be an open covering, and Sij E CE'(Ui rl Uj) be such that {Sij} E Zl (U, j[OO). Let {o:;} iEJ be a partition of unity relative to U, define Si E CE' (Ui) by S.i = LjEI G'jSij (where niSij is defined by (ajSij )(:1') = aj(:1')8ij(;r:) if x E Ui n [IJ, = if x E Ui - Ui n Uj). Then, as in the proof of the Mittag-Leffler theorem,
°
Sk
-
S£
=
L
=
j
L
O:jSk£
=
Ski
on
Uk
nUt -
j
E" --+ D AO,1 ("\ ell ows:'T Det- {, iJij } E' ZlfT'\ L!, vVe define the map H I ,~) E J\) as 10 (Sij E HO(Ui n , E)). Let 'Pi E (Ui) be such that 'Pi - 'Pj = Sij on [Ii n [li- Then B'Pi - 8'Pj = on Ui n Ui, and so the {8'Pi} define an element of A~l whose image in the quotient A~I(X)/8CE(X) is D({sij}).
°
\Ve check that this is independent of the choices made. First, if { \(1 T : it -> I is a refinement of {U;}, and 8 "13 = sT(")T(I3)I\!~ n 1;;3, we~may take 't/!v = \"T(a) as the solution of ljJ,;e - '~JI3 = sap, and vve see at once that {o1,L'a} define the same form as L
{a 0, :So 1
diD.
W
+ w)p(w)dw
/\ dtY! , z E U ,
where p( w) = 8~ ('p~w)), tV f= 0, p(O) = 0; p is Ceo and has support in the d~sc Iwl :So (in fact in the annulus ~E :So Itvl :So E).
E
8ii
1 =-2' 8'" /.. 1fZ
1 IC
8ex 0, the mth_ .tensor power L@m of L imbeds X in some projective space (i.e. if the corresponding map 'PL0= is an imbedding of X in jp'N, N + 1 = hO(L@m). It is called very ample if 'PL is already an imbedding. We have seen that if deg(L) > 2g, then L is very ample. Hence, if ample, since deg(L0m) = mdeg(L). Conversely, if L is ample, then an effective divisor D (some m > 0) since L0m must have at least section =f=- O. Moreover, D =1= 0 (since then L@m is trivial, and cannot mdeg(L) = deg(L0m) = clegD > O.
Proposition
> O.
4. A holomorphic
line bundle
L on X is ample
deg(L) > 0, Lis L0m ~ L(D) for one holomorphic imbed X). Thus
if and only if its degree is
Let X, Y be Riemann surfaces and 1 : X -+ Y a non-constant holomorphic map. .If a EX, b = I(a) and w is a local coordinate at b with w(b) = 0, we set orda(f) orda (w 0 j). The integer b(a, j) = orda (j) - 1 is called the ramification index of 1 at a; 1 is a local homeomorphism at a if and only if b(a, j) = O. Let now X, Y be compact Riemann surfaces, and let 1 : X -+ Y be a non-constant holomorphic map. We denote by gx, gy the genera of X, Y respectively. Let b = EX b( a, j); b is called the (total) ramification index of I. Let C be the set of critical points of I, i.e. C = {a E Xlb(a,j) > O} and B = I(C) the set of critical values (sometimes called the branching locus).
La
We triangulate Y by simplices in which all points of B = I(C) are vertices, and assume that the simplices are sufficiently small. We can then lift the triangulation by 1 to a triangulation of X. If we denote by eo(X) the number of vertices, by el (X) the number of edges (= I-simplices) and by eo (X) the number of faces (= 2-simplices) jn the triangulation of X, with similar notation for the triangulation of Y, we have e2(X) = de2(Y), el(X) = del(Y), eo(X) = deo(Y) - b [if ai E C, then each edge at bi = I(ai) lifts to b(ai, j) + 1 edges all ending in the same vertex ai; the cardinality of 1-1(B) = d (cardinality of B) -b]. Thus, we have 2 - b1(X) = d(2 - b1(Y)) - b. If we take Y = ]P'1, there exists a non-constant holomorphic map 1 : X -+ ]P'1 (= nonconstant meromorphic function on X). Moreover, we have 911" = 0, b1(]P'1) = O. If we denote by d the degree of I, we have
1= 0 be a meromorphic I-form on Y and let Wo = f*(w); we have deg(wo) = 2gx-2. If a E X, b = I(a) and we choose local coordinates z at a and w at b (z(a) = 0 = w(b)) so that near a, the map 1 is given by z zn = w, then n = orda(j). If w = h(w)dw Let w
f-t
near b, then
Wo
= j*(w)
=
h(zn)nzn-1dz
near a, so that
2gx If d is the number of sheets of 1 (= degree of j), this gives (summing first over a E and then over b) deg(wo) =
L: ( L: bEY
ordaU) )ordb(w)
+
L:
(ordaU) -
1-1 (b)
=
b1(X) ;
dimH1(X,0)
= dimH°(.X,O)
is a topological invariant
We pass now to a discussion of Weierstrass points. Let X be a compact Riemann surface of genus 9 = dimH1(X, 0). We have seen (Th.3 in §7 and the remark following) that if P E X, there is 1 meromorphic on X, holomorphic on X - P (but not at P) with a pole of order :s: 9 + 1 at P.
1)
bEY,aEf-1(b)
aEf-1(b)
in particular, the gen'us gx olX.
=
= ddeg(w) + b.
It is natural to ask if this result can be improved and the order of the pole reduced; as we shall see, this is only possible for special choices of P (finite in number). Given P E X, let (U, z) be a coordinate neighbourhood at P with z(P) = O. We call P a vVeie-rstmss point if there is a meromorphic function f on X. and constants Co, ... , Cg-l, not all zero, such that in particular, if there is a non-constant holomorphic map X -+ Y, then gx gx = gy :2: 1, we must have b = 0 and d = 1 unless gx = gy == 1.
:2:
gy; if
With the same notation as above~'let aI, ... ,aT be the points of C, let bj = f (aj)' denote by X(X), X(Y) the topological Euler characteristic of X, Y, so that, e.g. x(X) = dimcHO(X,C) b1 (X)
= dime
HI (X, C)
- dimcHl(X,C)
=
+dimeH2(X,C)
1st Betti number of X .
We
(i)
IIX - {a}
(ii)
f -
is holomorphic
L~=;z::+1
is holomorphic at P.
According to the analogue of the Mittag-Leffler theorem given in §IO, this is the case if and only if the following holds: There exist Co, ... ,Cg-l not all. zero such that g-l
= 2 - h(X),
resp
(L: z~:lw) 1/=0
= 0
(2:S k:S n) then W(1,g2,
... ,gn)
= W(~,
== O.
... ,~)
2:=~
constants C2, ... , Cn, not all zero, with Ck ~ that 1, g2, ... , gn are linearly dependent over iC.
==
0 on V, i.e.
By induction, there are
2:=; Ckg"
= constant, so
Returning to Weierstrass points, let WI,' .. , wg be a basis of HO(X,J.'l) as before, (U, z) local coordinate. Set W(Wl, ,Wg) = W(fI, .. ·,lg) ifwk = Ikdz. Then, since the Wk are iC-independent, W(Wl, ,wg) ¢ 0 on U, and we see that Weierstrass points are isolated, i.e. there are only finitely many Weierstrass points on X.
'30
g-l
resp (L
z~:l Wk) = COlk,O
+ cl/k,l + ... + cg-l/k,g~l
One further remark. If w = w(z) is another coordinate system on U, so that Wk = Ikdz = gk( w )dw, thcn Ik = gk (w(z)) ~"::' and we find that
1/=0 g-l
=L
c~li")(O).
1£
1'=0 v.
1(1') k -_
g-l
LCvliv)(O)
det
=1=
1+2+
(0, ... ,0); this is the case if and only if
Set W(fI,
... ,ln)(z)
called the Wronskian of the functions
If
+ ... + c,Jn ==
cl!l
0,
Ci
a linear combination of the columns
Ii")],
Let U be a connected open set in iC 10:;1' jp'1 and (z, w) f-+ z from Y to jp'l Moreover, since I :X -> jp'1 is not branched over 00 and the number of branch points is 2g+ 2, P is of the form c(z - Z1) ... (z - Z2g+2) where c 1= 0 is constant, and Z1, ... , Z2g+2 are distinct points in iC. We may assume that c = 1. Consider now the I-forms on Y defined by ==
Wv
Z
v-1
dz
-
W
°
Since 2wdu' = pI (z )dz on Y, and pI (z) 1= at the branch points Z1, ... , Z2g+1, we have 2zv-1 P~(z) is holomorphic at points on Y over jp'1 - {oo}. Near Z = 00, we have
Wv =
W = ±zg+1(l + 0(;)), so that Wv = ±zv-g-2 (1 + O(;))dz and this is holomorphic at z = 00. Thus W1, ... ,wg form a basis of HO(y, 0). Also, we see that Wo 1= 0 over C = jp'1 - {oo} and wg 1= 0 at the points over 00. Consider the holomorphic map 'PKy = 'P : Y -> jp'g-1 given by the canonical bundle of Y. We see that 'PlY - z-1(00) is the map (z, w) H (1, z, ... , zg-·1), and the image of Y is isomorphic to jp'l Moreover 'P(z,-w) = 'P(z,w). We have an isomorphism 7f X -> Y taking z to the function f. Moreover, the map 'PKx determined by the canonical bundle of X is intrinsically defined, up to a linear transformation of jp'g-\ it is called the canonical map 01 X. Thus, if I : X -> jp'1 is a function of degree 2, we see that it is isomorphic to the map 'PKx : X -> 'PJ<x(X) C jp'g-l (and 'PJ<x(X) c:::jp'1). We conclude that I ~s unique up to an o:utomorphism 01 jp'l, i.e .. that two functions of degree 2 differ only by a Mobi'U.s tmnsfomwtion f f-+ a, b, c, dEe, ad - bc 1= O. Now, if f : X -> jp'1 is of degree 2, then any branch point P of f is a Weierstrass point. This is obvious if f(P) = 00; if
~ft~,
f(P)
1=
00,
consider
(j -
1
f(p)r
We shall now show that conversely, any \Veierstrass point on the hyperelliptic curve X is a branch point of the (essentially unique) map f : X -> ][',1 of degree 2. viz the canonical map. We identify X with the Riemann surface of 'UJ2 - P(z) = 0, where
P = (z - Z1)'" (z - Z2g+2), the Zj being distinct. A basis of HO(X,O) is given by = d z, v = 1, ... , g. If W 1= 0 ·(i.e. P(z) 1= 0), and z 1= 0 the Wronskian of v zV-1/w, v = 1, ... ,g equals w-gW(l, z, ... , zg-1) = W-gcg (where cg = (v!)) [see proof of Lemma in §12, where we saw that W('Pfr,··· ,'PIn) = 'PnW(f1, ... , In); also ·W(l, z, ... , zg-1) is the determinant of a triangular matrix with I,ll, ... , (g - I)! on the diagonal]. If z = 00, then Wv = ±zV-g-2{1 + O(-~)}dz = =r=(.;y-v{l + O(~)}d(~) and the Wronskian at 00 is again the determinant of a triangular matrix with non-zero diagonal elements. Thus the points with P( z) 1= 0 and the points over z = 00 are not Weierstrass points, proving our claim.
z:~'
W
n~:i
Thus, if X is hyperelliptic, the Weierstrass points are exactly the branch points of the canonical map 'PJ<x : X -> 'P(J(x) C JP'g-1. There are 2g + 2 such points. When 9 > 2, we have 2g + 2 < (g - 1) .g. (g + 1), the bound on the number of Weierstrass points given before. It can be shown that non-hyperelliptic curves have more than 2g+2 Weierstrass points. For non-hyperelliptic
curves, the canonical map is an imbedding.
Theorem. Let X be a non-hyperelliptic compact Riemann surlace Then, the canonical bundle J(x is very ample, i.e. global sections common zeros and 'PJ<x : X -> jp'g-1 is an imbedding.
01 genus of J{x
g(? 3). have no
Proof. 1) Given P E X, 3w E HO(X, 0) with w(P) 1= O. If this were false. the map O.:.p -> 0 given by tensoring with the standard section sp of L(P) would induce an isomorphism HO(X,O_p) 08'; HO(X,O). Now, hO(O_p) - h1(rLp) = 1- 9 + (2g3)
=
=
=
1 (since, if there exists a non-constant I with --+ IF1 is an isomorphism). Hence hO(O), so that HO(X, ll_p) cannot be isomorphic to HO(X, 0).
9 - 2 and h1(O_p)
hoe Op)
(f) ? -P, f has a single simple pole and I : X hO(O_p)
= 9 - 1
4. Hence, it suffices to prove the theorem for n = 3. But in this case, the theorem is equivalent to Lemma 2. Before giving some applications of the general position theorem we introduce some terminology. Let X be a compact Riemann surface and L. a holomorphic line bundle on X. If 11' is a vector subspace of IIO(X,L), 11 1= {O}, we call the set of effective divisors {D [D = dives) for some s E 11}, the linear system (or series) determined by V. If V = HO(X, L), we call it the complete linear system of L. If L = L(D) for some divisor D. then, this complete linear system consists of all effective divisors D' ?: 0 linearly equivalent to D: D' ~ D, D' ?: O. This is called the complete linear system of D and denoted by IDI. We shall write dim ID[ = hOeD) - 1; it is called the dimension of the complete linear system, and jD[ is in (1 - I)-correspondence with the projective space (HO(X, L(D)) - {O}) ICC* = IP'(IIO (X, L(D))). If hOeD) > 0 and hl(D)_~ hO(O_D) > 0, we call D a special div'isor; this means that both D and Kx -D, where Kx IS a canolllcal divisor on X, are linearly equivalent to effective divisors. We begin with the following Lemrna 3. Let D be a divisor with hO(D) > 0, and .let r be an irdeger ?: O. Then dim IDI ?: r zf and only if, for any divisor II ?: 0 of degr'ee r, there is D' E ID[ with r D' ?: ll; in particular', if PI, .. " Pr E X, there is Dr E IDI with Pi E supp(D ) for i = 1,2,. ., T. If this cond-ition holds fo1' all Pi in a non-empty open set in X, the Pi being distinct, then dim IDI ?: T.
::c~
Proof. Suppose that dim HO (X, L(D)) ?: 'T' + 1, and let II = 'nvP'c· In terms of a local trivializatioll hv of L(D) at pv• and local coordinates (Uv, at Pv with
Zv(Pv) = 0, if 05 E HO(X, L(D)), then (05) ?: ~ if and only if (d~j"hv(s) Is=Pv = 0 for v = 1, , k. Thus, the condition is that 05 lie in the intersection of the kernels of the nr + + nk = 'I' linear forms 05 1-+ (d~Jf.J, hv(s) Is=P on HO (X, L(D)),
o :::: f-l < nv,
and this intersection
has dimension?:
dim HO (X, t(D)) -
'I'
?: 1.
v
The converse results from the following general fact. Lemma 4. Let X be a Riemann surface, L a holomorphic line bundle on X and V a vector subspace of HO(X, L) of dimension k. Then there are k points Pi>"" Pk EX such that if 05 E V and s(Pv) = 0, v = 1, ... , k then s == O. (In fact any k points in general posit'ion will do.) Proof. If k > 0, let Sr E V, Sr '/= 0, and let Pr E X be so that sr(Pr) i' O. Then Vr = {s E Vls(Pr) = O} is not all of V, sohas dimension k - 1. If k - 1> 0, choose 05 2 E Vr and P2 E X with S2(P2) i' O. Then V2 = {s E Vrls(P2) = O} = {s E Vls(Pt} = 0, s(P2) = O} has dimension k - 2. We have only to iterate this process. A consequence is the following important Proposition (i.e. hO(Di)
2. Let Dr, D2 be divisors i = 1,2). Then
which are linearly
equivalent
to effect'ive divisors
> 0,
Adding, we have 2hO(D) ::::d + 2, hO(D) :::: ~d + 1. Moreover, if equality holds, then dim IDI + dim If( - DI = dim IKI so that any divisor f(' ?: 0, J{ ~ J(', can be written J{' = Dr + D2, Di ?: 0 with Dr ~ D, D2 ~ J{ - D. Assume that X is not hyperelliptic, and consider X C !F'g-r imbedded by the canonical map. If H is any hyperplane transverse to X, then the points of H n X give us a divisor J{' ~ J{, f(' ?: 0, and we can write J(' = Dr + D2 with Dr ~ D, D2 ~ K - D, D1, D2 ?: O. Assume also that D1, D2 i' O. If [D;] is the linear subspace of !F'g-r generated of the Riemann-Roch theorem) dim[Dr] = degDr
- hO(Dr)
by Di, we have (by the geometric form
= d - hO(D)
dim[D2] = degD2 - hO(D2) = 209- 2 - d - h°(I( dim IDr!
+ dim
ID2
1
::::
dim IDr
Moreover, if equality holds, then any D E IDr + D2 written D = D~ + D; with D; E IDil, i = 1,2,.
1
1
(i.e. D ?: 0, D ~ Dr
+ D2)
can be
Proof. Ifri = dimlDiI and Pr, ... ,Pr1, Qr, ... ,Qr2 are any points in X, there is D; ~ Di, D; ?: 0 with Pi E supp(D;), Qj E supp(D;). Then D~ + Db E IDr + D2 and contains all (rr + '1'2) points Pi, Qj in its support; hence the inequality. 1
The divisors D~ + Db with D; ?: 0, D; ~ Di form an (Tr + 7"2)-dimensional subvariety of the projective space IDr + D2 = !F'( HO (X, L( Dr + D2))). If equality holds, this subvariety must be the whole projective space. 1
We come now to an important
theorem.
We denote by f( a canonical divisor on X.
CLIFFORD'S THEOREM. Let D be an effective special divisor on X (so that hO(f( - D) > 0). Let d be the degree of D. Then dimlDI::::
1
2d
1
= ~ degD
- D) .
+ D2
.
Since the assumption this gives
of equality dim !DI = ~d implies that hO(D)
+ hO(K
- D) = 09+ 1,
Hence both Dr and D2 span linear subspaces of dimension:::: 09- 3. If d ?: 09- 1, the points of Dr are linearly dependent, if d < 09-1, the points of D2 are linearly dependent. Since H is an arbitrary hyperplane transverse to X, this contradicts the general position theorem. Thus, if X is not hyperelliptic,
we must have Dr or D2 = 0, and the theorem is proved.
We now give another proof of Chifford's theorem not using Castelnuovo's tion theorem. vVe have only to prove the statement about equality. Proposition 3. Let D 2g - 2. Then dim ID[ :::: hypeTelliptic.
o
general posi-
be em effective divisOT of deo9Tee d. Assume that 0 :::: d :::: If D i' 0 and D f K, and if equality holds. then X is
Proof. Vie have hO(D) - hO(E - D) = 1- 09+ d ::::-~d + d (since 9 -- 1 ?: ~d) so that, if hO(K - D) = 0, we have dim IDI ::::·~d -1. Thus, we may assume that D is special, in which case the inequality is a conseq{;ence of Prop. 2 (as in the first part of the above proof of Chifford's theorem), and in either case, hO(D) + hO(K - D) ::::9 + 1. Assume that D is special and that hO(D)+hoU:
-D)
=
09+1 (i.e. that hO(D)
=
~d+l).
If d = 2, then harD) = 2 and there is a non-constant meromorphic function f with (I) 2: -D and f is of degree 2 so that X is hyperelliptic. We shall show that if deg D > 2 and K rf D, then there is a divisor Do 2: with deg Do < d such that hO(Do) + hO(K - Do) = 9 + 1. Since deg Do < d ::; 2g - 2 = degK, we have K - Do rf and we can continue till we obtain a divisor D' with deg D' = 2, harD') = 2, so that X is hyperelliptic.
°
°
Let D' 2: 0, D' ~ K -D. Then D' f 0. Choose points P E supp(D'), and Q rf. supp(D'). Since dim IDI = ~d > 1, we can replace D by a linearly equivalent effective divisor whose support contains P and Q; we assume therefore that D has this property. Let Do be the largest divisor::; D and::; D' (i.e. if D = 2:aD(a)a, D' = 2:aD'(a)a, then Do = 2:affiin(D(a),D'(a)). a. Clearly, Do(P) > 0, Do(Q) = 0 so that degDo deg D and Do f O.
"2el
,
+ 1::; hO(D)
=
1-
9
+ d,
g::; d - n .
Equality implies that hO(D) = n + 1, i.e. restriction to X of linear forms onpn give all .sections of OD. This means, of course, that hyperplane sections form a complete linear system. Corollary.
A smooth non-degenerate curve of degree n in pn is rational, i.e. 9 = 0.
In fact, it can be shown that the only such curve is the closure of the image of C under the map z f-t (l : z : Z2 : .•. : z") which we met as the canonical curve of a hyperelliptic Riemann surface. Another application of the general position theorem was made by Castelnuovo himself to estimate the genus of a curve of degree el > > n in P". Let X c pn be a non-degenerate imbedding of a compact Riemann surface in P"; let el be the degree of X. Then, as we have seen el 2: n. Let N = [~=i] (integral part). Let D be the divisor on X given by a general hyperplane section X n H. Then hO(kD) -hO((k-l)D) 2: l+k(n-l). Moreover, 'if equality holds for a ceTtain value of k, then HO(X, OkD)/ HO("X, O(k-l)D) is genemted by HO(X,D), i.e., the natuml map SymkHO(X,OD) --+ HO(X, OkD)/HO()C, O(k-l)D) is sv.Tjective.
Lemma 4. 1) Let 1::; k::; N.
2) If k > N, HO(X,
It follows from the exact cohomology sequence that hO(D)
so that D cannot be special, i.e. h°(I( - D) = 0. Hence
we have hO(kD) OkD)/ HOpI"., O(k-l)D)'
- hO((k
- I)D)
= el, and
HO(X,OD)
genemtes
Proof. vVesuppose that the hyperplane H is so chosen that it intersects X transversally and such that D = X n H is in general position, i.e. that no n points of D lie on a plane of dimension n - 2. If k ::; N, we have k(n - 1) ::; d - 1, 1 + k(n - 1) ::; d. Choose a set E of 1 + k(n - 1) points of D. If PEE, write E - {P} = E1 U ... U Ek where each Ej has n - 1 points. By Castelnuovo's general position theorem, the points of Ej (j = 1, ... ,k) generate a plane Bj of dimension n - 2 which does not contain P. Hence there is a hyperplane Hj with P rf. Hj, Eei C Hj, so that there is a linear form Aj on p7l with Aj(P) f 0, Aj(Ej) = 0. Let AI" = Al .. /\1;; then Ap is a hornogeneous polynomial of degree k vvith Ap(P) f 0, Ap(E - {P}) = O. Let 3(1") E HO(X,OkD) be the section ApIX. We claim that the images of the sections s(P), PEE, in HO(X,OkD)/HO(X,O(l._l)D) are linearly independent. In fact, If s D is the standard section of OD with diviwr D, if {cp } PEE are complex numbers such that
L cps(P) PEE
E 3D' HO(X,
(k -1)D)
,
then LPEE cps(P) = 0 on D, hence on E; but the value of the sum L cps(P) on Q E E is cQs(Q)(Q) [since s(P)(Q) = 0 if Q f= P]' so that, since s(Q)(Q) f= 0, we have cQ = 0 (VQ E E). Hence dimHO(X,QkD)/HO(X,O(k_l)D) ~ cardinality of E = 1 + k(n - 1). Since the sections s(P) are clearly E SymkHO(X,O) (since Ap is a product of linear forms), we have shown that the image of Symk HO(X, OD) in HO(X, OkD)/ HO(X, O(k-l)D) has dimension ~ 1+k(n-1). This proves both statements in part 1) of the lemma.
N
hO((r
+ N)D)
=
'2.JhO(kD)
I;
N
~ I;(1 + k(n = 1 + rd
1)) + 1 + rd 1
+ N + 2N(N + 1)(71 -
1
g::; (r + N)d - rd - N - 2N(N + 1)(71 -
.:!.!:.,. OkD-->CD-->O
hO(kD) - hO((k - 1)D) ::; dimHO(X,
>
-
N, H
O( r)
X,OD
CD)
generates
=
= C if
xED,
=
(by Lemma 4)
k=l
On the other hand, the exact sequence
This proves 2) and also that for k
(hO(kD)-hO((k-1)D)
k=N+l
~=i,
(where SD is the standard section, and CD,x = OkD,x/OCk-r)D,x otherwise) implies that
+ hO(O.D)
N+r
+
To prove part 2) we remark that if k > and P E supp(D) we can write supp(D)-P = E1U·· ·UEk where each Ej has at most 71-1 points. As in the proof above, we can construct a homogeneous polynomial Ap of degree k, Ap = Al ... Ak where Aj is a linear form with Aj(P) f= 0, Aj(Ej) = 0. Then, as in the proof above, the sections s(P) E HO(X, OkD), s(P) = AplX are linearly independent, in HO(X, OkD)/HO(X, 0Ck-l)D), and we find that hO(kD) - hO((k - 1)D) ~ d.
O-->O(k-l)D
- hO((k -1)D))
k=1
°
1 2
+ 1)(71 -
=
N(d - 1) - -N(N
=
N2(n - 1) + EN - 2N(N
1
1) .
1)
1)
+ 1)(71 -
1 1) = 2N(N
- 1)(71 - 1) + EN .
d.
HO(X,OI.n) ..~ H X, O(k-l)D)
O(
\.
Further, equality implies that hO(kD) - hO((k -1)D) = 1+ k(n -1) for all k ::;N, and the fact that HO(X, OD) generates HO(X, OkD) for all k ~ 2 follows by induction on k from the lemma. [Note that the function 1 E HO(X, OD), so that Symk-1 HO(X, OD) C
From this, we obtain
Symk(HO(X,OD)).]
Castelnuovo's genus estimate. Let X be a (smooth) nondegenerate Let d = deg(X), and set N = [t:i]· Define E (0::; E < 71 - 1) by
There are many beautiful geometric applications of this theorem of Castelnuovo. There is an excellent discussion of this circle of ideas in the book of Arbarello, Cornalba, Griffiths and Harris: Geometry of Algebraic OUTves, 1. (Springer-Verlag). We mention only one consequence, a famous theorem of Max Noether.
curve in jp'n
d -1 = N(n -1) + E.
1
1) + NE.
g::; 2N(N -1)(71 -
Proof. Let T be a large positive integer. Then h Roch theorem
NOETHER'S THEOREM. Let X be a compact Riemann surface of genus g ~ 3. Suppose that X is not hyperelliptic. Then, if J{x is the canonical line bundle of X and m ~ 2, the natural map
1
(( T
+ lV)D)
= 0, and by the Riemann-
Proof. vVe consider X C jp'g-l as the canonical curve. Then the hyperplane section D is a canonical divisor, hence deg D = deg J{ = 2g - 2. The integer N above is N = [2g1~23] = 2 if g > 3, = 3 if g = 3. If g > 3, E = 2g - 3 - 2(g - 2) = 1 and
!N(N
- 1)(71 - 1) + NE
= g - 2
+ 2 = g.
If g
=
3, E = 0, N = 3 and
.
!N(N - l)(n - 1) + NE= 3(g - 2) = 3 = g. Thus we have equality, and Noether's theorem follows from Castelnuovo's. It should be added that if (g 2': 3 and) X is hyperelliptic, the above result definitely fails. This follows, e.g. from the fact that J(~m is very ample for large rn, but the mapping 'PKx induced by J(x is not injective.
Before proceeding further, we recall some facts about compact oriented surfaces. We shall not prove them here; proofs can be found in, for example [6]. The basic theorem about the classification of compact orientable surfaces is the following: A compact orientable Ceo surface X without boundary with a finite number of handles attached.
is diffeomorphic to a sphere
The number 9 of handles is half the first Betti number of X; thus, if X is a compact Riemann surface of genus g, it is diffeomorphic to a sphere with 9 handles, and two such surfaces are diffeomorphic. A sphere with 9 handles can be described, up to diffeomorphism, as follows. Start with a convex polygon Li. with 4g sides aI, bI, ai, bi, ... , ag, bga~, b~ in C, oriented, as usual, "counter clockwise". If aI, ai are the directed segments pq, pi q', we identify aI, ai by a linear map of pq onto q' pi (i.e. one taking p to q' and q to pi). Thus, ai is identified with all. We make similar orientation reversing identifications of aj with aj and of bj with bj (j = 1, ... g) This identification is indicated schematically below
the above identification process. This gives us (piecewise differentiable) curves ai, bj on X. If 'P is a C= I-form defined in a neighbourhood of these curves, and is closed, we
and call these the a- and the b-periods of 'P. Let 0: be a C= closed I-form on X, 'P a C= closed I-form defined in a neighbourhood of U ai U U bj. We identify them with I-forms on .6.(= 6.) and on a neighbourhood of fJ.6. respectively. Fix Po E A and, for P E .6., set u(P) = 0: (.6. is simply connected).
J:a
We then have
Under this identification, .6. becomes a compact surface X diffeomorphic to a sphere with 9 handles. All the vertices of .6. map onto the same point Xo EX, and aj, bj map onto closed curves at XQ in X; we shall call these curves again aj, bj. The segments bl -1 b-1 . 1y. aj' j map onto aj , j respective I
Proof. Let PEak and let pi be the corresponding joining pi to P as shown.
point of a;,. Let 7 be a curve
These curves aj, bj in X form a basis of Hi (X, Z) over Z, and their intersection numbers are given by ai . aj = 0, bi· bj = 0, ai . bj = 8ij = -bj . ai (8ij is the Kronecker 8; 8ij = 1 if i = j, 0 otherwise). These curves are indicated schematically in the figure below.
Then u(P) 1
bk
,
- u(PI)
= r.o:; now, the image of ""y in X is a closed curve homologous to
so that, since a is ;losed,
If we slit a sphere with 9 handles along curves aj, bj as shown (which have only Xo as a point of intersection of any pair of them), we obtain a simply connected polygon .6. with 4g sides. Let now X be a compact Riemann surface of genus g. We fix an identification (diffeomorphism preserving orientation) of X with a surface obtained from a 4g - gOll6. by
It(Q) - U(QI) =
1
0:
ak
= Ak(a)
.
Then, the matrix (AjkhSj,kSg
It; ~(L 1~L +
UyJ =
=
tj k=1
+
(u(P)
In fact if Aj is the g-vector (fa, Wj, ... , Jag Wj), then, if .EcjAj .ECjWj are zero, so that .ECjWj = 0, Cj = 0 Vj.
+ l)uyJ +
- u(PI))yJ(P)
t1
. In view of this corollary, we can choose a basis
(u(Q) - U(Q'))yJ(Q)
h
k=1
Uk
1
Q E bk and pI, Q' are the corresponding points on
(with the notation above: PEak, a~, b~ respectively)
is invertible.
Wj = fJkj
of HO (X, fJ) such that
(Kronecker fJ) .
Uk
basis of HO(X, fJ) [relative to the choice ai, bj of basis
We shall call this a normalized of HI(X,Z)].
Theorem. (Riemann's bilinear relations). genus g > O. Let WI, ... ,wg be a normalized
which proves the lemma.
, wg
WI, ...
= 0, the a-periods of
Let X be a compact Riemann basis of HO(X,fJ). Set
surface
of
We deduce from this the following basic Proposition 1. Let X be a compact Riemann surface of genus g notation introduced above. If
W
is a holomorphic 1-form on X,
# 0,
W
>
O. We use the
we have
Then, the complex definite.
matrix
B = (Bjk)
is symmetric,
and its imaginary
Proof. have
t,.
have the same meaning as before, and let Uj(P)
r
UjWk =
paTt is positive
9
< O.
1m LAdw)Bk(w) k=1
Let aj,bk,
1M
r Wj /\ Wk
lx
(Stokes' theorem)
= J~Wj.
We
= 0;
on the other hand, (by Lemma 1),
r
1M
.tLW
r
=
lt;
du /\
w=
If (U, z) is a local coordinate on X, and z = :r f E O(U),
1
W /\
u
w
=
1.u Ifl
2
t
W /\
lx
UjWk = t(Av(Wj)BvCWk)
8t;
+ iy,
dz /\ dz = -2i
1
w.
we have setting
J Ifl
2
d:r /\ dy .
W
=
f
= Bj(Wk)
dz on U, Thus, B is symmetric. Prop. 1,
- Bv(Wj)Av(Wk))
v=1
Now, let
- Bk(wj)
CI, ...
,
since
L A,/(.EckWk)Bv(.EckWk)
W=(Wl,""Wg),
Suppose there is a meromorphic function f with (f)
L~=l C"W"
7 E 21fiZ
with c" E C. Further, I/'
".£, niA(Pi) i=l
.
=
.- Qk),
D. Then
the Pk, Qk being
7=
~~=1
WPkQk
+
V closed curves I on X (not containing
any of Pk, Qk)'
re
are such that
I/' 'P E 21fiZ
then (1) = D where f(P)
defined because of the condition on
V closed curves (, where 'P = =
expU:a 'P) (expU:a 'P)
is well-
I/' 'P).
We assume X identified with a convex polygon 6. as described earlier by slitting X along curves ai, bj which do not pass through Ph, Qk. Let "
(where all integrals are along c). If c' is another curve from Po to P, there is an element { E HI (X, Z) with Ie Wk = Ie' Wk + I/' Wk for all k, so that the map is well-defined. [In the intrinsic description, A(P) = class of the linear form W f-7 lei on H°(.X, fl).]
".£, niPi
A,
,,=1 }Po
L~=l WPkQk + L~=l C"W", Fix a base point Po E X. We define the Abel-Jacobi choose a curve c from Po to P and set WI,.'"
Qv
Proof. Since D has degree 0, we can write D = L~=l(Pk points of X (and no Qk being one of the P's).
Conversely, if c" E
}Po
equivalent
the integration being along some curve from Po to P", and from Po to Q" respectively (the curves being, for each v, the same for all Wk)·
called the Jacobian of the Riemann surface X.
A(P) = ( (
on X of degree O. Then D is linearly
'P =
9
".£, WPkQk
+ ".£,
k=1
c"W"
.
v=l
Then, I/' 'P E 21riZ for all closed curves in X - U{ Ph, Qd if and only if A,,( cp) = Iav cp E 21fiZ and B,,(cp) = Ibv'P E 27fiZ for all v = 1, ... , g; in fact, if Ck, C~ denote small circles around Pk, Qk respectively (with respect to coordinate neighbourhoods around these points), then ( is homologous to an integral linear combination of 0.", bu, Ck, C~) and J~" 'P = +1, 'P = -1 Vk since wPQ has residue +1 at P and -1 at Q.
Jc~
Thus, there exists a merom orphic function
f
with (f) = D if and only if:
Now, Av( A- A(D) is a holomorphic bijection between complex manifolds. It is a standard fact in complex analysis that such a map is biholomorphic. We note two further consequences of these results. Let Div(X) be the set of all divisors on X and let P(X) be the subset of those divisors (of degree 0) which are linearly equivalent to O. We set Pic(X) = Div(X)/P(X) . If Divo(X) is the set of all divisors of degree 0 on X, we set Pico(X) = DivO(X)/ P(X). Then, we have
15. The Jacobian and Abel's Theorem Th~orem 3. The Abel-Jacobi abehan groups)
map A : Div(X) A: Pico(x)
-+
-+
J(X)
leX) .
Proof. That A : DivO(X) -+ J(X) is a homomorphism of abelian . Abe.l's theorem asserts that the kernel of this map is exactly P(X)' gtrhoutpsIS cle~r. the mduced A po o(X) J(X) , ' , so a we obtam follows fro~ ;~;eor~m I~; if D~ sg(X) ~~a~~V~n~~l: t;:::v;o:~:t it~s;UJective. Th~s the base pomt m the definition of the Abel-Jacobi map A : X -+ ~(Y) a~d Po IS has degree 0 and maps onto (. " , en - gPo
(tj'
Theorem
4. If the genus 9
> 0,
the Abel-Jacobi
map A- . X ~ . J(X)
. an zm . beddzng. . ~s
Pro~f. If P, Q E X and A(P) = A(Q), then, by Abel's theorem tl' . . functIOn f on X with (f) = P _ Q' h' .h 1 . .' Iere ISa melomorphlc " . ' I.e. w IC Ias a smgle sImple pole As we h . seen, thIS ImplIes that X is isomorphic to jp'I and the genus is 0 Th A'. '.. . aye . . ,. us, IS mJectIve. A ISgIVen by (JxPo WI,·· ., JX) 't s tangent map at P E X is given b (J (P) Po Wg ane1 1 fg(P)), where Wk = fkdz in terms of a local coordinate UTeha Yth 1 h',···' ve seen at t e (1Ience tlIe f) k cannot all be zero at the same point so that. V\ d A I'Sa,Iso lllJectlve. '" Wk , '.':1 We en.d this section with a remark on Theorem 2. We have treated th f h A JacobI sg( X) () . . e case 0 t e bel'. map" -+ J X because It IS the most important. However th h and ItS proof generalise as follows. ' e t eorem Theorem 5. Let.1:::; k :::;g, and consider the map A : Sk(X) -+ J( Y' Jf D _ PI+."+PkESk(X),thenthejibreA-IA(D)isasmooth b .. "). J zsomorphic to jp' (HO (X 0 )\ T11e k 1tl t su mamJold, analytzcally . ' D J' ran 0 Ie angent map to A at Del k r I Outszde a proper analytic subset of Sk(X), the map A is injective. qua s -(Jm DI· If k > g, these statements, except the one about generic il1jectivi"y fA' Tl f' ' Co, remam true. Ie proo gIven when k = 9 that the rank of dA at D is 9 - dim 'Dr d general ease verbatim. I exten s to the
I
As for the injectivity statement, it is sufficient to show that the set {D rank of d 4 atD = k} (k :::;g) is non· empty, i.e. the set {D E Sk(X) dimlD' _ . ~ thIS follows from the fact that if D' > O' fl' I - O} I 0. But D' " , , IS a eegree 9 and dim ID'I - 0 and 't = D + D" with deg D = k, and D" 2: 0, then dillljDI = O. '-, we wn e
I
Let A be a lattice in Cg, i.e. a subgroup of Cg which is discrete and of rank 2g; the quotient 1'1'[ = Cg / A is a compact complex manifold, called a complex torus. Let L be a holomorphic line bundle on 1'vI, and 11" : Cg -+ M the projection. A well-known theorem in complex analysis asserts that any holomorphic line (or even vector) bundle on Cg is holomorphically trivial. Let h : 11"*(L) -+ Cg x C be a trivialisation. If A E A and z E cg, then the isomorphisms 7r*(L)z -+ C and 11"*(L)z+>. -+ C differ by multiplication by a constant since 11"*(L)z = 7r*(L)z+>. = L,,(z); if we denote this constant by \O>.(z), then for /\ E A, Z >-+ \O>.(z) is a holomorphic function without zeros, and we have, for A, 1-' E A, \Op.(z
+ A)\O>.(Z)
= \O>.+p.(z) , z E .(z)} is called a factor of automorphy. Conversely, any such family, i.e. any factor of automorphy, defines a holomorphic line bundle on M, obtained from cg xC by identifying (z,t,) and (w,v) if there is A E A with w = Z + A and v = V>.(z)u. A section of this line bundle can be interpreted as a holomorphic function f on Cg with fez + A) = \O>.(z)f(z) VA E A. Such functions are called m'lJltiplicative holomorphic functions. Let X. be a compact Riemann surface of genus 9 2: 1, and let J(X) = Cg / A be its Jacobian. 'oNe use the notation of §15, so that A has as a basis the vectors ek = (0, ... ,0,1,0, ... ,0) (1 in the k-th place) and Bv = (Bvl, .. ·, Bvg), where Bvk = Wk·
J~v
There is a unique factor of automorphy e-27fizk -rriBkk, k == 1, ... , g. Definition. Let T ?:': 1. be an integer. hololllorphic on Cg such that e(z + ek)
{\O>.} with \Oek(z)
==
1, and \OBk(Z)
A theta function of order T is a function e(z), and e(z + Bk) = e-2"ir(Zk+~Bkk;e(z),
=
e
=
k = 1, ... ,g. Thus, a theta function of order r is a holomorphic section of L@r, where L is the line bundle all .l(X) defined by the factor of autolllorphy given by Vek = 1, 'PBk (z) = - 11"iBkk). We now construct the Riemann
theta function:
B)
=
it is given by
L exp{11"i(n,
Bn)
+ 211"i(n, z)} ,
nEZg
Here B = (Bvk) is the matrix Bvk = fbv Wk; it is symmetric and has positive definite imaginary part. "vloreover, if 2 = (ZI, ... ,Zg), '11) = (WI,""Wg), 'w) = :EZiwi is the standard bilinear form on Cg.
Lemma 1. The series defining 'I9(z) is uniformly and '19 is a theta function of order 1. Further, '19 Proof.
We have le1ri(n,Bn)
> 0 so
there is 5
I
i=
convergent on compact 0, and 'I9(z) = '19 ( -z).
subsets
ofC9,
= e-1r(n,Im(B)n).
Now, since Im(B) is positive definite, 2': 5(u, u) = 5 11112 Vu E IRn. Thus
that (1l, Im(B)u)
Hence an+ek = eiBkk+21ri(n,Bk) Applying this to
f - ao'l9,
Let L be the line bundle on J(X) defined by the factor of automorphy 'Pek == 1, 'PBk = e-21rizk-1riBkk. Lemma 2 asserts that HO (J(X), L) has dimension 1, and {J defines a non-zero section of L. Let be the divisor on J(X) defined by this section: = div( '19). Locally on J(X), is defined by the equation 'I9(z) = 0; more precisely, if a E J(X) and Zo E C9 maps under the projection r. : C9 --7 J(X) onto a, if V is a smallneighbourhood of Zo and 7f(V) = U, en U is defined by (u, {J 0 (r.!V)-1). Set theoretically, e is the image in J(X) of {z E C9 'I9(z) = OJ. It is called the theta-divisor of J(X).
e
e
I
The convergence follows. Clearly 'I9(z + ek) = 'O(z): '19 is clearly periodic of period standard Fourier series. We have
+ Bk) =
f == ao'l9·
we conclude that
.The Riemann theta function is a powerful tool in the study of the relationship between X and J(X). The first use we shall make is to the proof of a famous imbedding theorem of Lefschetz. We begin with some preliminaries.
e
'I9(z
an' It follows that if an = 0 for some n, then an+ek = 0 f == 0 if and only if ao = O.
Vk, and hence that an = 0 for all n. In particular,
L e1ri(n,Bn)+27ri(n,z)+27ri(n,Bk)(Bek
1 in each variable,
being a
We need a slight generalisation
of Lemma 2.
Lemma 3. Let r be an integer 2': 1. The vector space Vr of theta functions of order r has dimension r9; in particular it is finite dimensional.
= Bk)
nEZ'
=
L exp(r.i((n
+ ek),
+ ek)) + 27ri(n + ek,
B(n
Note: The finite dimensionality of HO(lVI, E), where M is a compact complex manifold and E, aholomorphic vector bundle on M, can be proved exactly as in the proof of §7, Theorem 1.
z)
nEZg
- r.i(ek, Bek)
. = e -21rizk -1riBkk"'U.O( Z ) ( smce n
+ ek
- 2r.i(ek, z)) 9
runs over Z when n does) .
That {J i= 0 follows from the fact that a Fourier series whose coefficients are not all zero cannot vanish identically. ThatO(z) = '19 ( -z) is obvious if we replace n by -n in the series defining {J.
Proof of Lemma 3. Let f E Vr; then, can be expanded in a Fourier series: f(z)
L
ane27ri(J1,Bk)e27ri(n,z)
f
= f(z
is periodic of period 1 in each variable, and = LnEZ' ane21ri(n,ZI. we have
+ Bk)
= e-27rirzk-'7rirBkk
f(z)
nE'Z9
Lemma function.
2. Any
theta function
of order 1 is a constant
multiple
of the Riemann
== ,----- a e-7ri.rBkke27fi{n-reklz) L-J 12
theta
=:
nEZg
Proof. Let f(z) be a theta function of order 1. Since each variable, it has a Fourier expansion f(z)
=
f
is periodic, with period 1, in
so that an+i'ek = e1rirBkk+27ri(n,Bk)an' n = (nj, ... ,n9) with 0::; nj < r, then Lets=(sj,
L
.. ,Sg)E7L9,0::;SjriBkk
+ Bk) f(z)
==
e-1fiBkk
e21ri(n,ZI .
2':ane21Ti(n-e,l,;,z)
The series converges uniformly for z in any compact set in C9 as in Lemma 1, and one verifies, as in Lemma 1, that {J,.,s E Vr "Is. Since the non-zero Fourier co'}fficients of {Jr,s are at the lattice points {s + mln E Z9} and these sets are pairwise disjoint for
0::; Sj < r, it follows that {'l9r,. hence form a basis of Vr.
Is
=
(Sl, ... ,Sg)
E
Sj < r} are independent,
z,g,O::;
Consider now a basis 8 = (80, ... , 8N), N + 1 = 3g of the space of theta functions of order 3; as we shall see, the functions 8j do not have common zeros; moreover, if), E A, 8(z + ),) = ew.\(z)8(z) where w), is a polynomial of degree::; 1 (as follows immediately from the definition of theta functions and the fact that ek, Bk generate A over il). Hence, 8 defines a holomorphic map, which we denote again by 8.
Proof
of Lemma
The map 8 : J(X)
-+ ]p'N
i.e.
"I
Zo
E
+ Bk))
=
g,
For any
1/
e
Suppose that
U'l,
W2
E
e
g
e
W E
e9
is such that
.
if
),
=
= ek
or
og oZv (z)
),
= Bk
"I), E
,k
=
1, ... g ;
A ,
Hence g(z
+ ek) - g(.z) 27riwk
=
= = -
+ C121 + ... + CgZg
.
Ck = 27rink , and
+ Bk) - g(z)) + 27rimk L CvBvk + 27rimk = -27ri L nvBvk + 27rimk
(g(z
.
v
Thus 10 = - Lv nvBv
1J~(;)z)
is holomorphic
e is
and nowhere 0 on
e
g,
left invariant by translation
Of course, this lemma and our remark above imply that W1-102 is injective.
]p'N
+ 27rimk
-27riwk
+ Lk mkek
E A as desired.
We next show that the tangent map de of 8 is also injective, i.e. that 8 : J(X) is an immersion.
Equivalently, if ( E J(X) and the theta divisor = + (, then ( = 0 in J(X).
( :e e
=
oZv
og oZv (z +),)
t?9(W2 + b)'l9(U'2 - 2 - b) 19(Wl + b)1J(Wl - Z - b)
V>lenow use the following lemma. 4. If
+ Bk) - g(z)
g(z) = Co
~i:~~~i
and so is holomorphic and non-zero on U.
Lemma wE A.
+ z)
so that 5'JL88ZV defines a holomorphic function on the compact connected manifold J(X), and so is constant. Hence, there exist constants Co, Cl,' .. , cg so that
e
'!J(Wl + z) '!J(W2 + z)
e-27ri(Zk+Wk)-rriBkk'l9(1O e-2rrizk-rriBkk'!J(1O) 2riwk eexp(g(z)) .
such that
0 = -.!L(z)
+),)
and
We claim that this implies the following: the function z f-+ is holomorphic and nowhere 0 on g. In fact, given Zo E g, we can choose b E 9 so that '!J( 10j + b) =1= 0 and '!J( Wj - Z - b) =1= 0 j = 1,2 for all z E U, where U is a small neighbourhood of Zo; we then have
e
such that
1 ::; v ::;g, it follows that
o -.!L(z oZv
This follows from the following remark: if..,'J is Riemann's theta function and a E g, then 'l9(z + a)19(z + b)'!J(z - a - b) then fez) = 'l9(z + a)'!J(z - a) E V2. Also, if a, bE is a theta function of order 3. It follows that V3 has no base points either (i.e. the basis functions 80, ... ,8 N have no common zeros).
e
mk
g(z
a theta q,
We now show that 8 : J(X) -+ ]p'N separates points. that 8(wd = t8(W2), t =1= O. Then
=
defined by
e :J
I(;g
·Since 'l9 is periodic with period 1 in each variable, there exists, for 1 ::; k ::; g, an integer nk such that
Hence, there exist integers
Proof. We start by showing that V2 has no base points, function f of order 2 with f(zo) =1= O.
holomorphic function g on
'!J(w + z) _ g(z) tr'g 19(z ) - e ,z Ell.-.
exp(g(z THE LEFSCHETZ IMBEDDING THEOREM. theta functions of order 3 is an imbedding.
a
4. There exists
E A, so that 8: J(X)
then by
-+
If a E 1(;9, the injectivity of d8 at 7r( a) [71: iC9 -+ J(X) to the following statement: the rank of the matrix
80(a) ( ~(a)
I \
8~~('a) 8z9
-+
]p'N
being the projection] is equivalent
equals 9
+ 1.
Suppose that the rank were that
< 9 + 1.
Then, there exist Co, ... ,c9 E C, not all zero, such
f;
In this section, we study the influence of the theta divisor on the Riemann surface X. The results were given by Riemann in his fundamental paper on abelian functions. The proofs given here are not very different from Riemann's.
8
9
Co
('I9(a+1l)-z9(a+v )'I9(a-1l-v))
If we set r.p( z) =
=
(2:~=1 gz~(z)) C,;
y 8z ('19 ( a+u)'I9(a+v y
C
)'19(0.-1l-
v))
V1l, V
E
([9
.
/ 'I9(z) this can be written
A priori, r.p is meromorphic and has poles at the zeros Z of '19in C9. However, given a and 1lo E C9, we can find a neighbourhood U of 1lo and v E ([9 such that a + v ~ Z and 0.- It - V ~ Z for 1l E U. Thus r.p is holomorphic on C9. Moreover, we have
r.p(z
+ ek) =
r.p(z)
and
r.p(z
+ Bk)
- r.p(z)
=
t
Cy 8~y (-2Jrizk
- JriBkk)
= -2Jrick
.
//=1
Let L be the line bundle on J(X) e-2rrizk -rriBkk; the 'I9-function is the divisor of the section '19.We [addition in J(X)]. It is defined
defined by the a holomorphic denote by B( by the section
factor of automorphy r.pek == 1, r.pBk (z) = section of L, and the theta divisor B is = B + ( the translate of B by ( E J(X) 'I9(z - () of the translate L( of L by (.
Let A : X --+ J(X) be the Abel-Jacobi map; the function P 1--7 '19(A(P) - () is a section of the pull-back A*(L() of L( by A. If we choose a basis aj, bj of HI(X,71) as in §14 and slit X along these curves, we obtain a polygon (simply connected) ~ c C and '19(A(P) - () may be thought of as a holomorphic function on ~. Note that the aj: bj can be chosen to avoid any given finite subset of X; in what follows, we shall tacitly assume that this has been done. The sides of the 4g - gon ~ will be denoted, as in §14, by ay,by,a~,b~ (a~,b~ map onto the curves a~l,b~1 in X).
rt
Theorem 1. Let ( E J(X) be s1lch that A(X) B( [the set {( E J(X)IA(X) c Be} is clearly a proper analytic s1lbset of J(X)j. Then, counted with multiplicities, the intersection A(X) n B( consists of 9 points; more precisely, the divisor of the section 'I9(A(P) - () of A*(Ld has degree. g: ('I9(A(P) - ()) = Pi(()·
2:I=1
9
But r.p is periodic, of period 1 in each constant. But then
2:~=1 y gz~
Zj;
hence
Cl'j
I> (P
= 0 for j = 1, ... , g, so that r.p is
i(())
= (-
~
i=l
where If. E J(X) is a point independent of ( (it depends only on the base point Po EX chosen to define the Abel-Jacobi map).
hence r.p(z) = C /'19== 0, and Co = r.p(a + 1l) + r.p(a + v) + r.p(a -u - v) contradicts our assumption that not all the co, ... , cg are 0, and proves that plY is an immersion.
= O. This
e : J(X)
--+
Proof.
Jav
Let
Wk = byk·
w
= (WI, ... ,wg), where WI, ... ,wg is a normalized basis of HO(X, On ~, the Abel-Jacobi map is given, modulo A, by
0)
If g, is a function on 8~, we define functions g,± on the edges aj, bj of 86. by iJ?+ = iJ?, iJ?-(P) = iJ?(PI) if P E aj or bj and pI is the corresponding point of aj, bj. If P E a
y,
we have, as in §14, Lemma 1, (see figure next page)
-1-1
== ~
dlogP+
27ri a.
&-:"
27ri
log 19(A+(,6) - () 19(A+(a) - ()
1 -log - 27ri
19(A+(a)-(+el/) ~-----~ 19(A+(a) - ()
=
,
mod;Z
= -
0 mod;Z.
/
- ,-~----!.. /
....•..•.••
1
(AtdlogF+
- A;; dlogF-)
= canst for v = 1, ... ,g.
O'V
At(P)
- A;;(P)
A;;(Q)
=
=
J;'Wk
= -
Jb. Wk
=
-Bvk,
Jg, Wk = Ja. Wk = Dvk' Thus, if A± A + - A - = ev
on
bv,
=
while if Q E by, we have A.t(Q)(At,···,
A + - A - = - Bl/
We may assume that 19(A(P) - () f:- 0 if PEal::". 19(A(P) - () in I::" is given by
-:A r dlogF(P) ~7r~1M Now, if P E bl/, F+(P)
Ai),
t(1 27r~v=l
on
al/'
The number of zeros of F(P)
=
a.
r )dlog lb.
= 19(A+(P) - () = 19(A-(P)
a (A(x)-,
F:(P) . F (P)
- (+ ev) = F-(P),
while, if
PEaI" F+(P) = 19(A-(P) - (- Bl/) = e21fi(Av(P)-'.)+rriB··19 (A-(P) - () so that 1og F-(P) F+(P) -- 27r2'A I'(P) - 2? 7rh,v + 7ft'B w, and we have dlog F+· F- = 27r2WI/on av. Hence, the number of zeros of Fin .6. equals l:~=l W// = g. This proves the first part of the theorem.
Ja.
For the second part, let P1((), ... , Pg(() be the zeros of 19 (A(P) - () in 1::". We shall denote by const a term which is independent of (. We have tAk(PI/(()) v=l
=
-:A r Ak(P)dlogF(P) ~7r! 1&",+
= 2~i t(1 v=1
all
r)
Jbll
Consider the integral over al/' We have A;; 27riw,,; hence (AtdlogF+
1 av
- A;;dlogr)
=
=
(AtdlogF+ At
+ Bl/k,
- A;;dlogF-)
1 a.
(
= exp -27riAv(x)
11
'9--: ~m
~
- 7riBw
dlogF +-= (1/ - AI/(x)
9
LAdPv(()) v=l
+ 27ri(v
)
,
so that
~:i~l =
so that
1
- -;;Bvl/ ~
mod;Z
9
=L
Dvk(v
+ const
= (k
+ const
v=l
which proves the theorem.
+ 27ri
1
dlogF+
+ const
.
dlogF-
A;;w//
av
If 0 < k:S; g, and Sk(X) is the k-th symmetric power of X, we denote by Wk the image in J(X) of the map A : Sk(X) ---+ J(X) [A(P1,· .. , Pk) = ~A(Pi)]' Wk is thus the set Wk = {A(D)ID effective divisor on X of degree k}. Wkis an analytic set in J(X).
.
=
while dlogF+
(At - A;;)dlogF+
= -Bvk
.)
+ BI/,
Before proceeding to the next theorem, we need some preliminaries.
1 av
If x,y denote the ordered endpoints of bl/' we have A(y) = A(x) 19 (A(xH+Bv)
+
= ~
we have
+
Let M be a compact connected complex manifold with dime 1\.-1 = n. A divisor D on !'vI is a finite linear combination D = l:~=l nkYk, bk E ;Z, where the Yk are irreducible analytic :mbsets of l\1.of dimension n - 1 (i.e. codimension 1). On a complex manifold M, codimension 1 analytic sets Y c 1\1 have local equations, i.e. Va E lvI, there is a neighbourhood. U of a and .f holomorphic on U such that, if J; E U and 9 is a holomorphic function near x vanishing on Y near x, then 9 is a multiple of .f by a holomorphic function near x. .
n
If U c M and fk are local equations for }lie on U, set fu = f~k. If V is another such open set and fv the corresponding meromorphic function on V, then fu = guv fv on un V, where guv is holomorphic and nowhere zero on un V. The {guv} form transition functions for a holomorphic line bundle L = L(D) on M. It comes with a standard section (meromorphic) SD defined by the function Uu}· If D = I:nkYk is a divisor, the set UndO Yk is called the support of D, and written supp(D). If nk 2': 0 for all k, D is called effective. The standard section SD is holomorphic exactly when D is effective. Meromorphic sections s of a holomorphic line bundle L define divisors on M. If Y is the analytic set of zeros and poles of s, and Y = U Yk its decomposition into irreducible components, let s be represented locally by a meromorphic function F and ik be local equations for Yk. Then F = u· f~k, u a holomorphic function without zeros. The nk are constant along Yk and we set (s) = I:nk Yk. The integer nk is called the order of Sk along Yk [order of zero or pole according as nk > 0 or nk < 0; the order of the pole is Inkl if nk < 0].
n
where
K
so that ( = A(D') +
K
E Wg-1
+
K.
If 1i(A(x) - A(P) - () == 0 \lP, let k be the largest integer such that 11(A(Do) - A(D1)() = 0 for all effective divisors Do, D1 of degree k. We have k < 9 since 5g(X) --4 .I(X) is surjective. Let Eo, E1 be effective divisors of degree k + 1 with 1i(A(Eo) - A(E1) - () =1= O. We may suppose that supp(Eo + Er) consists of 2k + 2 distinct points. Let Eo = P + Do where Do 2': 0 has degree k. Then, x f-+ 19(A(x) + A(Do) - A(E1) - () "CJ 0 (it is =1= 0 for x = P); let D be the divisor of this function. Then D 2': 0 has degree g. Further, if x E supp(E1), 19(A(x) + A(Do) - A(E1) - () = 19(A(Do) - A(E1 - x) - () = 0 since E1 - x 2': 0 has degree k. Hence D 2': E1, and we can write D = E1 + E2 with deg(E2) = 9 - k - 1. Now, by Theorem 1, A(E1) + A(E2) (-K = A(E2+Do) with deg(E2+Do)
=
=
A(D) g-k-1+k
= (+
A(E1) - A(Do) - K, so that Thus, supp(8) c Wg-1 +K.
= g-1.
We saw at the beginning of this proof that if ( = A(D) + K, with D = I:Pi with the Pi distinct in general position, then D is the divisor of zeros of 19 (A( x) - (), ( = A(D) + K, so that the zeros of 19(A(x) - () are simple. It follows that
is the constant in Theorem 1.
In other words, the divisor 8 has the form 1·Y where Y is in'educible of dimension g-l (.50 that the theta function has only simple zeros at a general point of 8); moreover, 8 consists exactly of the points A( Pv) + K, PI, ... , Pg -1 EX.
L~:i
Proof. We start by showing that vVg_1 + K c supp(8). Let D = PI + ... + Pg be a divisor of degree 9 with distinct Pi in general position so that D is the unique point of 5g(X) mapping onto A(D) in .I(X). Further, we may assume that A(X) rt 8(, where ( = A(D) + K (since A : 5g(X) --4 .I(X) is surjective). Let Ql, ... ,Qg be the zeros of P f-+ 1i(A(P) - (). By Theorem 1 we have I:il( Qi) = ( - K = A(D), so that, by choice of D, we have D = I:Qi = I:Pv. In particular t9(A(Pg) - () = 0, so that 0 = 1i(- L~:iA(Pv) - K) = 1i(L~:i A(Pv) + K). Since D can be chosen to satisfy the above conditions arbitrarily in a non-empty open set in 59 (X), it follows that 11(A( D') + K) = 0 for all D' in a non-empty open set in 59-1 (X), so that ·t9IWg-1 + K = O. To prove that supp(8) such that
C Wg-1
+
K,
Theorem 3. If on X, we have
Proof.
K
is the constant in Theorems 1 and 2, and Kx
is a canonical divisor
'vVebegin with a rem.ark which we shall use later on in these notes as well.
Let D 2': 0 be a divisor of degree 9 -1. Then hO (D) 2': 1. By the Riemann-Roch theorem h0(I<x-D) = hO(D)-(l-g+degD) = hOlD) 2': 1, so that Kx-D is linearly equivalent to a divisor D' 2': 0 which must have degree 9 - 1. Hence A(Ex - D) E Wg-1. Hence A(Ex) - 1Vg-1 c Wg-1. Further, A(D) = A(Ex) - A(D') E A(Ex) - vVg-1
let ( E 8, and suppose first that there is P E X
11(A(:r;.)- A(P) - () "CJ 0 in x, in this case, if D = div( 1I(A(x) - A(P) - ()), then D = P + D' where D' 2': 0 has degree 9 -1 (P E supp(D) since1l(-() = 1i(() = 0); moreover, by Theorem 1,
Since 8 is not left invariant by translation 4), we obtain
by a non-zero element of .I(X)
(§1(:;, Lemma
Theorem 4. Let ( E J(X). Then A(X) c e( if and only if (- '" = A(D) where D ~ 0 is an effective divisor of degree 9 with dim IDI > 0; in other words D is a special divisor of degree g.
connected polygon 6. as in §14, with the av, bv avoiding a suitable finite set of points in X: Consider the function on 6. defined by
11
iI(A(x)
- A(Pk) - () - A(Qk) - ()
+ rDo) =
~(Pk - Qk)
r
Proof. A(X) c e( if and only if A(P) - ( E e 'elP E X, i.e. (- A(P) E e = W9-1 + r; 'elP E X. Thus, the condition is that ( - '" = A(D) where D has degree 9 and P E supp(D). Now, D is determined up to linear equivalence; if we fix Do with A(Do) = ( - "', the condition is that there is D ~ 0 linearly equivalent to Do and containing an arbitrarily given point P E X. This simply means that dim IDol> o. Corollary. If ( E J(X) is such that A(X) rt e(, then there is a unique divisor D ~ 0 of degree 9 such that A(D) + '" = (. D is given by the divisor of zeros of iI(A(P) - ().
F(x) = Its divisor = (~Pk + rDo) - (~Qk function defined on X.
iI(A(x)
= (f).
If x E bv and x' is the corresponding point of b~, then A(x') F(x) = F(x').
It is not, however, a
= A(x)
If x E av and x' is the corresponding point of a~, we have A(x')
+ ev,
= A(x)
and we have
+ Bv
and
This follows from Theorem 4 and §1.5,Theorems 1,2. This corollary gives a complete answer to the so-called Jacobi inversion problem, viz to describe the inverse of the birational transformation A : S9(X) -> J(X). We shall give another application of these results. Consider the map A : S9(X) -> J(X), (PI,"" P9) H ~A(P;), and let Y C S9(X) be the set of critical points, i.e. Y = {D E S9(X) rankD(dA) < g}. Y is an analytic set of dimension :S 9 - l. Further, if DE Y, then A-1A(D) C Y (by §15, Theorem 2 and Abel's theorem) and the dimension of A-I A(D) at any of its points is dim IDI > O. Hence Y' = A(Y) is an analytic set in J(X) of dimension :S 9 - 2. In particular, no finite tmion of translates of Y' can contain e.
I
Let now P E X and let x be a variable point on X. By Theorem 4, if A(P) + (- '" rf. Y', then the function x H iI(A( x) - A( P) - () has exactly 9 ze~os PI, ... , P9, and ~A( P;j = ( + A(P) - r;; further ~Pi is the only divisor ~ 0 of degree 9 satisfying this equation.
e,
If we assume, in addition that (E
(= A(Q~)
9
= A(P)
so that ~Pi = P Thus, if ( given by
E
+ I:A(QJ), j=l
1
+ ~QJ.
e and
( rf. -A.(P)
+ + Y', K.
then the zeros of :r H lJ(A(x) - A(P) - () aTe
(P, Q~, ... , Q~-l) where Q~, ... , Q~-l
- Av(Qk»)
th
= v
I:
component of
9
rJjej
+
j=l
1
I:
mjBj
j=l
9
= 'nv
+ I:mjBjv. 1
Now if WI,
cp(x)
=
1) If
J; E
...
J;o
W
,w9 is the normalized basis of Hal-X, 0), and W = L;=1 mjwj, and we set (Po fixed), we find that e21fi 0 so that .6.g-1 + P ~ t.~_1 + Q. This gives, if w = A(Q), (u - b) + x = AX(t.9~1) +-Ax(P) = AX(t.~_l)+W E Wg-1,w; since Q E X is arbitrary, t~-b+x E WEVllHI W 9 -1.,W = W*g-2
n
A~)(D(b))=b-C1'
C1
aconstant.
We write D(b) = Do(x) + Dl(x,y), where Do(x) consists of the part of D(b) which maps into vi n TVr,a+x under Ay, and no point in D1 (x, y) maps into vV",a+x' We now claim that Do(x) has degree 1, i.e. that Do(x) consists of a single point which occurs with multiplicity one in
III
n Wg-1,b·
First, suppose that deg Do(x) 2': 2. Then, if we fix x and let y run overWg_l_r - Z(x), the image of DI (x, y) in J would lie in a fixed translate (Vg_ 2) ( ) of Vg-2. But -Ay
Do(x)
the image of DI(x,y) is a fixed translate of b - Ay(Do(x)), hence is a fixed translate of -y (depending on x). Since Z(x) =!= Wr-I-r, it follows that Riemann's singularity theorem expresses the order of vanishing of the 'lJ-function at a point ( E in terms of dim IDI, where D 2': 0 is a divisor of degree 9 - 1 with ( - K = A(D). Riemann proves this by relating this order to the vanishing of'lJ on sets of the form W,. - Wr -- ( (Uber das Verschwinden der Theta-Functionen).
e
n -VE\I~-2,.B
Vg-1,v C
n
Wg-I,v+c
By Lemma 1, the term on the left is a translate translate of W;, contradicting the definition of r. Thus degDo(x)
(if Vg-I = VV:q-1
+ c) .
-VEW;_l_T
of VI, while that on the right is a
..:;l.
Now, degDo(x) 2': 1; if this were not the case, D(b) = DI(x,y) would have its support in a finite set depending only on y [viz the set in Y whose image under Ay is 5 n VI; 5nV1 is finite being contained in VI nWg-I,b, VI rt Wg-I,b]' But then A~) (DI(x, y)) = a + x - Y - CI would be independent of x for x in some non-einpty open set in WI.
The theorem has been and generalised to Wk, of Riemann, Annals of and this whole circle of Cornalba-Griffiths-Harris
formulated more geometrically (using the tangent 2 ..:; k ..:; 9 - 1, by G. Kempf: On the geometry Math. 98 (1973), 178 - 185. For a discussion of ideas, one cannot do better than consult the book
cones to e) of a theorem this theorem of Arbarello-
[10].
We start with two lemmas which we have essentially proved before in connection with the Riemann factorisation theorem. 1. Given P EX, there exists ( E 'lJ(A(x) - A(P) - () is not == O.
Lemma
e such
that the function
Thus, deg Do (x) = l.
x
As remarked above, if y E Wg-1-r - Z(x), DI(x,y) has its support in a finite set depending only on y. Hence we can find infinitely many points Xv E TVI (1/ 2': 1) so that DI (xv, y) = DI (y) is indepeIJ-dent of 1/. Thus Ay (Do (xv )) = a+xv-y-coAy (DI (y)), and Ay(Do(xv)) - Ay(Do(XI)) = Xv - Xl, 1/ 2': 1 .
Proof. If Y C 59(X) is the set of critical points of the map A : 59(X) -;. J(X), Y = {DE 59 (X) rank of dA at D is < g}, then AIY has no isolated points in any of its fibres (§15, Theorem 2) so that Y' = A(Y) has dimension":; 9 - 2.
Clearly, Ay(Do(xv)) - Ay(Do(XI)) E VI,t, t = -Ay(Do(XI)), and Xv - Xl E WI,-x,. Thus, the curves Vl,t and WI,-3;, intersect in infinitely many points, and so must be equal. This proves the theorem.
1-+
I
°
Now, if x I-+O(A(x)A(P) - ()== 0, then (+ A(P) = A(D) + n, where D 2': has degree 9 and dim IDI > 0, i.e. D E Y (§15, Theorem 2 again). Thus, we have only to choose ( E e, ( rf- n, - A(P) + Y'. In what follows, we denote by Fp(P suitable line bundle on X).
E X)
the section x
1-+
'u(A(cc) - A(P) - () (of a
e;
Lemma 2. Let ( E if P is such that Fp(x) 1= 0, then div(Fp) = P + Do, where Do 2': 0, degDo = 9 - 1 and Do is independent of P. (It can depend on (.)
°
Proof. If D = div(19(A(x) - A(P) - ()), then D 2': 0, degD = 9 and dimlDl = (since Fp 1= 0); moreover Fp(P) = v(-() = v(() = O. Hence D = P + Do, Do 2': 0, degDo = 9 -1 and dim IDol = 0. Let Q be such that FQ D'
=
1= 0; then div(FQ)
=
Q
+ D1,
DI 2': 0 ,deg DI
=
9- ] .
We have A(D) = A(P) + ( - 1'(" so that A(Do) = ( - I~; similarly, A(D1) = ( - K, and since Do, DI have degree 9 - 1, Abel's theorem implies that Do ~ DI, so that, since dim IDol = 0, we have Do = DI·
-"f'"
I
3. Let ( E e, and suppose that there is P E X such that Fp -=j. O. Then, there are at most 9 points Q E X with FQ == O.
Lemma
Choose Xo so that 19(A(xo) - A(P) - () 1= O. The function y f-+ 19(A(xo) A(y) - () is not == 0, and its divisor is div (19(A(y) ~ ('») (where (I = -( + A(xo», since the 19-function is even, so has degree g.
Proof.
1. Let (E
Theorem
e. Then Fp == 0 'liP
if and only if
EN}
(() = 0
-
az
E X (i.e. 19(A(x) - A(P) - () = 0 'IIx, 'liP)
for
1I
= 1, ... , 9 .
y
Proof.
Suppose that 19(A(x) - A.(P) - () = 0 'IIx, 'liP. If we differentiate
with respect
to x, we obtain, since dA(x) = (W1(X), ... ,W9(X»),
f; az 9
I:9819~(-()Wy(P) v=l
0,
1I
(W1, = 1,
, w9)
e, let
r = r( be the largest integer such that19(Wk
~ Wk- ()
==
0 for k
< r;
< g.
Theorem 2. . (= K, + A(D),
Let s be an integer> 0, and ( E e. Then r( where D ;:::0, deg D = 9 - 1 and dim IDI ;:::s - 1.
;:::s
if and only if
Let r = r(. Then, there are effective divisors Do, D1 of degree r with 19(A(Do)1= o. We may suppose that supp(Do + D1) consists of 2r distinct points. Further, for fixed D1, the set of Do E sr(x) satisfying this condition is a non-empty open set in sr(x).
Proof.
A(D1) - ()
We write Do = P + t..o where deg t..o = r - 1. The function F: x f-+ 19(A (x) + .4.(t..o)A(D1) - () is -=j. 0 (it is 1= 0 if x = P). If x E supp(D1), then A(x) + A(t..o) - A(D1) = .4.(t..o) - A(D1 - x) E Wr-l - Wr-1. Hence, since 19(Wr-1 - Wr-1 - () = 0, F(x) = 0 if x E supp(Dd. Hence the divisor D of F, which is of degree g, can be written
a.{)
y (A(x) - A(P) - ()wy(x)
since
Given ( E we have r
=0
= 0;
'liP;
'-
