Compact Riemann Surfaces (Lectures in Mathematics. ETH Zurich)

Lectures in Mathematics ETH Zurich

Department of Mathematics Research Institute of Mathematics Managing Editor: Oscar E. Lanford

Raghavan Narasimhan Compact Riemann Surfaces , LTP,D 02 C;~f,jGIAS 81 B '_I C i

Ill,'

T

-=-

~

'.

',i

"

A0 ~

0

RJ~is!;rf)2.G_,_ ..2_if\:),"_"",_,

~LoC(Tig"l).;.I':.\J3'1_1-.Q.O.~-

Algebra

Cien

liI~lij~lllilrllll ~I~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 1>0 0 0 0 3 9 7 0 0 5

Birkhauser Verlag Basel· Boston· Berlin

Raghavan Narasimhan Department of Mathematics University of Chicago Chicago, IL 60637 USA

These notes form the contents of a Nachdiplomvorlesung given at the Forschungsinstitut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. K. Chandrasekharan and Prof. Jurgen Moser have encouraged me to write them up for inclusion in the series, published by Birkhiiuser, of notes of these courses at the ETH. Dr. Albert Stadler produced detailed notes of the first part of this course, and very intelligible class-room notes of the rest. Without this work of Dr. Stadler, these notes would not have been written. While I have changed some things (such as the proof of the Serre duality theorem, here done entirely in the spirit of Serre's original paper), the present notes follow Dr. Stadler's fairly closely. My original aim in giving the course was twofold. I wanted to present the basic theorems about the Jacobian from Riemann's own point of view. Given the Riemann-Roch theorem, if Riemann's methods are expressed in modern language, they differ very little (if at all) from the work of modern authors. A CIP catalogue record for this book is available Washington D.C., USA

from the Library

of Congress,

Deutsche Bibliothek Cataloging-in-Publication Data Narasimhan, Raghavan: Compact Riemann surfaces I Raghavan Narasimhan. - Basel; Boston; Berlin: Birkhauser, 1992 (Lectures in mathematics) ISBN 3-7643-2742-1 (Basel ... ) ISBN 0-8176-2742-1 (Boston)

It is a great pleasure to express my thanks to the ETHfor its hospitality, to Prof. J. Moser for his encouragement, and to Dr. A. Stadler for the enorIIlous amount of work he undertook which made these notes easier to. write. But. ~peci[;;tlthanks are due to Prof. K. Chandrasekharan. But for him, I wouldr!bthavebElen'~~~rhe ETH, nor would these notes have been written without his advice andencouragergi\nt.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, pennission of the copyright owner must be obtained.

© 1992 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Printed directly ii'om the author's camera-ready manuscript on acid-free produced from chlorine-free pulp. TCF = Printed in Germany ISBN 3-7643-2742-1 ISBN 0-8176-2742-1

I had hoped to follow this with some of the extensive work relating theta functions and the geometry of algebraic curves to solutions of certain non-linear partial differential equations (in particular KdV and KP). Time did not permit pursuing this subject, and I have contented myself with a couple of references in §17. These references fail to cover much other important work (especially of M. Mulase) but I have not tried to do better because the literature is so extensive.

paper

1. Algebra,ic functions

.

2. Riemann surfaces

.

3. The sheaf of germs of holomorphic functions

.

4. The Riemann surface of an algebraic function 5. Sheaves

.

6. Vector bundles, line bundles and divisors

.

7. Finiteness theorems .... 8. The Dolbeault isomorphism

.

.

9. Weyl's lemma and the Serre duality theorem 10. The Riemann-Roch theorem and some applications 11. Further properties of compact Riemann surfaces 12. Hyperelliptic curves and the canonical map 13. Some geometry of curves in projective space 14. Bilinear relations .... 15. The Jacobian and Abel's theorem 16. The Riemann theta function 17. The theta divisor 18. Torelli's theorem ..

.

. .

Let FE tC[x,V] be an irreducible polynomial in two variables (with complex coefficients). We assume that its degree in V is ~ l. Recall that by the so-called Gauss lemma, if we identify tC[x,V] with tC[x][V], and if F is irreducible, it is also irreducible in iC(x )[v], the polynomial ring over the field of rational functions in x. Moreover, tC[x, V] is a factorial ring (i.e. a unique factorisation domain). An algebraic function is, intuitively, "defined" by an equation F(x, V) = 0 (where F is irreducible in tC[x, yD· To make this statement more precise, we begin with the following. The implicit variables

function

x, V defined

theorem.

on {(x, V) E

Let f

be a holomorphic function of two complex rl, r2 > O. Assume that

rc2 Ilxl < rl, Ivl < 'r2},

of

".(0,0) =J 0 . uV

Then, there exist positive there is a unique solution x f-7 V(x) is holornorphic

II

numbers 10, 8 > 0 such that for anu xED 6 = {z E rc zi < E:}, v(x) of the equation f( x, V) = 0 with Iv( x) 1 < 8. The function on Dc.

Proof. Since %t(0,0) =J 0, we can choose 8 > 0 such that f(O, y) =J 0 for 0 < Ivl :S 8. Choose now 10 > 0 such that f(x, V) =J 0 for Ixl :S 10, Ivl = (; (possible since f is non-zero on the compact set {O} x {V Ilul = 8}). By the argument principle, if

Ixl < 10,

J

1 of -;:;: {".(x, uy ~1f/.

y) / fCE, y)} dy

Ivl=8

is an integer 11(X) equal to the number of zeros of the function y f-7 by our choice of 5, 11(0) = 1. On the other hand, since f(x, y) =J 0 for the integrand, and thus also the integral, is a continuous {unction of x for 11(:1:) = 1 for Ix! < 10, which means precisely that there is a unique zero with ly(x)1 < 5. That.

y(x)

is holomorphic follows from the formula 1

Y

(

(x\--I

-

')'

•./ff'l,

"

Ivl=o

li(T c')

l~dl/

y f(x', V)

,

(which is an immediate consequence of the residue theorem). Let F(x, y) = ao(x)yn + al(X)yn-l + ... + an(x) E iC[x, y] be an irreducible polynomial with n 2: 1; the polynomials ao, ... , an E iC[x] have no non-constant common factor

bk-2Qk-3 = .1k-lQk-2 + Qk"I, it would follow that P divides bk-2Qk-3 and hence Qk-3' Repeating this argument, P would divide all the Qj (j 2: 1), hence also ~F and F, contradicting the irreducibility of F. Thus Qk = Qdx) E iC[xj is t O. y . If now a,b E iC and F(a, b) = 0 = ~~(a,b), we see from the above equations that Ql(a,b) = 0, then that QAa,b) = 0, ... ,Qk(a,b) = Qk(a) = O. Since Qk to, the set

since F is irreducible. Lemma 1. Let a E iC be such that ao(a) =f 0 and such that there is no b E iC with F( a, b) = 0 = ~~(a, b). Then, there is E: > 0 and n holomorphic functions

Ilx - al < E:} with the following Ix - al < E:, Ix' - al < E:; moreover

Yl (x), ... , Yn(x) in the disc {x (i) Yi(X) =f Yj(x') if i =f j,

E

C

properties:

{x E iC I:ly E iC with

F(x,y)

=0=

~F (x,y)} Y

I

C {x E iC Qk(X)

Before proceeding further, we insert some toplogical preliminaries. we consider will be Hausdorff.

= 0, Ix - al <.c:, then?') =

(ii) if?') E iC and F(x,?'))

Proof. Since ~~ (a, b) =f 0 for all solutions b of F( a, b) = 0, the polynomial F( a, y) has exactly n roots bL •.. , bn. If E: > 0 is small and Yi(X) the holomorphic function on Ix - al < E: with Vita) = bi and F(X,Yi(X)) == 0 (which exists by the theorem above), then the Yi have property (i) if E: is small enough, and property (ii) since the equation F(x,'I)) =0 has at most n solutions.

.

F(x,y)

All topological spaces

Yi(X) for a unique i between 1

and n.

Proposition 1. Let F E such that the equations

= o}

Definition. A continuous map p : X -t Y, where X, Yare locally compact (Hausdorff) spaces, will be called proper if, for any compact set KeY, the inverse image p-l(K) is compact in X Lemma 2. If X, Yare locally compact, a proper map p : X i. e. take,~ closed sets in X to closed sets in Y.

-t

Y is necessarily

closed '

Pro.of. Let A C X b: ~losed, an~?o.E Y. Let K be a compact neighbourhood o.f Yo mY. Thenp(.1)nI'I. -p(.4.np (fl.)) IS compact (smce A IS closed andp-l(I1.) IS compact), hence closed in K.

of,,

= 0 = '7}(x,y)

UY

Remark. A continuous map p : X -t Y between locally compact spaces X,)f is proper, If and only If, for any locally compact topological space Z, the product

Proof. By the division algorithm, there are polynomials bi E iC[x] (i 2: 0) with bo ao [F = ao(x)yn + ... + an(x)] and polynomials .1j, Qj E iC[J', y] (j 2: 1) such that n

bof

,oF = 1'11'7} uy

of b -0 1

y

= .12Ql

+

Q

1,

degy Q 1 < degy

of = n a Y

1

+ Q2

'vVemay suppose that degy Qk = 0, i.e. that Qk E iC[x] (since we can otherwise continue the division process). We claim now that Qk(X) t O. If, in fact, Qk == 0, then from the last of the above equations, any prime factor P of Qk-l with degy P > 0 would divide bk-lQk-2, hence Qk-2 (since bk-1 E iC[x] anel elegyP > 0). From the equation

is ~losed. If X, Y have countable bases, this call be seen by using the following remark: If tXl,""Xn, ... } is a sequence of points in X, without limit points and such tha.t {pc xn) } n2:1 converges in Y, then the image of the dosed set {( ;];n,' ~) n 2: 1} in X x JR IS not dosed in Y x R

I

The property in this remark can be used to define proper mappings between spaces which are not locally compact. Remark. Let p : X -t Y be a proper map between locally compact be a locally compact space (with the induced topology). Then IS agam proper.

pi

Let Z c Y : p-l(Z) -t Z

Lemma

3. Let Cl, ... , cn'E iC. Let w E C and suppose that wri

Then

+ Clwn-l + ... + Cn

= O.

v

Iwl < 2maxlcvll/

If IzI 2:

Izi < 2,

>

O. If z

= ~,

Z

<

we have zn

+ 7z,,-1 + '" + ~ =

0, so

< _

'21

+ . . . + 2"1 < 1,a

'I. ra d'lCt'Ion. Th us con

2c.

ao(x)y" + ... + an(x), ao t- O. Let V = I F(x,y) E C I ao(x) = O}. Let'if: V --+ C be the projection (x, y) f--t x. Then 'if ['if-l(C - So) --+ C - So is proper. Proposition {(x,y) E C2

qx, V], F(x, y) = o} and So = {x

2. Let F E

=

Let K c C - So be compact. Then there is 5 > 0 so that lao(x)1 2: 5 and lavCx)1 ::; t for x E K. If (x, y) E V, x E 'if-l(K), we have

Proof.

y

n

alex)

+ ao ()yx

,,-1

+

...

an(x) - 0

+ ao ()x -

so that, by (1.8), Iyl ::; 2maxv 5-2/v. Thus 'if-l(K) (K x C) n V is closed in C2, 'if-l(K) is compact.

is bounded.

C

Proof. If p is a finite covering, if yoE Y and V is an open neighbourhood of Yo which is evenly covered by p, then p p-l(V) --+ 'V is clearly proper. It follows easily that p is proper.

I

Conversely, let p be a proper local homeomorphism, let Yo E Y and let p-l(yO) = {Xl, ... , xn}. Let Uj be an open set with Xj E Uj and such that p Uj is a homeomorphism onto the open set V; = p(Uj). Since p is proper and X - U~Uj is closed in X. E = p(X - U~Uj) is closed in Y. Clearly, Yo t/. E. Let V = Y - E. Then p-l(V) C U{ u ... u U~, and we have V C VI n ... n Vn. If we set Uj = Uj n p-l(V), then p-l(V) = U~Uj and pi Uj is a homeomorphism onto V.

I

Let F E qx, y] be irreducible, F(x, y) = ao(x)yn + ... + an(x). Let So = {x E C I ao(x) = O} and Sl = {x E C I :Jy E C with F(x,y) = 0 = ~~(x,y)}. Then, if V = ((J:,y) E rc2 i F(x,y) = O} and 'if : V --+ rc the projection (x,y) f--t :1', then

, Since clearly 'if-l(K)

=

Definition. Let X, Y be (Hausdorff) topological spaces and p : X --+ Y, a continuous map. p is called a covering map if the following holds: VYo E Y, there is an open neighbourhood V of Yo such that p-l(V) is a disjoint union UjEJ Uj of open sets Uj with the property that pi Uj is a homeomorphism onto V Vj E J. The triple (X, Y,p) is then called an (unramified) covering. We also say that X is a covering of :1". An open set V

3) Let X, Y be locally compact, let p : X --+Y be a local homeomorphism (i.e. Va E X, . :J an open neighbourhood U of a such that V = p(U) is open in Y and p I U is a homeomorphism onto V). Then, p isa finite covering if and only if it is proper.

1 + ... + W1 2, we wou ld h ave 1< _ iZT i.e. [wi

of

2) p : C --+C*, p( z) = e is an infinite covering of C*.

v

Proof. Let C = maxv Icvll/v that, since Icvl ::; cV,

It is a standard fact in the theory of covering spaces that any connected n-sheeted covering ,6.* is isomorphic to Pn.

Y with the property in the definition is said to be evenly covered by

p.

It follows from the definition that the cardinality Ofp-l(y) is a locally constant function on }'. (With the notation in the definition, the cardinality of p-l (y) is that of JVy E V.) Thus, if Y is connected, "the number of points" in p-l(y) is independent of y E '1". The covering is said to be finte (infinite) if the cardinality of p-l(y) is finite (infinite). pis called an n sheeted covering if p-l(y) contains exactly n points for y E Y. If p : X --+ Y, p : X --+ T are two coverings of Y, they are Baid to be isomorphic if there exists a homeomorphism 'P : XI --+ X such that po 'P = pI Examples. 1) Let ,6. = {z E C Ilzl < I} and ,6.* = ,6. - {O}. Then, if n Pn :,6.* --+,6.* given by p,,(z) =zn is an n-sheeted covering.

2:

1, the map

is a finite covering (of n sheets). This follows from Proposition

2 the implicit function theorem.

Before proceeding to show how the set V can be modified over the points of So U51 and the point at 00 in rc to define the algebraic function completely, we shall introduce the notion of a Riemanll surface and some related topics.

I

2. Tori. Let T E C, Im(T) > O. Let A = {m+nT m,n E Z}. A is an additive subgroup of C. Consider the quotient group X = C/ A and let 7r : C -+ X be the canonical projection. With the quotient topology, X is a compact Hausdorff space, and C -+ X is a local homeomorphism. [These statements are easy consequences of the following two remarks: if a E C, and we consider the set U = {a+'\+f1T f1 E lR:, < A, It < U is open and maps bijectively onto an open set in X; further X is the image of the compact set [J (closure of U) for any a E C. 7r is actually a covering map.]

I '\,

Let X be a 2-dimensional manifold (i.e. X is a ~aus~orff space and any point in X has a neighbourhood homeomorphic to an open set III lR:-). Consider pairs (U, 'P) where U is open in X and'P : U onto an open set in C.

-+

'P(U)

C

iCis a homeomorphism

Two such pairs (Ul, 'PI), (U2, If'2) are said to be (holomorphically) compatible ifthe map 'P2 0 'Pll : 'Pl(Ul n U2) -+ 'P2(Ul,nU2) is holomorphlc; Its lllverse IS also holomorphlc by a standard result in complex analySIS. A complex structure on X is a family S of pairs {(U, 'P)} which a.re pairwise compatible mid such that U U = X; there is then a unique maxnnal famIly of paIrs WIth the~e two properties and containing S; we shall usually assume that the com.plex structure IS . 301 The elements (U '.0) of this (maximal) complex structure are called charts or maxnn . , ,.. . 'f U- . h coordinate neighbourhoods. In a coordinate neighbourhood, we.usua~ly ldentl y Wit 'P(U) 3.-fldwrite z for 'P as one does with the usual complex vanable m iC. A Riemann surface is a connected 2-dimensional manifold X with a complex structure S. Vyeshall also assume that X has a countable base of open sets, although a theorem of Rad6 asserts that this is automatic (for a proof, see e.g. [4]). If

f

nc

X is open (X is a Riemann surface) ~nd f : n -+ C is cO=iinuous, is holomorphic if for any chart (U, If') of )(, the functIOn f 0 If' : If'(

say ~~lat. u) --+ ~ IS

Volf:

nn

T

holomorphic. If X. Yare Riemann surfaces, f : X --+ Y a continuous map, f is called holomorphic if, · any '. c11a·t (T( ",',) of v the fl'nction .,!> 0 f . f-l(V) ---; 'IjJ(V) c iC is holOlllorphic. I , \', for (j.

-

",

'.

'i'.

.

Non-constant holomorphic maps between Riemann surfaces are ?pell._Also, a b~ject~ve holomorphic map f : X --+ Y has a holomorphic inverse. :} --+ X. Such bIjectIve holomorphic maps are called analytic isomorphisms (or blholomorpillc maps). Examples 1. The complex projective line = Riema.nn sphere. Let jp'1 be the one-point compactification IC U {oo} of IC. We set U1 = jp'l - {oo} = IC, 'P1 : U1 --+ If..: being the identity; if z E IC - {O} = IC* if z = ex The map 'P2 0 'P11 is the map z f-> 1/ z of If..:* into itself, so that these two ch~rts define a complex structure on 1P'1. This Riemann surface is called the proJectlve lllle or the Riemann sphere.

-!

+!},

As charts, we use pairs (U, 'P) obtained as follows: let V be any open set in C such that 7rW is a homeomorphism onto an open set U in X; set 'P = (7rW)-l : U -+ 1/ C C. Two such charts (Ul, 'PI)' (U2, 'P2) are holomorphically compatible: we clearly have 7r('P20 'Pll(z)) = 7r(z) for z E 'Pl(Ul n U2) thus 'P2 0 'Pll(z) - z E A 'c/z E 'Pl(Ul n U2), so must be constant on connected components (because 'P2 0 'Pll is continuous and A is discrete). The Riemann surfaces X constructed above are called tori or elliptic curves. 3. Surfaces of "higher genus". Let 9 be an integer> 1, and let 0 < r < 1. Let 6. = {z E C izl < I}. There is a unique bijective holomorphic (= biholomorphic) map T : 6. -+ 6. such that T(r) = re31ri/2g and T(re"i/2y) = re21ri/29. Let (J : 6. --+ 6. be the rotation z f-io ze27ri/49.

I

and denote by l' the group of biholomorphic maps of 6. generated by Ak, Bk('c/k

E

Z).

A special case of a theorem enunciated by Poincare (for the theorem and its proof, see the elegant article by G. de Rham: Sv.r les polygi3nes generatev.rs de gr01.'pesFv.chsiens, L'Enseignement Mathematique, 1971, pp. 47-61) implies that there exists an r, 0 < r < 1, such that r acts freely (without fixed points) and discontinuously on i'l., and the quotient 6./1' is compact. One sees that the canonical projection 7r : 6 --;. 6/1' is a covering map, and obtains a complex structure on 6./1' for which the map 7r is holomorphic as in the case of tori. 4. Let Y be a Riemann surface, X a connected 2-dimensional manifold and p : X --+ Y a local homeomorphism. There is a unique complex structure on X for which the map p is holomorphic. obtained as follows: Let U be an open set in X such that plU is a homeomorphism onto an open set 1/ in Y such that V C Tj for some j. where {(V;, ~Jj )jEJ} is the given complex structure on Y. Let 'Pu U --+ IC be the map 'PU = 'l/Jj 0 p. It. is easily checked that two such pairs (U. ,'Pu') are holomorphically compatible, so that one obtains a complex structure on X for which p is holomorphic. The uniqueness is a consequence of the following remark: let U c: X be open and pIU, a homeomorphism onto 11 C Y. Then, if pis holomorphic, the map (piU)-1 : 11 --+ U is again holomorphic.

be a sequence of points in X with

In this case, tl = n U {P} is a neighbourhood of P in X containing no other boundary points of X. Since plO -+ De - {a} is isomorphic to the map Pn defined above, there is a homeomorphism

Let De = {z E «:: Ilz - al < t:} if a E «::, and let Dc = {z E «:: Ilzl > ~} u {oo}if a = 00. Then, for all sufficiently small t: > 0, all but finitely many of the {xv} he in the same connected component of p-l(De).

Set X = XU{ algebraic boundary points of X}. We can extend the complex structure on X to one on X by taking as a chart containing an algebraic boundary point P E X - X the pair (tl, 1, then p is not a local homeomorphism at P.

Consider now a Riemann surface X and a holomorphic map p : X -+ «:: which is also a local homeomorphism. We consider «:: as the complement of 00 E jp'1,and p as a local homeomorphism X -+ jp'1. We shall define boundary points of X. Let {xv the following properties:

}v:2;l

1)

{xv} is discrete (i.e. has no limit points in X);

2)

{p (x v)} converges to a point a E jp'1;

3)

Two such sequences {xv}, tyv} are called equivalent if the sequence 7

_

~v -

{X(V+l)/2 Yv /2

for for

l/ l/

odd even

again has the three properties above [i.e. limp(xv) = limp(yv) = a say, and the connected components of p-1 (Dc) containing all but finitely many of the Xv, Yv respectively are the same]. A boundary point of X (relative to the map p) is then al! equivalence class of sequences {xv}v>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,
La

. f(k)(a)

d = (-) dz

-a

defining

forp-11._ _-0

if

(V,I)

is a pair

la·

=~~o

If we choose a chart as above, then the map la f---+ *,-l~n)(a)zn is an isomorphism (as iC-algebras) of 0 a with the ring iC{z} of power series with a non-zero radius of convergence. Let Ox = UaEX Oa (disjoint union). We will sometimes write simply 0 for Ox· We define a map p: Ox ---+ X by p(f) = a if f E Ox·

La

La·

{Lx

Let now E Ox, and (U, f) a pair defining the germ We set N(U, fJ = I x E U}, the set of all germs defined by f at the different points of U. vVe define a topology on (')x by the condition that the sets f)} form a fundamental system of neighbourhoods of when (U,!) runs over all pairs defining

La

With

this

topology,

1o·

Ox is Hausdorff

a.nd the map]J

Ox --; X is a loca.l

homeomorphism.

La'

Moreover, if U is open in X and f holomorphic on U, then p(N(U, pIN(U, f) is injective, having the inverse x f-+ Lx (= germ of f at :1').

I))

, The properties of p stated in the lemma follow from this. Remark. If X is given with a countable base for its open sets, it can be proved directly that any connected component of Ox has a countable base. This is a consequence of the so-called Poincare-Volterra theorem (for a statement and proof of which one may consult [7]). vVe can now construct the "Riemann surface of an analytic function" .

k

Let Oa be the set of all germs at a. Oa is clearly a ring, even a re:-algebra. The set ma of germs f with f (a) = 0 is an ideal; the complement 0" - ma consists of the units of 0" (J -l~as an m~erse in Oa ¢=} f (a) =f 0) so that ma is the unique maximal ideal -a -a in Oa.

Lemma.

Thus, Ox is Hausdorff.

La

Proof. Let fLb E (')x, and suppose that i fLb' We must show that they can be separated by open neighbourhoods, and consider two cases.

La'

1) a =f b. Let (U,fJ, (\I, g) be pairs defining fLb respectively (a E U,b E V). We can find such pairs with un V = 0, in which case N(U, f) n N(\I, g) = 0

Let X = iC and consider Oe. Let IvI be a connected component of 0e, and p : 1111 ---+ iC C jp'l the restriction to 1111 of the map f f-+ a constructed earlier; p : M ---+ iC is a local hOITleOmorphisill, so there is a uniqu: structure of a Riemann surface on ]\;J for which p is holomorphic; p is then a local analytic isomorphism, i.e any a E l'v1 has an open neighbourhood U such that plU is an analytic isomorphism of U onto the open set p(U). We define a holomorphic function h on IvI by h(f ) = f (a) [if (U, f) is a pair defining f with U connected, then N(U, I) C M and-h(f -x )-~ f(x) "Ix E U, so that h is _0. holomorphic]. Intuitively, this "universal" function h describes all the germs that can be obtained by "analytic continuation" of a fixed germ E AI.

La

Let now 1',;! = NI U {algebraic boundary points of M} and]) : phic map extending p : IV! ---+ iC we constructed before.

---+

jp'l be the holomor-

Let U be a connected open set in iC and f, a holomorphic function on U. Vie assume that (U, f) defines a germ 1" E AI for SOllle a E [, (and hence for all a E U, since U is connected) . The set E = -- AI is a discrete .set in and we Let be the union of 111 with those points PEE of h (i.e. points where h has either a holomorphic a meromorphic function h f all X f with h f 11\11= h. of f;

have our universal function h on lU. which are not es.e,entialsin·f~ularitiE;.s exten.sion or a pole). Thus, there is Let p f : X f ~+ jp'1be the restriction

If F E Xf - 1\11,and ]Jf(F) = a, then near F, the map Pf is equivalent to the map :; f-+ a +:;n ('11 2': 1) or z f-+ z-n Thus, if z is a local coordinate at F on Xf and 'w

denotes a local coordinate at a on pl, the local description of the maps Pj, hj can be written:

Let F(x, y) = 0,0 (x)yn + 0,1 (x )yn-l + ... + an(x) E iC[x, y] be an irreducible polynomial, let V = {(x,y) E CZI F(x,y) = a}. Let So = {x Eel ao(x) = a}, Sl = {x E C !3y E C with F(x,y) = = ~~(x,y)} and let S = So U Sl U {oo} C pl. Let 7r: V ---7 C be the projection (x,y) f--+ x, V' = V - 7r-l(S) =V - If-l(SO U Sd and 7r' = 7rlVl We have seen that. if Dc is a small disc around a E S (Dc = {[zl > 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 <

are meromorphic I'

a,.(w)

ao(w)

at

near II -< ~ Iwl"'

"W

W

'tV

= 0, so that

=

there exist constants

= O. By Lemma 3, we have

for some constants Cl, k. Hence

vVe now claim that Viis connected. r-sheetecl covering with 1 S; T < n.

0,

ao( w)

J

l)j

has a merommphic

If this were not the case, 1'1

lV,

--+

C >

< extension

!pI - S is an

For x E pI - S, let bv(x) (u = 1, ... , T) be the uth-elementary symmetric function of Yl,' "., Yr, where the "Vj are the values taken by 171 at the points of by the

definition of 7)1, the Yj are values of the second projection P2 at points (x, y) E V, so that F(x,Yj) = 0, j = 1, ... ,r. We claim that bv extend to meromorphic functions on 1P'1. In fact, since the Yj are values of the function 7)1 which is meromorphic on WI, in the neighbourhood of a E 5 we have an estimate of the form Let X be a topological space. A presheaf data.

:S: C11x [(PI, ... , Pr) =

7f

1

1 (x)]

ai-£' (resp. C1!xl--e'

of abelian groups on X consists of the following

1) An assignment U f-+ F(U) of an abelian group F(U) to each open set U C X (F(0) = {O} = abelian group with just one element) if a = 00)

.

2) A family (p~)vcu open sets with V

Thus, the bv are meromorphic on 1P'1, and are therefore rational functions of x.

(a)

Let G(x,y) = yr + b1(x)yr-1 + ... + br(:r). Then, if x E pi - 5, the roots of G(x,y) (viz Yl,oo.,Yr), are also roots of F(x,y). Hence G divides F in C(x)[yJ, and, since degy G 2: 1, F is not irreducible (Gauss' lemma).

(b)

Thus V' is connected, and W, the algebraic completion of V', is a compact Riemann surface. W carries a meromorphic function 7), and if 7f : IT! -t pi is the extension of 7f' : V' -t pi - 5, we have

This construction is, of course, a special case of the construction of the Riemann surface of a holomorphic germ, once one has proved the connectedness of Vi This statement is equivalent to the following: Let a E pi - 5, and let YI, ... ,Yn be the germs at a satisfying the equation F (x, Yj (x)) = O. Then, for any j, there is a closed curve/, in pI - 5 starting at a such that analytic continuation of YI along ~Ileads to Yj·

pg

c

of group homomorphisms p~ : F(U) -t FW) whenever U, V are U (called restriction maps) having the following properties:

= identity on F(U) 'if open U;

if TV eVe

U are three open sets, then

If the groups F(U) have additional structure (rings, vector spaces, C-algebras, ... ) we shall speak of presheaves of (rings, vector spaces, C-algebras, ... ) if the restriction maps respect this additional structure. Example. Let X be a Riemann surface, and, for U C X open, let O(U) denote the C-algebra of functions holomorphic on U. If V c U, the map p~ : O(U) -t O(V) will be restriction: f f-+ fill. A presheaf (F(U), p~) will be called a sheaf if the following two conditions are satisfied. Let U C X be open, U = UiEI Ui where the {Ui} are again open (i) If f,g

E

F(U)

and pgi(f)

= pgi(g)

Vi, then f = g.

(ii) Given, for each i, an element Ii E F(Ui), 3f E F(U)

with

/6, (f)

We shall usually denote of a mapping).

if

pg;nuj (Ii)

=

//;:nu/fj)

'ifi, j E I, then

= fNi.

pt;(f)

by

fW

(as if we were actually dealing with restriction

'vVecan associate a sheaf to any presheaf (by the construction used to define germs of holol1l0rphicfunctions). Let F = (F(U) , p~) be a presheaf on X, and let a E X. On the disjoint union UaEU F(U) we introduce an equivalence relation as follows: f E F(U), 9 E FW) are equivalent if 3 an open set TV, a E TV c Un v' such that. flTV = gIVV. The set Fa of equivalence classes is called the stalk of the presheaf .1' at. a. [It is also the direct limit of the directed system (F(U), p~).J We can introduce a topology on IFI = UaEX Fa (disjoint union) by taking as a fundamental system of neighbourhoods

of f E Fa the following sets: let f E F(U), a E U, be a representative of the equivalence clas~ f ,and let N(U, I) = {f Ix E U}, where f . is the equivalence class in Fx defined by (U--:1). IFI does not haveto be HausdorfCbut the projection map p : IFI --> X, p(j) = a if f E Fa is local homeomorphism. If we set IFI (U) = {set of sections of IFI over U, i.e. the set of continuous maps s : U --> IFI such that po s = identity}, and let 'r~(s) be the restriction of the map s : U --> IFI to V c U, then (IFI (U), 'r~) isa sheaf, the sheaf associated to the presheaf :F. We now define morphisms between presheaves. Let F = (F(U), p~) and 9 = (9(U), r~) be presheaves on X. A morphism a : F --> 9 is the assignment, to each U open C X, of a morphism au : F(U) --> 9(U) such that, if V C U, the diagram

1,,;;

If au is an isomorphism for all U, then a : F

If a : F presheaf

9

-->

is a morphism of presheaves,

-->

9 is

called an isomorphism.

we define the kernel, ker a, of a to be the

to

101 is an isomorphism,

Definition. If a : F --> 9 is a morphism betweerl the sheaves F and 9, we shall denote by Im( a) the sheaf associated to the presheaf {im( au ), r~ I im( au) }. Given morphisms the sequence

a : [;

-->

F and f3 :F

-->

9 between

sheaves [;, F,

9

on X, we say that

[; -!!:-
(a) f3u 0 au = covering {U;}iEI

9(V)

°

sheaf Ox of X.

This is often called also the structure

is exact (at F) if the sheaves ker((3) and Im(a) This amounts to saying the following:

9(U)

commutes.

functions on X" as defined in §3. Since the natural map from we shall not distinguish between the two.

°

(in the sense just defined) are equal.

VU and (b) if f E F(U) and f3u(j) = 0, then, there exists an open of U such that flUi E im(auJ Vi E I.

Let now X be a topological space and F a sheaf of abelian groups on X. Let U = {Ui}iEl be an open covering of X. Then, for q 2': 0, we define the group of q-cochains of F (relative to U) by

II

Cq(U, F) =

F(Uio

n··· n Ui,)

.

(io, .. ,iq)Elq+l

If F and 9 are sheaves, so is ker( a).

We define the coboundary b : CO(U,F) --> C1(U,F) Cij =Ji!Ui n Uj - fjlUi n Uj. We also set

If we define the image im( a) to be the presheaf

Example. Let "\. = e = c - {OJ, let O(U) = {set of functions holomorphic on U}, O*(U) = { set of functions holomorphic and nowhere zero on U}. If exp : 0 --> 0* is the morphism defined by expu : f f-+ exp(21ril) (J E O(U)), then im( exp) fails to satisfy the second condition in the definition of sheaf. Namely, if U1 = C - {.r E ~Ix ,,:=. OJ, U2 = C - {x E ~ix ::::OJ, and we set == z on th, h(z) == .0 on U2, then fi E im(expuJ since U1, U2 are simply connected, but there is no f E im(expu,uu,) with flUi = li (i = 1,2) (the function.z has no single valued logarithm on = UJ UU2).

Finally. let B1(U,:F) Zl(U,:F).

= Image(b

: CO(U,F)

-+

by b((fi)iEI)

C1(U,F)):

we have Bl(U,F)

C

e

Remark. If F is a presheaf and we construct IFI the sheaf associated to F, we have a morphism a : F -+ IFI defined as follows: for f E F(U), au(f) is the section of IFI over U defined by a f-+ f = element of Fa induced by (U, I), a E U. It can be checked directly that if F is a sTi~af, then a is an isomorphism. If we start with the sheaf U f-+ O(U) = {space of functions holomorphic on U} on a Riemann surface X, then the space 101 is simply the "sheaf of germs of holomorphic

the first

cohomology

gTOllp of

F

relative

to U.

We also set HO(U,:F)

= {(filiEl

E CO(U,F)

I b(fi)iEI

= o} :

by the sheaf axioms, the map F(X) --> CO(U,F) defined by f f-., isomorphism of F(X) onto HO(U, F) for any open covering U. Elements HO(U, F) are also called (global) sectiolls of :F.

induces an of F(X) =

Let V == (VaJoEA be a refinement of U; there is thus a refinement map T: A that Vo C UT(o) Va E A (V is also an open covering of X). T induces a map

-+

I such

-+

I are two refinement

mQps (i.e. Va

C

UT(a)

n U(7(a)

== ~, and ~ E BI(U, F).

This proves Proposition

as follows. If ~ == (Cij)i,jEI E ZI(U,F), we define T*(~) == ha(3)o(3 by ~fa(3 == CT(O)T((3) lVa n V(3. Clearly r* (BI (U, F)) C BI (V, F) so that it induces a map (denoted again T*) of HI(U,F) to HI(V,:F). Proposition 1. If r, (J : A A), then the induced maps

Thus 8{(hi)}

Va E

2.

We now define the cohomology group HI(X, F). Let U, V be open coverings of X, . U == {UihEI, V == {VO}OEA, V a refinement of U. Then, there is a map T(U, V) : HI(U,F) -+ HI(V,F) (defined using a Tefinement map r : A -+ I, but independent of the· choice of this map). If W is a refinement of V, we have r(U, W) == r(V, W)or(U, V). We define HI (X, F) as the direct limit of the system (HI (U, F), T(U, V)) which is the following: Let R be the equivalence relation on the disjoint union Ilu HI (U, F) defined by: ~ E HI (U, F) is equivalent to 7) E HI(V, F) if there is an open covering W which is a refinement of both U and of V and such that r(U, W)~

Proof.

If (fij)i,jEI

fT(a)T((3)

E ZI(U,F),

- f(7(o)(7((3) ==

we have, on Vo

(1T(o)(7(a)

+ fO"{O)T((3»)

Then, HI(X, F) == IlHI(U, F)j R. For any U, there is a map T(U) : HI(U,F)

n V(3, -

== r(V,

(1(7(a)T((3)

+ fT((3)O"((3»)

== ga - g.6 ,

~ under

HI(X,F)

(T(U)~

== equivalence

class of

R).

Proposition

where go == fT(O)O'(O) I Vo· Hence {fT(O)T((3) - fO"{0.)O"((3)} E BI(V,F).

-+

W)7) .

2 is equivalent to the statement

that T(U) is injective.

We shall need the following special case of a theorem of Leray. LERAY'S THEOREM. Let F be a sheaf of abelian groups on the topological space X. Let U == {UihEI be an open covering of X. Suppose that HI(Ui,F) == 0 Vi. Then, the natural map Proof. Let r A -+ I be a refinement map, and let ~ and suppose that T*(O E BI(V, F). Thus, 3ga E F(V",) 9a lVa n V(3 - g(31 v::' n V(3. Let i E I and x E Ui. Choose define hi(X) == ga(x) + fiT(a)(X), If 13 E A is such that x E

== {(fij)i,jEI}

E ZI(U,F), with fT(O)TUJ) I v::' n V(3 == a E A so that x E V~, and ViJ, we have

Proof. It is sufficient (because of Proposition 2), to show that for any refinement V == {Va}oEA of U, the induced map r* : HI(U,F) -+ HI(V,F) is surjective; here T: A -+ I is a map with 1/0., C UT(a)Va. Let {ca(3}0.,(3EA E ZI(V, F). Now {ca,/3IUi} E Z"(UinV, {Uirl\o~,}oEA of Ui. Since HI F) == 0 by hypothesis, exist [lia E F(Ui n Va) such that

fiT(a)(J;)

- fiT((3)(X)

== -(1T(a)i(.1:)

+ fiT({3)

== - fT(a)T(6)(X)

(a;))

(since f, E ZI (U, F))

Hence, the above formula defines hi E F(Ui). If x E Ui n Uj, and we choose

0'

with x E Va, we have

.

Now, on Ui n Uj n Va n \0$ we have gio - [li.(3 == ca{3 == [lja - gj{3, i.e. [lia - [lja == gi(3 - [lj(3: hence (by the 2nd sheaf axiom), there exist elements lij E F(Ui n Uj) so that lij == gi0. - [lja on Ui n Ui n IoTa. Clearly, lij + lik == lik on Ui n Uj n Uk· We have, 011 v::' n Vp (C U ((3)) IT(a)T(p) + cap == +([IT(a)a - [IT((3)<,) + (9T((3)a - 9T(l3)(3) T

hi(x)

- hj(x)

== [la(X) == fiT(O)

+ fi,(a) (:r) (X) + f,(o)j(X)

F), where uinv is the covering Proposition 1 implies that there

ga(X)

- fjT(O)(X)

== hi(X)

.

== -9T((3)(3

+ 9,(a)a

.

Since gT({3){3 E F(UT({3) n V{3) = F(V{3), this shows that {c<>,{3} and the same element in HI (V, F) .but the latter is the image, under proves the theorem.

{-'"1'T(<»T(,6)} T*,

of { -'ij}.

induce This

The relevance and usefulness of Leray's theorem in the theory of Riemann surfaces stems from the following: MITTAG-LEFFLER'S THEOREM. Let D be an open set in C. Then HI(D, where 0 is the sheaf of germs of holomorphic functions on D.

0) = 0,

3. Let D be open in C and let

1E coo (D).

+ iy, x, y

Proof define

of Proposition

+ Z 8y

1 has

1 11(z+11') . ---d11' 27fz"= 11'

= -.

compact

support

in D, and

/\ diiJ .

.

hm

If(z+h+W)-fCz+11')l

_

, n

l(z)

d11' .

11'

Iwl=e

jwl=c

Since f(z+u~~-f(z) is bounded the result follows.

as a function of

this last integral

11',

-t

0 as

E: -t

0, and

To prove Proposition 3, let {Kn}n>l be a sequence of compact sets in D with Kn C Kn+1 (interior of Kn+1), U Kn = D a~ld such that D - Kn has no connected component relatively compact in D.

1 .E Coo(D)

and let 'Un E C':>O(D) be such that ~ = 1 011 a neighbourhood of Then un+1 -- Un is holomorphic on a neighbourhood of Kn, so that there is hn, holomorphic on D so that IUn+1 - Un - hnl < 2-11 on Kn (n ~ 1). Let

Then u E COO(C) and ~~ = f. First, to see that U E Coo(C), remark that 1~'1 is integrable on any compact set in C, (polar coordinates at 0). so that, for instance, the existence and continuity of ~~ follows from the fact that

l~E~ "=

+ 11') -

n

3. Case 1. Suppose that u(z)

l(z

Let be open in C and let KeD be compact. Let L be the union of K with those connected components of D - K which are relatively compact in D. Then L is compact and has the following property: any function holomorphic in a neighbourhood of L can be approximated, uniformly on L, by functions holomorphic. on D.

.8)

[)z = 28x

J

2 1T"~ '1(Z ) +

To complete the proof of Proposition 3, we need the following form of Runge's theorem; we shall not prove this here. A proof is given e.g. in [7].

real, then

[) 1(8

11'=

Then, 3u E Coo (D) such that

~~= 1· Recall that, if z = x

l(z+11')d

~-~11'

If now 1 E C'oo(D), if we apply this special case to the function 'Pf where 'P E C8'"(D) and = 1 on a given c:ompact set KeD, we obtain the following: If 1E Coo(D) and KeD is compact, there is U E Coo(D) such that ~~ = f on K

To prove this, we first prove the following result. Proposition

J

=

w

f8f

dw/\d:w=

jI[;

1

a(z+w)-dw/\dw x w

_

J{n-

Define It = Un + Lm>n (Um+l - Urn - hm) - hI - ... - hn-1 uniformly on K n' I'Ve have U

=

Un

+ (Un+1

-ltn

- hn)

-+-

L

(Um+1

on Kn; the series converges

- 1{,m _. hm) - hI --.

,-

hn-1

m2:n+l

since, f having COlllIJact support, lilllh--.O f(z+h~-f(z) on C. We have only to iterate this argument. If c OU=_l_lim 27fi C~O

OZ

J

= ~;(z), uniformly

O~(z+w)~dw/\d(i>;

OZ

J

Iwl:>E

10

J

1

",_(z-+-w)-diu/\dw=

uZ

0

,.

lO

~(z

+ w'

~(L_.-J)d1U/\dii) u'UJ

J

L

+

(Um+l - Urn - hm) - h1 - ... - hn ,

so that this defines a function on D, Since Lm>n+1 on

'U.)

Iwl:>, =-

= u'n+1

m'2:n+l

Iwl:>E

,. 01

and boundedly

> O.

f(z -+-w)dW) d ( ---'W

]('<-1-1

:J J{n and

D'(~~+l =

f

on En, we have

Proof of Mittag-Leffler's theorem. covering 11 of D (D open in C). Given 11 = {U;}iEf,

Coo(D), support If i

$~= 1on Kn

Vn, i.e.

011

n.

We shall prove that H1(U, 0) = 0 for any open

let {a;}iEf be a partition CUi, the family {support

of unity with respect to U, i.e. ai E is locally finite, and ai == 1.

(ai)}iEf

is hololllorphic on Ui n Uj and Cij -+- Cjk = Cik 011 Ui n Uj n Uk for alIi, j = k we find that C'ii = 0 Vi, now taking k = i, we find that Cij = -Cji'

Cij

=

(a;)

(Um+1 - Urn - hm) is holomorphic

L

.i, k,

taking

Now, given i,j, we define a COO-function on Ui by setting it = ajCij on Ui n Uj, = 0 on Ui - (Ui n Uj). Since support (aj) C Uj, aj = 0 in a neighbourhood of (8Uj) n Ui, so that this function is Coo on Uii we denote it simply by ajCij' Set 'Pi = I:jEI ajCiji this is Coo on Ui since {support (aj)} is locally finite. Given k, eEl, we have, on Uk n Ue, 'Pk - 'Pe

= 2.: aj(ckj

- Cej)

=

2.: aj(ckj

j Now, ~

1/J[Uk Let

= U

-

atz"

W Vk

= ~

E

+ Cje) =

2.: ajcke

j = 0 on Uk

n Ue.

= Cke .

j Thus, there is a Coo-function

'IjJ

on D with

I.

E COO(O)and ~~

=

1/-' on D, and let hk

~ - ~~= 0 on Uk, so that hk E OtUk)' This proves the theorem.

On Uk

=

n Ue,

'Pk -

U

on Uk·

we have hk - he

We have

= 'Pk -

'P.'

=

fJa~

=

have fi - IiIUi n Uj E ker((3u,nuj) = im(au,nuj) by the lemma above, i.e. 3eij E £(Ui nUj ) with aUinUj (eij) = fi - fjlUi n Uj. Clearly eij + ejk - eik maps to 0 under 00, so is 0 on Ui n Uj n Uk. Thus {eij} E Zl(U,£), so defines an element 6(g) E H1(X,£). We have to check that this element is independent of the choices: viz of U and the .Ui}' If V = {V"'}"'EA is a refinement of U (given by T : A --+ 1) and we choose the elements f~ = fT("') IV"" we dearly get the same element of H1(X,£) (in fact simply the restriction of the cocyde {eij} defined above to V). Hence we consider the same covering U and possibly different liftings fI E F(Ui) with (3Ui UI) = glUi. As above, we see that there are elements ei E £(Ui) such that aUi (ei) = fI - k If eij E £(Ui n Uj) map to ii - fj on Ui n Uj, we have ei - ej = eij - eij (since the image under a is 0), and {eij}, {eij} define the same element of H1(X,£).

Cke·

One more general construction and theorem will be required from sheaf theory. This is the exact cohomology sequence, and we shall develop just the part of this sequence which will be needed. Let 0 --+ £ ---'=-+ F ~ 9 --+ 0 be a short exact sequence of sheaves of abelian groups on the topological space X. (Recall that the exactness at g, i.e. the surjectivity of (3 amounts to the surjectivity of (3x : Fx --+ gx Vx EX, where Fx, gx are the stalks of F and 9 at We have

which maps cocydes into cocydes and coboundaries to coboundaries. hence induces a morphism oo~ : H1(U,£) --+ H1(U,F). This in turn induces a1. . Definition of .B1• This is the map H1(X,F) with a above). Theorem (The Exact

Cohomology

£x

--+

Fx,

H1(X,i])

induced by (3: F

--+

9 (as

Sequence).

Let 0 --+ £ ---'=-+ F ~ 9 --+ 0 be a short space X. Then, the sequence

Proof. Given the injectivity of ax: of the sheaf axioms for £.

--+

exact

sequence

of sheaves

on the topological

the injectivity of ax is simply the first

The exactness at F(X) asserts that if f E F(X) and there is a covering {Ui} of X such that flUi E im(aui) Vi, then f E im(ax). Let ei E £(Ui) be such that QUi(e;) = flUi. We have (ei - ej )IUi n Uj = 0, (since a is injective). By the second sheaf axiom, there is e E F(X) with elUi = eNi. Clearly ax(e)lui = av;(ei) = flUi, so that axle) =

Remark. One can define (~ech cohomology groups Hq(X,F) for all q 2: 0 and extend the exact sequence above when X is paracompact [see e.g. Serre--Faisceaux algebriques coherents, Annals of Math. 61(1955)].

Recall also that H°l-X, £) = £(X), .... We define (to : H°(.X, £) --+ HO(X, F) by aO = ax (and similarly, /3° : HO(X, F)--+ H°(.X, i]) by /3° = /3x). We now define morphisms b: HO(X,9) --+ H1(){,£) and c/ H1(X,£) --+ H1(X,F). : HI ,F) --+ H1(X,g) as follows.

Proof. Exactness at HO(X,9). First, if 9 = /30U), definition of o(g), we can take any covering {Ui} and f; = we have o(y) = O.

r

Definition of covering {Ui}iEI

o.

Let g E H°(.X, g) = g(X). Since /3 is surjective, there is an open = U of X and elements f; E F(U;) such that (fi) = gilUi Vi. We

f E HO(X,Q), Since fi--

flUi-

f,

then, in the = O.

-

Conversely, suppose that 6(g) = 0; it is represented, with respect to a suitable covering U = {Ui}iEI by {eij} E Zl (U, £) with au,nuj (eij) = fi - Ii !Ui n Uj and (ji) = glUi· Since o(g) = 0 (and H1(U, £) --+ H1(X, E) is injective), there are elements Ci E E(Ui)

with ei - ej IUi n Uj = eij' Let II = f; - au, (ei); then fi - fj = Ii - fj - aUiilUj (eij) = 0, so that there is f E F(X) with flUi = k We have (3Ui(f) = glUi-((3oabi(ei) = glUi; thus 9 = (3x(f). Exactness at H1(X,£). Let U = {Ui} be a covering and {eij} E Zl(U,£). Then a1(0 = 0 [~being the class in H1(X, £) of {eij}] ~ ::I{f;} E CO(U, F) with fi- fjlUin Uj= aUiilUj (eij}; this condition is, by definition, satisfied if ~ = 8(g). Conversely, ifthis holds, then fi - fjlUi n Uj E ker((3), hence there is 9 E HO(X, Q) with glUi = (Ju, (f,), and, again by definition, ~ = 8(g). 1 Exactness at H1(X,F). We could define a map ((300')1 lf1(X,E) -7 H (X,Q) induced by (300' = 0 as we defined 0'\ clearly ((30 aj! = O. But we see at once that (/30 a)l = ;31 0 al, so that im( (1) C ker(p1 ).

Conversely, let {fij} E Zl(U, F) [F a suitable covering of X] and suppose that (3(fij) = gi - gjlUi n Uj where gi E Q(Ui). By shrinking the covering U, we may suppose (since (3 is surjective) that ::If; E F(Ui) with (3(Ii) = gi But then, if fij - (fi - fj)lUi n Uj = i[j' {I[j} represents the same element of H1(X,F) as Uij}, and f[j E kerPUiilUj = imauiiluj' If aUiilUj(eij) = fij' we see at once that {eij} E Zl(U,£) and its image under a1 is the class of {lij} = class of Uij}.

Let X be a t.opological space. Suppose given another topological space E and a continuous map 1f : E -7 X, and that each fibre 1f-1(a) = Ea, a E X, is provided with the structure of a C-vector space of dimension n. We call1f : E sense:

-7

X a (continuous)

vector b'undle if 1f is locally trivial in the following

Va E X, ::I an open neighbourhood U of a and a homeomorphism hu : 1f-1(U) with the following properties:

-7

U

X

cn

(i) The diagram

(where pru : U

X

cn is projection on the first factor) commutes.

(ii) Va E U, the map 'Pa : Ea -7 isomorphism of C-vector spaces.

cn

defined by hu(x)

=

(a, 'Pa(x)),

x E Ea, is an

If X,E,1f are Ceo (resp. complex analytic) and if hu can be chosen also Coo (resp. biholomorphic), we shall speak of Ceo (resp. holomorphic) vector bundles. The integer n is called the mnk of the vector bundle. If the rank n = 1, we call1f : E -7 X a line b'undle. The map hu is called a trivialisation of E on U (or a local trivialisation of E at a). It is also sometimes called a linear chart for E on U. If 7f : E -7 X is a holomorphic vector bundle 011 the complex manifold X, we can find an open covering U = {Ui}iEI and hololllorphic trivialisations

()n of

n

[Ti

u)

en

1

the

11la,p

h"i

0

n)

X

{en -~

u)), where, for fixed x E Ui n Thus, there is a holomorphic map gij : Ui

H·

(.r,

'U

n

H

n Uj

-7

x

(~n

ha.s the

fOnI)

v) is a C-linear i~olnorphism GL(n, such that

(multiplication in G L(n, q). The {gij} are called transition bundle (corresponding to the local trivialisations hi)'

functions of the vector

If we replace {hi} by other trivialisations U<}, then h~ 0 h;1 : Ui x ren is of the form (x,v) f-'o ((X,'Pi(X)V), where 'Pi : Ui -+ GL(n,q is holomorphic; the corresponding transition functions are 'Pi gij 'P;

E, E' are iSOlnorphic if there are morphisms tL : E -+ E' and u' : E' -+ E such that u 0 u' = identity on E' and u' 0 u = identity on E. A bundle 7f : E -+ X is called trivial if it is isomorphic to the "trivial bundle" pr x : X x ren -+ X (where pr x (x, v) = x). Isomorphisms, and trivialisations, can be continuous,Coo, holomorphic, ... ; this simply .means that, e.g. the morphisms tL: E -+ E', u' : E' -+ E have this property.

1.

Conversely, given a family of holomorphic maps gij : Ui n Uj -+ GL(n, q satisfying the cocycle condition gij gjk = gik on Ui n Uj n Uk, we can construct a vector bundle as follows. Let jj; = lliEI Ui x ren (disjoint union), n- : jj; -+ X the map defined by (x, v) f-'o x. Call (x,v) E Ui x ren equivalent to (y,w) E Uj x ren if x = y and v = gij(X)'W. This is an equivalence relation because of the cocycle condition, and two points in the same Ui x ren are equivalent only if they are equal. Moreover, n- induces a map 7f : E -+ X where E is the quotient of jj; by the equivalence relation. The projection from jj; to E induces a bijection h;1 from Ui x ren to 7f-1 (Ui), and one checks easily that 7f : E -+ X is a holomorphic vector bundle. If 7f : E -+ X is a vector bundle, a (continuous, Coo, holomorphic ... ) section of E is a (continuous, Coo, holomorphic ...) map s : X -+ E such that 7f 0 s = identity on X. If we consider local trivialisations hi : 7f-l(U;) -+ Ui x ren as above, then hi 0 sex) = (x,f;(x)) for x E Ui, where fi is a map Ui -+ ren. Since hi 0 sex) = hi 0 h;1 (x, fj(x)) = (x, gij (:r)Ji (x)) if x E Ui n Uj, we have

The converse also holds, as is checked at once. Thus a (continuous, Coo, holomorphic ... ) section of 7f : E -+ X can be identified with a family {Ii hE! of (continuous, ("')0, holomorphic ... ) maps fi : Ui -+ ren such that fi = gij ij on Ui n Uj. Sections of E over open subsets of X are defined in the obvious way. Let now X be a Riemann surface and 71 : E -+ X a holomorphic vector bundle on X. A rnemmorphic section of E is defined as follows: let 5 c X be a discrete set and let s : X - 5 -+ E be a holomorphic section. Then 8 is said to be a meromorphic section of E if, Va E S, there is a neighbourhood U of a and a coordinate, on U with = 0 such that Un 5 = {a} and for some integer IV ;c: 0, 8 is the restriction to U - {a} of a holomorphic section of E over U.

c

n are trivialisations, If U = {UihEI is an open covering of X and hi : 71-1 -, Ui X we can write hi 0 8(or) = (x, f;(x)) for x E Ui - 5; Ii is holomorphic on Ui - 5, and the section 8 is meromorphic if and only if Ii is meromorphic on Ui for all i if f; has at most a pole at points of Ui n 5).

If 71 E -+ X, 71' : E' -+ X are vector bundles (continuous, Coo, holomorphic ... ) a morphism'tL : E -+ E' is a map such that 71' 0 U = 7f and u : 7f-l(a) -+ 7f'-I(a) is Clinear. It is called continuous, Coo, holomorphic, ... if u has this property. Two bundles

We proceed now to define divisors on a Riemann surface. Let X be a Riemann surface. A divisor D on X is a map D : X -+ Z such that the support of D is locally finite, i.e. VK c X compact, the set {P E XID(P) # O} is finite. We usually write

L D(P)P;

D=

PEX

if X is compact, the sum is finite. We define the sum and difference D1 ± D2 of divisors D1, D3 by (D1 ± D2)(P) = D1 (P) ± D2(P). We say that a divisor D is effective if D(P);C: 0 VP E X. Given divisors D1,D2, we say D1 ;c: D2 if D1- D2 is effective. The set {P E XID(P) # O} is called the support of D and written supp (D). Recall the following notation: if f is meromorphic in a neighbourhood U of a point a on a Riemann surface X and z is a local coordinate on U with z( a) = 0, we set orda (f)

= {

c::

if f == 0 near a if f(z) = anzn

~;;:'=k

Let' now 8 be a meromorphic section of a holomorphic vector bundle E on a Riemann surface X, s =;E O. For a E ,X, choose a coordinate neighbourhood (U, z) with zeal = 0 and a trivialisation h :'EIU -+ U x ren Then h08(:r) = (x,f(:c)),:r E U. where f is an n-tuple of meromorphic functions on U. There is an integer k such that. f = zk g, where g is an n-tuple of functions holomorphic near a, (1,nclg(a) # O. Vve set k = ordo(8). One checks easily that this is independent trivialisation.

of the local coordinate

and of the lou),l

If 8 is a merom orphic section of a holomorphic vector bundle E, the divisor a i.e. the divisor orda(8)a .

f-'o

orda (8),

L

aEX

is called the divisor of -5 and denoted by (8) or div(8). In fact, any divisor on a Riemann surface can be obtained as the divisor of a meromorphic section, even of a line bundle (i.e, vector bundle of rank 1), and we turn now to this construction npP (where Let X be a Riemann surface and D a divisor on X; let D = # O} ancl let Up = D(?) E Z and the sum is locally finite). Let 5 = {P E {Up, zp} be a local coordinate system at P E 5 with zp(P) = 0: we assume that the

r~--~-~--I

I

30

6. Vector Bundles, Line Bundles and Divisors

open sets Up are so chosen that Up n UQ = 0 if P =1= Q, P,Q E S. Let U* = X - S, let f. == 1 and let fp = z~P on Up, PES. If I ={*} US, let 9ij = /i//j, i,j E I, 9ij being defined on uinuj [9ij = 1 by convention if uinuj = 0]. Now, if uinuj =1= 0, gij is holomorphic and nowhere 0 on this intersection. Moreover, gij gjk = 9ik on Ui n Uj n Uk (if this latter intersection is =1= 0). Thus, the {gij} form a system of transition functions for a line bundle L(D). Moreover, the functions {f;}iEl define a merom orphic section of L(D) since, by definition, fi = gijfj, i,j E I. We shall denote this section by SD. Since SD is defined by fi on Ui, we have orda(sD) = orda(f.) = 0 if a E X - S, and orda(sD) = ordp(z~P) = np if a = PES. Thus (SD) = D. If we use a different local coordinate (Up, (p) at P (with the same Up), then h p = ((p / Zp )np is holomorphic and non-zero on Up, and if we set h. = 1, and denote by {gij} the transition functions obtained from the functions (~P, then gL = higijhjl. If L' is the line bundle defined by the {g;j}' there is therefore an isomorphism of L(D) onto L' taking SD to the corresponding section of L' defined by the {fI} (f; = 1, fF = GP). Some general remarks. 1) If S is a meromorphic section iE 0 of a holomorphic vector bundle on X, then s is holomorphic if and only if div(s) is effective [s may have zeros, but no poles]. 2) Given a divisor D, and U C X open, let OD(U) = {f meromorphic on U I div(f) 2': -D on U, i.e. orda(f) 2': -D(a) Va E U}. The assignment U 1-7 OD(U) is clearly a sheaf, denoted OD· If r(U,L(D)) denotes the space of holomorphic sections of L(D) over U, and SD E r(X,L(D)) is the standard section with div(sD) = D, then the map OD(U) -+ r(U,L(D)), f 1-7 fSD, is an isomorphism, in fact diver) 2': -D = -div(sD) ~ div(fsD) 2': O. Thus, the sheaf of germs of holomorphic with OD.

sections

of L(D)

can be canonically

identified

There is a unique isomorphism u : L(Dr) -+ L(D2) taking SD, to fSD2; x outside the zeros and poles of f and the supports of D1 and D2 by

is defined for

it extends holomorphically to X because, on any open set U C X, the section x 1-7 A(X lSD, (x) is holomorphic if and only iCe 1-7 A(x)f(x)SD2(X) is holomorphic [div(sD,) n U = div(fsDJ n U]. Conversely, if u : L(D1) -+ L(D2) is an isomorphism, then u 0 SD, is a meromorphic section of L( D2). If f is defined by u 0 S D, = f S D2, we have (f) = D1 - D2 (since orda(sD,) = orda(u 0 SD,) Va EX). Remark. In terms of the sheaves 0D, and 0 D2, the isomorphism above is simply the map: 'P E OD,(U) 1-7 f'P E OD2(U). Some further remarks. If Jr : E -+ X is a holomorphic vector bundle, and E* = UaEX(Ea)* (E~ is the dual of the vector space Eo. = Jr-1(a)), we can make E* into a vector bundle in a, natural way as follows. Let U c X be open and hu : Jr-1(U) ---7 Ux a trivialisation. We define hu : UaEU E~ -+ U x by hu( v) = (a, (hU,a)-l( v)), v E E~, where hU,a : Ea -+ is the isomorphism induced by hu Ea -+ {a} x and h[r,a : -+ E~ is the dual map. If gij : Ui n Uj -+ GL(n, q are the transition functions of E, those of E* are t gij 1, (t IV! denotes the transpose of a matrix !vI).

en

en

en

en

I

en

If E1, E2 are vector bundles on X, we define a vector bundle Jr : E1 0 E2 -+ X with Jr-1(a) = E1,a 0 E2,0; if {gij)} : Ui n Uj -+ GL(nv,q are transition functions of E~ (v = 1,2), E10E2

has transition functions giJ) ®g~]) (Kronecker product).

if L1, L2 are line bundles with transition transition functions

IlVealso remark that if Jr : L -+ X is a line bundle, and if so, 81 are two mewmorphic sections of L (with 80 iE 0), then there is a meromorphic function f on X with 81 = .f.so.

U

g)J) . g~T

functions gg),

(multiplication

In particular,

g~]) respectively, L1 0 L2 has

in C* = C - {O}).

vVenow define linear equivalence of divisors. Two divisors D1, D2 are said to be linearly equivalent if there exists a merom orphic function f on X, f 't 0, such that

If D1,D2 are divisors, we see directly from the construction that L(D1) 0 L(D2) is isomorphic to L(D1 + D2). Moreover, L( -D) is isomorphic to L(D)* for any divisor D. If L is a line bundle and if it has a section s with s(x) =1= 0 Vx E X, then L is trivial; in fact, the map X x -+ L given by (,e, A) 1-7 AS{cT) is an isomorphism. It follows that if L is a line bundle on X, then L 0 L * is trivial: if v E L,", 'V =1= 0, there is a unique linear form eon L" ,such that = 1 [and the form corresponding to Cl1, C E C, is ~e]. Thus x 1-7 v e is a nowhere zero section of L @ L'·.

Lemma. Two divisors D1, D2 cite linearly equivalent L(D1) and L(D2) are (holomoTphically) isomorphic.

If L is a holomOl'phic line bundle on X and s is a meromorphic section of L, s 't 0, then L ~ L(D), where D = div(.s). In fact, if 3-D is the standard section of L(-D). then s .o--D is a nowhere vanishing holomorphic section of L @ [Alternately, the isomorphism 1-7 A.o D (x) defined outside the support of D extends to an isomorphism of L onto L(D).]

'if and only if the line bunclles

Proof. Suppose D1 and D2 are linearly equivalent, and let f be meromorphic on X with (.n = D1 - D2. Let SD 8D2 be the standard sections of L(D1), L(D2) respectively. "

e

1, ... , N. Then 11.s111/ ::; 2-k Ilsf;

in fact, # is holomorphic

SUPOUi1#1 = sUPaui l.sil ::; II.sf·

2-k Iisf. Theorem 1. Let X be a compact Riemann surface and 7T : E -+ X a holomorphic vector bundle on X. Then the space HO(X, E) of global holomorphic sections of E on X : HO(X, E) = {s : X -+ E I 7T 0 S = identity on X, s holomorphic} is a finite dimensional vector space. Proof. We choose finitely many coordinate following properties: (a)

Zi :

Ui

-+

6. = {z E C Ilzl

(b) If V; = z;l{z E C Ilzl

<

neighbourhoods

{Ui,

Zi};=I,

..N

with the

Hence, if.s E HO(X,E)

=

Let s E HO(X, E); then s is represented Si : vVi -+ such that

cn

Thus, if 2k

> e, it

follows that s

==

O.

If Oai is the ring of germs of holomorphic functions at ai and mf, the ideal in Oai k generated by zf, then Oa)mr is a C-vector space of dimension k. Moreover, if 2 > 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 <81 8 : HO(W, E) 0o(w) COO(W) --+ HO(W, E) 0o(w) 'AO,l(W) (note that 108 is well-defined since 8 is O(W)-linear). It follows easily that if V is any open set in X, there is a unique O(V)-linear map 8E,v : CE'(V) -:! A~I(V) such that, if U c V and EIU is trivial, then, for any S E CE'("V), 8E,v(s)lu = oE,u(sIU). We shall denote this map simply by 8E : E --+ A~I or just 8. We can now state the main result of this section.

We shall assume that the reader is familiar with the language of exterior algebra and di£, ferential forms on a manifold. We shall briefly review the aspects which are of relevance to our discussion. Let X be a Riemann surface and T~ the complex tangent bundle of X (i.e. the bundle of complex valued tangent vectors). Its dual T.i
= space of I-forms on W of type

If f E COO(W), the exterior derivative elf can be written uniquely as df = (}f + 8 f with of E A1,O(W), 8f E AO,I(W). In local coordinates, we have of = %f dz, 8f = %:d2. If 0: E A1,O(W), we set 00: = 0, Bo: = da (exterior derivative) and similarly, if (3 E AO,I(lV) oj] = dl] and 8,13 = 0. In local coordinates, if 0: = adz, ;3 = bd2, we have 80: = II dz = II d2 and oj] = ~.~dz II dz. Let 11' : E --+ X be a holomorphic vector bundle on the Riemann surface X. If H' c X is open, we set CE(lV) = space of Coo sections of E over TV the .space of COCo maps s : W --+ E with 11' 0 s = identity on W). If EIW is trivial (i.e. if there is a holomorphic trivialisation h : 1I'-1(W) -> 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
If 'Pi E C'E(U;j is another solution of 'Pi - 'Pj = Sij, then 'Pi - 'Pi = 'Pj - 'Pj on Ui n Uj, so defines 'P E C'E(X); clearly if w,w' E A~I(X) are defined by wlUi (resp. w'!Ui) = a'Pi (resp. a'P;), we have a'P = w' - w.

Since aC'E(X) has finite codimension in A~I(X) [by the finiteness of dim HI(X, E) and the Dolbeault isomorphism], this is a consequence of the following standard theorem in functional analysis.

We check that D is injective: given {Sij} E Zl(U,E) and 'Pi E CFf'(Ui) with'Pi-'Pj on Ui n Uj, and if'P E CFf'(X) is such that a'P = a'Pi on Ui, then, we have

Let V, W be Frechet spaces and 'u : V --+ liV a continuous linear map. .If dimc(Wlu(V)) < 00, then u(V) is closed in W. (Note. It is not true that a finite codimensional subspace of a Frechet space is closed.)

= 8ij

Finally, D is surjective. In fact, given w E A~I(X), we can find an open covering U = {Ui}iEI and 'Pi E C'E(U;) with a'Pi = wlUi [because, if 0 c rc is open and if f E [,,00(0) then, by case 1 of proposition 3 in §5, if J( c 0 is compact, then we can find u E COO(O) with ~~ = f on K]. Moreover, if Sij = 'Pi - 'PjlUi n Uj, then aSij = 0 on Ui n Uj, so that 8ij E HO (Ui n Uj, E). Clearly {Sij} E Zl (U, E). By construction, D({Sij}) = class ofw. Note. We could simply have used the short exact sequence of sheaves 0 --+ JE --+ JEoo --+ A~l --+ 0 together with the lemma (HI(X,JEOO) = 0); the exact cohomology sequence then implies the theorem. However, the actual construction of the map D given above is often useful. The Dolbeault isomorphism and the finiteness theorem of §7 have an important consequence. First, we define topologies on C'E(X) and A~I(X), Let a E X, and let U be a neighbourhoocl of a carrying a coordinate .:;(z : U --+ z(U) c being an isomorphism) and such that EIU is trivial. If 'P E A~l(X), then 'PIU can be identified with an expression.

ce

Let X be a compact Riemann surface. We introduce a topology of complete metric space (even a Fn',chet space) on A~I(X) by the following requirement: A sequence 'P(v)

E

V

sequence ('Pl

)

converges in

iJxf?;ym U].

,f.

+ In

, •.•

A~l (X) converges if and only if for any U as above, the corresponding ,'P~))

TIn copies coe(U)

=

k),

[i.e., for any differentiation

the sequence {Dk'P}V)}

k

of order k (D

=

converges uniformly on compact subsets of

\Ve introduce, in the same way, a topology on CFf'(X). Schwartz) topologies on A~l , CFf'.

Dk

These are called the Coo (or

We outline t.he proof. First, since ker(u) is closed, 1/1 ker(u) is again Frechet, so we may assume that 1t is injective. Let VVo c iiV be a finite dimensional space such that the projection Wo --+ WI 1t( V) is an algebraic isomorphism. Now, Wo is closed in W. [If WI,.'" wk is a basis of Wo, then the map k --+ TVo, (Xl, .. " xkl f-7 Xi1Ci is continuous and bijective; it is easily seen that it is a homeomorphism because the image of {I: 12 = I} is a compact, hence closed subset of TVo so that there is a neighbourhood of 0 disjoint from this image. It follows that Wo is complete in the induced topology from W, hence closed.] The map vVo EB V ----+ W, (w, 1') f-7 "W + l((V) is a continuous bijection, hence a homeomorphism (open mapping theorem). Since V is closed in Wo EB 'V, its image is closed in W.

ce

I:

We shall end this section by describing the canonical bundle of a Riemann surface "Y. ~et W C X be open. A holomorphic I-form on W is a I-form w of type (1,0) such that 8w = 0 (i.e. such that clw = 0). If (U, z) is a coordinate system, amI wlW n U = fclz, then, if w is holomorphic, so is f. Meromorphic I-forms are forms defined outside a discrete set in W which, for any coordinate system ([T, .z) with U c Ware of the form frlz with f meromorphic on U. Let n = 11x (= l13<) be the sheaf U f-7 Ox (U) = {space of holornorphic I-forms on U}; it is called the sheaf of holomorphic I-forms. There is a line bundle K = Kx on X such that, for any open set U C X, we have HO(U,Kx) = Ox(U). This line bundle can be described intrinsically by analyzing the complex cotangent bundle T~~,'cand decomposing complex covectors into those of type (1,0) and those of type (0,1). Here we shall simply give one system of transition functions defining it. Let {(Ui, be a covering of X by coordinate neighbourhoods. Clearly, there are functions g;j E O(U; n Uj), holomorpbic andnovvhere zero on U; n Uj such that

Let Ex be the line bundle defined by the transition function {gij}· If vI' c X is open and .3 E HO(Vr, J(x), then.3 is given by a family {f;}, Ii E O(VV Il Ui) such that

Consider the hololllorphic I-forms Wi = fi dZi on VV n Ui; we have, on j;T" n Ui n Wi = Ii dZi = gijfj clzi = fJ clzj = Wj, so that they define wE fh(U). The map 8 H W is clearly an lsoInOrphiSll1.

The line bundle K = Kx is called the canonical (line) bundle of X. Under the correspondence described above, meromorphic sections of Kx correspond to merom orphic I-forms. We may thus speak of the divisor of a meromorphic I-form w; such divisors are called canonical divisors. Note also that the above construction, applied to Coo sections of B.-x over an open set We X gives an isomorphism of C~'«(W) with A1,OCW).

What is usually called Weyl's lemma is a regularity the operator

a.

theorem for the Laplacian

rather

Let X be a compact Riemann surface and Jr : E --+ X a holomorphic vector bundle on X. Let Jr* : E* --+ X be the dual vector bundle and let K x be the canonical line bundle on X. We define a bilinear form ( , ) HO(X, E* ® Ex) x A~\X) --+ as follows.

e

Let s E HO(X, E* ® Kx), rp E A~l(X), and let (U, z) be a coordinate on X. Let w be a holomorphic I-form on U without zeros (i.e. if w = at every point of U). Then s, '.p can be written uniquely

where A

E

/\(x)(oo(x»),

f

neighbourhood dz, then f f 0

H°(U, E*), a E C'E(U); we define a Coo function (A, a) on U by lA, oo)(x) = x E U IA(x) is linear form on Ex = Jr-l(X) and oo(x) E Ex], and the 2-form

is independent of the choice of w on U [if 1;./ = fIN' with .f holomorphic and nonzero on U, then (JA) <SJw' = slU, and rplU = (700) ,Li', so that (J/\' 7(0)1;/ !\ w' =

1./12 (A, oo)f fw /\ w

=

(A, oo)w !\ w]. This defines a 2-for111(s, rp) on X, and we set (S,lp)

=

r (s,',o).

Jx

Note that if VV c X is open and supp( ',0) HO(W, E* <SJEx).

c

I'll, then

(.s, ',0) is defined for all 3

E

Let (U. z) be a coordinate neighbourhood such that there is a trivialisation hu Jr-I -7 U X and let h[! Jr*-l (U) --+ U X be the corresponding trivialisation of the dual (= inverse of the transpose). Then, if v E Ex and A E E;, E [7, :.uld hu(v) = (:r,vI, .. ,V,J. AI, .. ·,A,,), we have A(V) = r:,'l'),kVk. If s E HO(X. E* Kx) and '+' E and if supp(rp) C U, we can write

en

en

sIU=/\@dz rp=o:

dz

with

h'U(/\(x»)

with

hU(O:(;l!))=(X,

= (;l),Al(X),

...

An(X»)

We make the following remark. If f E CE'(X), and s E HO(X, E* ® Kx), then (s, aj) = O. To see this, f can be written (partition of unity) as :L~=liv, where each fv has its support in a coordinate neighbourhood (U zv) on which EIUv is trivial. We therefore suppose that p = 1 that supp(j) c U, with (U, z) as above. If

This is equivalent to showing that there exist holomorphic functions Al,· ., ,An on U such that

V,

hu(J(x))

(x,h(x), ... , fn(x)), then af = a®d2 with hu(a(z)) hence, if hu(s(z)) = (z, Al(Z), ... , An(z)) on U, we have =

WEYL'S LEMMA: The regularity

theorem

for

=

(z,

W,···,£If) VCoo functions al, ... ,an on U with supp(aj) compact in U, (i.e. Vaj E Cgo(U)). If we set for aj E Cgo(U), G(al,"" an) = F(a ® d2), hu (a(z)) = (z, al(z), ... , an(z)) the condition FlaCE'(X) = 0 implies that G(8fz', ... , iJ.ff) = 0 if fJj E Cgo(U). It is sufficient to show that Vk, 1 :So k :So n, there exists Ak E O(U) such that

8.

Let X be a compact Riemann surface, let 1r : E ---t X be a holomorphic vector bundle on X. We equip A~l(X) with the Coo topology (described in §8), viz: the topology of convergence of all derivatives on compact subsets of coordinate neighbourhoods in X.

Thus, Weyl's lemma is a consequence of the following result, usually known as the regularity theorem for

Suppose that F : A~l(X) ---t
Theorem. Let U be open in
= O.

tz'

---t


(1) If a(v) E Cgo(U) is a sequence with supports in a fixed compact set in U and converging in the Coo topology to a E Cgo(U), then T(a(v») ---t T(a). Remarks. 1. If E is the trivial bundle of rank 1, and X is an open set in
E

of Weyl's

HO(W, E*

®

lemma.

We begin with the following remark. If W c X is open, with supp(p) c W, then u = O.

K) and (u, <.p)= Q V<.pE A~l(X)

Hence, it suffices to prove the following: Let (U, z) be a coordinate neighbourhood such that there is a holomorphic trivialisation

hu : 1r-1(U) ---t U x
(2) T(~~)

= 0

if (3

E

Cgo(U).

Then, there exists /\ E O(U) s'uch that T( a)

=

Ju Aa dz

/\ d2 Va E cgo (U).

Proof of the theorem. We may assume that U is a bounded open set in E}. Let <.pE Cgo(
1 a(z) = -.

1

27T1

oc(z) = 8~ (z)

8z

+~ 211~

oc(z

!C

Ier a(z

Izi <

~E,

<.p(z) = 0 for

<.p(w) + W)-, -dw

1\

Izi

2':

E,

0 :So

E

~Q

> 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
lim-.

b ~O

1

J

8r.x
21fz IC-tl.,

-

J

J (

IC-tl.,

the first term =


J

IC-tl.,

8
o:(z+'.;;,l
Iwl=b

8ii8Z

1 r.x(z) - -2'

=

1fZ

ex(z)

8ii 8z

=

1 + -2'

1fZ

1

ex(z

=T

(8ii) 8z

= -2

1

7rZ

13 0= T (8 8-) Z

=

1 -2' 1fZ

1 U

f, E H1(X, E) .

.

ex(w)p(w - z)dw 1\ dw.

1

1 u ex(w)),(w)dw 1\ diu, + 27ri

1

8/3

-8- ),(w)dw 1\ dw

U

Kx),

+ w)p(w)dw 1\ dw ,

ex(w)),(w)d'U! 1\ dw

U

@

IC

in the Coo topology when h -t 0; we can iterate this argument). Thus, we have shown that there is ), E Coo (Ue) such that

T(ex)

s E HO(X, E*

r.x(z)
21fi

where, for wE U,,, ),(w) = T(z 1---+ p(w~z)). Moreover,)' E Coo(U£) (since, e.g. if h # 0 is real and -t 0, A(W+h2-A(w) = T(P(W+h-zj;-p(w-z)) and the term under T converges

1

Consider the bilinear form ( , ) : HO(X, E* @ Kx) X A~l(X) -tC defined above. We have seen that (s, aj) = 0 if f E C£,(X), s E HO(X, E*@Kx), hence this form induces a bilinearform HO(X,E*@Kx) x [A~l / aC£,(X)] -t C, which we denote again by (, ). If we denote by D the Dolbeault isomorphism D : H1(X, E) -t A~l(X) / aC£,(X), we define ( , )E by (s, f,) E = (s, D(f,)),

Now, if we approximate this integral by Riemann sums, these sums converge to the integral in the Coo topology on U (since p E COO) and have support in a fixed compact set in U since r.x E CO'(U£) and pew - z) = 0 if Iw - zl ::::c. Hence, by the continuity hypothesis (1), we have

T(ex)

Let X be a compact Riemann surface, 1f : E -t X a holomorphic vector bundle on X. We define a bilinear form (, )E : HO(X, E* @Kx) X H1(X, E) -t C as follows.

W

since f3 E CO'(U£) is arbitrary, it follows that theorem is proved.

=

L7rZ

g~ = 0 on

fI. : A~l(X) / aC£,(X) -t C be any C-linear map. We have seen that ac'jf(X) is closed in A~l(X) (for the Coo topology) and is of finite codimension. Since any linear form on a finite dimensional (Hausdorff) topological vector space is continuous, the C-linear map F : A~l(X) -t C given by F(
A~\X) / 8C'jf(X), is continuous for the Coo topology, and is zero on ac'jf(X). Weyl's lemma, there is s E HO(X, E* @ J(x) such that

By

Since D is an isomorphism this proves that 8.E is surjective. To prove that 8.E is injective, given s E HO(X, E* @ J{x), we must show that if (s,
.

1 U

Proof. Let

Moreover, T ( ~~) = O.

Vex E CO'(U,)

1 -;:;-:

SERRE'S DUALITY THEOREM. The map 8.E : HO(X, E* @J(x) -t H1(X, E)* is an isomorphism for any holomorphic vector bundle on the compact Riemann surface X.

f3-8), _ dw 1\ dw , 8'U'

Ue, i.e. that),

E

O(U£). The

In view of the fact that HO(X, E) and H1(X, E) are finite dimensional, the Serre duality theorem can be stated as follows:

is non-degenerate (or perfect). If F is a holomorphic vector bundle, and we take E = F* ® Kx, then E* ® Kx F ® K'X ® Kx = F (since, for any line bundle L, L* ® L is canonically trivial). Hence HO(X, F) is isomorphic to the dual of H1(X, F* ® Kx) for any vector bundle F on X. We shall end this section by singling out the special case of these results when E is the trivial bundle of rank 1 [so that A~l = AO,l(X) and H1(X, E) = H1(X, 0)].

Throughout this section, X will be a compact Riemann surface. We begin with some definitions.

Proposition. Let'P be a form of type (0,1) on the compact Riemann surface X. Then, 3f E COO(X) such that af = 'P if and only if, for every holomorphic 1-form w on X, we have

Let D = 2:~=1niPi be a divisor on X. The integer d = D [and written degD].

2:~=1ni

is called the degree of

to.

Let w be a meromorphic I-form, w If a E X, we denote by resa(w) (the residue at a of w) the following: If (U, z) is a local coordinate with zeal = 0, and w = f dz, then resa(w) = residue of f at a = the coefficient of ~ in the Laurent expansion 2:"::N cyZY of .f. It is independent of the coordinate system. In fact, if r is a piecewise differentiable 1 curve in U - {a} whose winding number. (= index) at a is +1, then resa(w) = -2XI . W. 7

J

Proof. Let a1, ... , am be the poles of w. Choose coordinate neighbourhoods (Uj, Zj) about aj (zj(aj) = 0) and let /::,.j = {x E U IZj(x)1 < E} (E > small). Let U = X - Uj:1 /::,.j. By Stokes' theorem, we have

°

I

ju

dw = -

2:" J J

w = -27':i

E!t:.;

2:.

resa; (w) .

J

But, a holomorphic I-form on a Riemann surface is closed: if w = dz

+ %f

dz i\ dz =

Corollary divisor (f)

°.

1. Let f be a meromorphic function, of f is zero.

In fact, deg(l) = 2:aEX resa(w) where w =

f t

f

die, dw = ~~ dz i\

0, on X. Then the degree of the

1df.

Corollary 2. If D1 and D2 are divisors on X and if D1 is linearly equivalent to D2, then deg(D1) = deg(D2). In fact, if D1

-

D2 = (I), we have cleg(D1)

-

deg(D2) =deg(f).

Let D be a divisor on X; we introduced the sheaf OD: U 1-+ OD(U) = {f meTomorphic on U I (f) 2: -D on U}. OD is isomorphic to the sheaf U 1-+ HO(U, L(D)) of holomorphic sections of the line bundle defined by D. An element f in the stalk OD,a is given by a (collvergent) Laurent series

f

=

L: n2'-D(a)

n

cnz

Let D1,D2 be divisors with D1 :::; D2. Clearly OD,(U) C OD2(U) for all U open in X and we obtain an injective morphism of sheaves 0D, -+ 0D2' [In terms of the line bundles L(D1), L(D2) and the isomorphism L(D1) @ L(D2 - Dl) c:::L(D2), the map is given by f E HO(U,L(D1)), f 1-7 f @SD2-Dll where SD2-D, is the standard section of L(D2 -Dl) with (SD2-D,) = D2 - Dd Let be the sheaf associated to the presheaf U

OD2(U)/OD,(U).

t-+

We have. (sg~)a

8g~ = 0 if

D1(a)

= D2(a)

(in

particular if a is not in supp(D1) Usupp(D2)). If D1 (a) < D2(a), then (sg~t is a finite dimensional vector space, isomorphic to the vector space {"Z-D2(a)$n<-D1(a) cnzn Cn E iC, z an indeterminate}.

I

Lemma 2. If Dl :::;D2 are divisors (1)

dimHO(X,Sgn

(2)

HI (X, sgn

(HI (X, sg~) = 0 by Lemma 2). If V is the image of HO(X, ODJ in HO(X, sg~), have two exact sequences O--+!f°(X, O--+Ho(X,

S&)/V

ODo)--+Ho(X, --+H1(X,

we

ODJ--+V--+O ODo)--+HI(X,

ODJ--+O,

so that hOeD,,) - hO(Do) = dim V and degD" - degDo - dim V = h1(Do) Adding these two equations gives

- hl(Dv).

on X, we have i.e. XeD,,) - degD" = X(Do) - degDo for v = 1,2. The lemma follows.

=degD2-degDl,

We shall denote by 9 the integer dime HI (X, 0); 9 is called the genus of the Riemann surface X.

= O.

sg~

Proof. (1) is a sheaf supported at finitely many points (E supp( D1) U sUPP(D2)). We therefore have dimHO(X,Sgn

= dim

II(OD2,a

/OD"a)

aEX

=

L: (D

2 ( a)

- D1 (a ))

=

deg D2

-

deg D1 '

aEX

(2) Given any open covering U, there is a refinement V = {V,,} such that, if 0: of (3, we have V" n V13 n (supp(D1) U supp(D2)) = 0. Clearly then ZI(V, = 0, and (2) follows.

Proof. By Lemma 3, we have XeD) - degD = x(O) - degO (0 is the divisor 0). Now, X( 0) = dim HO(X, 0) - dim HI (X, 0) = 1 - 9 (since the only holomorphic functions on all of X are the constants by the maximum principle).

Some notation. If D is a divisor, we set hieD) (i = 0,1), and XeD) = hOeD) - hi (D).

Let D be a divisor on X. We shall denote by DD the following sheaf on X: DD(U) = { meromorphic I-form w on U I (w) :::: -D}, U open in X. DD is isomorphic to the sheaf of holomorphic sections of J(x @ L(D).

sgn

= dime Hi(X,

OD) = dime Hi(.X,

L(D))

By the Serre duality theorem, we have dimeHl(X,L(D)) L(D)*) = dime HO(X, Kx @L(-D)) = dimeHO(X,D_D)' the Riemann-Roch theorem as follows.

Proof. Let Do be a, divisor with Do :::;Dv, 1/ = 1,2. vVe have a short exact sequence of sheaves --+ ~PD --+ 0 Dv ~. --+ 0

°

sg~

and hence a cohomology sequence 0--+ --+

HO(X, ODD) --+ HO(X, ODJ --+ !f°(X, H1(X,

ODD) -~. !fl(X,

ODJ

--+

0

sg~)

--+

RIEMANN-ROCH THEOREM, If D is a divisor we have dime HO(X, OD) - dimHo(X,lLD)

= dimeHO(X,Kx @ Thus, we may formulate

on the compact = degD

Riemann

+ 1-

s~lrface X,

g.

Thus: the number of linearly independent meromorphic junct'ions f on X to-ith (1) ::::- D is equal to deg D + 1 - g+ the nwnber of linearly independent memmorphic l-forms w on X w-ith (w) :::: D, We have seen (Theorem 3 in §7) that any holomorphic line bundle L on ;.0. compact Riemann surface X is isomorphic to L(D) for some divisor D on X; moreover, two

such divisors are linearly equivalent. We therefore define the degree deg(L) of the holomorphic line bundle L to be deg(D), where D is such that L e:=L(D), and write hi(L) = hi(D). By the Serre duality theorem, hO(Kx) definition of the genus. Thus: there exist exactly 9 linearly By the Riemann-Roch

=

h1(K'X ® Kx)

independent

=

holomorphic

h1(0)

=

i-forms

W. If a E X, and i is such that a E Ui, set resa(w) because Wi - Wj is holomorphic on Ui n Uj].

We can also (Dolbeault isomorphism) find C= forms of type (1,0)

on X.

on Ui such that

and the 2-form 'P on X given by 'PIUi = aai = dai represents the cohomology class D({Wij}) in A~~(X) under the Dolbeault isomorphism. We have Lemma. Jx D( {Wij})

is 2g - 2. Equivalently, of W is 2g - 2.

if W =I- 0 is

One further remark: if L is a holomorphic line bundle and hO(L) > 0, then deg(L) ~ O. Further, if hO(L) > 0 and deg(L) = 0, then L is trivial. In fact, if 05 E HO(X, L), s =!=- 0, then L e:=L(D) where D = div(s) ~ 0, so that degL = degD ~ O. If degL = 0, then . the divisor of s equals 0, i.e. s has no zeros and L is trivial.

= 21TiLaEX

Let D.k,e be a small disc of radius

r

} X

EO

.

around ak (k = 1, ... , r). Then

J

if = 6..-;-0 lim X

>

resa(w)

Proof. Let {3 = Wi - ai on Ui; {3 is a C= form on X - {a1, ... , an}, where S = {a1, ... , ar} is the set of poles of w (S n Ui = set of poles of Wi)' Since dwi = 0 on Ui - S, we have 'P = D({Wij}) = -d{3 on X - S.

This remark and the Serre duality theorem give us the following. VANISHING THEOREM. Let D be a divisor on X and let deg(D) H1(X, OD) = O. If degD > 0, then H1(X, DD) = O.

ai

9 (the genus) by

theorem

thus: The degree of the canonical line bundle Kx any meromorphic I-form, the degree of the divisor

[this is independent of i

= resa(wi)

-Uk

J

d{3 = - €--+O lim '" L.-t

Cl.k,e

k

{3.

8Cl.k,e

2g - 2. Then Now, if

EO>

0 is small, and i is such that ak E Ui, we have

J

Wi = 21Tiresak(wi)

=

8Cl.k,e

Proof. h1(D) = hO(K - D) (where K is any canonical divisor); since deg(K - D) = 2g - 2 - deg(D) < 0, hO(K - D) = O. Also, deg(DD) = 2g - 2 + deg(D).

21Tiresak (w) and lim£~o J ai = 0 (since ail is smooth). Hence Jx 'P = 21Ti Lk resak (w).

We shall denote by M the sheaf of meromorphic functions on X, i.e. the sheaf U >--+ M(U) = {space offunctions meromorphic on U}; and by Dm the sheaf of meromorphic I-forms on X.

Thus, Lk resak (w) depends only on the cohomology class ~ of {Wij} in H1(X, not on its representation as a coboundary of Dm); we denote this by Res(~).

8Cl.k,e

D) (and

Since HO(X, 0) = HO(X, K'X ® Kx) = iC, the pairing in the Serre duality theorem for E = Kx is given by (A, {Wij}) >--+ A' D({Wij}) Thus we have

J

Proof. Let U = {Ui}iEl be a finite open covering of X, and let {l~}be open sets with 11; CUi and UV; = X. Let {Iij} E Zl(U,M). We can clearly choose a divisor D > 0 with deg(D) > 2g - 2 such that (Iij) ~ -Don lo:; n lj (because 1/; n fj is compact in Ui n Uj. By the vanishing theorem, there are meromorphic functions fi on Vi, (f;) ~ - D on V;, with f; - fj = on V; n Vj. In particular, the image of {Iij} in Zl ({V;}, M) lies in B1 ({V;}, M). The proof for Dm is similar. We can now give another interpretation H1(X, OD) ---+ C.

of the duality pairing HO(X, D-D)

X

Let {Wij} E Zl(U, D). Because of the corollary above, we can find meromorphic I-forms Wi on Ui such that Wi - Wj = Wij on Ui n Uj. Denote the cochain {w;} E CO(U, Dm) by

Theorem. (Residue version of Serre duality). Let D be a divisor on the compact Riemann surface X. We have a natural pairing ( , )D : HO(X,D_D) x H1(X,OD)---+ HI (X, D) defined by (w, {Iij }) H {fijw} [Here, W is a merom01phic i-form with (w) ~ D and fij is meromorphic on Ui rl Uj with (Iij) 2: - D, so that is a holomorphic i-form on Ui n UjJ. The duality pairing ( , )(D) in the Serre duality ular, the bilinear form

theorem

equals 21TiRes( . )D: in partic-

is non-degenerate. As a consequence, we have the following analogue of Mittag-Leffler's pact Riemann surfaces.

theorem for com-

HO(X,OD)-+HO(X,CD)-+HI(X,O)-+HI(X,OD)

Theorem. Let {U;}iEf be an open covering of the compact Riemann surface X, and let Ji be meromorphic on Ui. Suppose that fi - fj is holomorphic on Ui n Uj. There exists a meromorphic function f on X with f only if, for any holomorphic i-form W on X, we have

Ji

holomorphic

stalk CD,a can be identified with finite sums L~l~)~ dz (where z is a local coordinate ata with z(a) = 0). The exact cohomology sequence is

on U/Vi if and

'since dimHI(X,O) = 1 and HI(X,OD) ~ HO(X,O_D)* = 0, it follows that image HO(X,OD) has codimension 1 in HO(X,CD) ~ cdeg(D). But, since if W is a global 1 form, we have resa(w) = 0, the image lies in the set

La

such that LaEsuPp(D) resa(wa) equal. where Wo E CO(U, Om) is the O-cochain (Jiw). [The condition is that if lia E X we choose ita) with a E Ui, then LaEX resa(Ji(a)W) =

0.]

The corresponding theorem for meromorphic forms (rather than functions) is the following. MITTAG-LEFFLER'S

=

O. Since both spaces have codimension 1, they are

The vanishing theorem HI(X,OD) = 0 = HO(X,O_D) if degD > 2g - 2 and the Riemann~Roch theorem show that hO(D) = degD + 1 - 9 is determined by the degree of D if it is large. The integer i(D) = hl(D)

Proof. The existence of f is equivalent to saying that the cohomology class ~ = {Ii - fj Wi n Uj} is = 0 in HI (X, 0); by the above theorem, this is equivalent to the residue condition.

;

= hO(I( - D) is called the index of speciality

of D.

We give some further applications of these results. Propositio'n 2. If D is a divisor on X with degD > 2g - 1, then, liP E X, 3s E HO (X, L(D)) such that s(P) =I O. Equivalently, there is a divisor D' ~ 0, linearly equivalent to D, not containing P in its support.

THEOREM for forms on a compact Riemann surface.

Let U = {U;}iEf be an open covering of X, and let Wi be a meromorphic I-form on Ui such that Wi - Wj is holomorphic on Ui n Uj. Let W = {W;}iEf, and, for a E X, set resa (w) = resa(Wi) where i is such that a E Ui. [This is independent of the i chosen since !..tJi - Wj is holomorphic on Ui n Uj.]

given by f f-+ f <2> Sp is not surjective (since, by the remark above, hO(D ~ P) = deg(D - P) + (1- g) < deg D + (1- g) = hO(D) (since deg(D - P) > 2g - 2). Its image consists precisely of sections of L(D) vanishing at P.

Then, there exists a merom orphic I-form W on X with W - Wi holomorphic on Ui Iii if and only if

Proposition

3. Let L be a holomorphic line b'undle on X with deo9L > 209. Then

(a) if P, Q E X, P =I Q, 3s E HO(X, L) such that 3(P) = 0, 3(Q) =I 0 (b) if P E X, 33 E HO(X,L) Proof. The existence of w is equivalent to saying that the cohomology class of {Wij} in HI(X,O) is zero; since, by Proposition 1, the map Res: HI(X,fJ) ~ C is injective, this is equivalent to saying that a resa (w) = O.

L:

This can also be deduced from the duality theorem as follows (without knowledge of the precise duality pairing). Let D ~ 0, D =I 0 be a non-zero effective divisor on X. Let 3D be the standard section of L(D) with (3D) = D. Consider the exact sequence of sheaves o ~ fJ .!.Eo, 0D ~ !CD~ 0. The sheaf !CD is zero outside supp(D); if a E supp(D), the

such that ordp(3)

+ 1.

Proof. We consider the bundle L <2> L( -P); since any line bundle is ~ L(D) for some divisor, Prop. 2 implies that 33' E HO (X, L <2> L( - P)) with s'( Q) =I O. Let s = 3' <2> Sp, where Sp is the standard section of L(P). Then, if P =I Q, we have s(Q) # 0, s(P) = 0. If P = Q, we have ordp(sp)

= 1.

The imbedding theorem. Let L be a holomorphic line bundle on X with deg L > 2g, and let IV = hO(L) - 1 = degD - g. We define a holomorphic map 'PL : X -> jp'N as follows.

Let So, ... ,SN be a basis of HO(X,L); if a E X choose a neighbourhood U of a and (l E HO(U, L) with (l(x) =1= 0 "Ix E U. We set 'PL(X) = point in projective space IP'N with homogeneous coordinates Note that ~ is a holomorphic function

e;gi : .., : s::r,~x,}).

on U; the point in IP'N is independent of (l for if (J' is another such section and (J' = h(J where h is a holomorphicfunction on U, hex) =1= OVx, then ~ = h'!j;; 'PL is, to start with, defined only outside the common zeros of So, .. · ,SN, but, by Prop. 2, these sections cannot have common zeros. Note. If L is a holomorphic line bundle on X and HO(X,L) =1= 0, it defines a holomorphic map 'PL: X ---t IP'N (N = dimHO(X,L) -1) on all of X. If So, ... ,SN is a basis as above and A = {x E X Sj(x) = 0 Vj}, the map is defined as above on X-A. If a E A, and (U,z) is a small coordinate neighbourhood of a with zeal = 0, 'PL U - {a} is the point in jp'N with homogeneous coordinates (fo : ... : f N) where fo, ... , f N are holomorphic functions on U and are not 0 outside a. We can write Ii = zk gj where k = minj orda(fj). 'PL on U is then given by the point in IP'N with homogeneous coordinates (go: ... : gN).

I

I

This construction only works because dim X = 1; in higher dimension, points of indeterminacy of 'PL corresponding to the so-called base points of L, where all S E HO(X, L) vanish, cannot be avoided in generaL We have: The Imbedding in jp'N.

Theorem.

If deg L

>

2g, then 'PL is an imbedding

of X

Proof. (i) 'PL is injective. Let P, Q EX, P =1= Q. Choose S E HO(X, L) with s(P) = 0, s(Q) =1= 0; if S = L~ CvSv, then 'PL(P) lies on the hyperplane LCvzv = 0, 'PL(Q) lies outside this hyperplane. (ii) The tangent map of 'PL is injective. Given P E X, choose S E HO(X,L) such that ordp(s) = 1, and let k, 0::; k::; N, be such that Sh(P) =1= 0, and let Co,.·· ,CN E C be such that S = L Cv8v' Then

and the functions fv = ~, v

=1=

k, give the inhomogeneous coordinates of 'PL(X)

x near P: 'PLCx) = (Jo(x), ... ,l,fk+l(x), dfv(P) =1= 0 for at least one v =1= k.

.. ,fN(x)).

Since ordp(1;)

=

for

1, we have

A well known theorem of Chow implies that 'PL maps X onto the set of common zeros of finitely many homogeneous polynomials in the homogeneous coordinates of jp'N. Thus X is analytically isomorphic to a smooth algebraic curve in jp'N. Moreover, an algebraic variety in jp'N (irreducible and set of common zeros of finitely many homogeneous polynomials) if it is connected, of dimension 1 and a submanifold

of jp'N, is obviously a compact Riemann surface. We shall therefore not distinguish between compact Riemann surfaces and (connected) smooth projective curves. A holomorphic line bundle L on X is called ample if, for some integer m > 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'<1'1' z l:: U. This is l
= det(li1'\z))

fI, ... ,In'

E iC, Ck =1= 0, then the column

IiI')

(0 :S v

< n)

with k

=1=

e

IiI')

[viz -

v < n) 2:=k'l'£ ~ 1;V) (0 :S

is =

so that the determinant is O.

To prove the converse, we start with the following remark. Let 'P let gk

= 'Plk,

k

=

1, ... , n. Then gi") = 'Pfi1')

det(gi,")) = det('Pfi"))

W,Z(

= 'Pndet(lk"))

))

+ 2:=1'<1'

A~/k/l)

[the matrix (gi"))

+ '" L.J' \1'I'gk(1') ( W ( z ))

[,\~

E

=

O(U),

'P ¢ 0 and

(~)'P(v-I')]

so that v is obtained from ('Pfi )) by

... + 9 =

~g(g

... , In) =

(~,,::)NW(gl,'

.. , gn) where N =

+ 1).

Let (Ui, be an open covering of X by coordinate neighbourhoods. The transition functions gij = ~:" define the canonical bundle Kx of X, and the assignment i f-+ Wi = W(!l,i, ... , Ig,;) (Wk = Ik,idzi on Ui) satisfies Wi = g[JWj. Thus, the Wi define a holomorphic section W of K~N, N = ~g(g

+ 1).

Theorem.

section

that

Proof.

gk

with the A~ independent of k. Thus W(fI,

'!lfe now make a few remarks on the Wronskian. E O(U).

1'+1 (v) (

1'<1'

I (IiI') (0)) O:;v
and fI, ... ,ln

(dW) dz

~2Z~ gk(W(Z))

k = 1, ... ,g

= 0,

1/=0

has a solution (co, ... , Cg-l)

+

= (~:)2g~(w(z))

There

the zeros

is a non-zero

of l'V are exactly

holomorphic the

Weierstrass

Thus we have the

points

W

of K~N,

N = ~g(g

+ 1), s'uch

of X.

Note that deg(div(W)) = NdegKx = (g - l)g(g + 1), so that there are at most (g _ l)g(g + 1) Weierstrass points; if 9 > 1, IY must have some zeros, and there must exist Weierstrass points. We also have the following: WEIERSTRASS GAP THEOREM. Let X be a compact Riemann surface of genus 9 > 0, and let P EX. Then, there exist exactly 9 integers 1 = nl < n2 < ... < ng :S 2g - 1 such that no merom orphic function on X, holornorphic on X - P, has a pole of order TIk, k = 1, .. , g, at P.

to the vth rowl. Thus, we have

Thus, except for finitely many points P of X, there exists a meromorphic function on X, holomorphic on X - P, with a pole of order m at P if and only if m 2: 9 + 1.

We now prove that if W(fI, .. , .In) == 0, then h,.·., .In are linearly dependent by induction on n. The result is trivial for n = 1 (WUd = fI)·

Proof. Let Dk = k . P, k = 0, 1,2, .... There is .I meromorphic on X, holomorphic on X - P, with a pole of order k at P if and only if c?'f E qX), (.f) 2:--Dh, (f) l -Dk_·1; if -sp is the stand2.rd section of L(P) with (sp) = P, this is equiv2Jent to saying that

adding multiples of rows fk/l) (1 :S k:S n) with ~, < W(gl, ... , gn) = cp1'lW(fI, .. ·, .In).'

If

fI ¢ 0 and

l/

V = U - {set of zeros of h}, it is enough to show that 1,

dependent (on V) over iC. Now,W(l,

t" ... , t,)

= flnW(!l,

*

t", ..,t, are

... ,.in) = O. If gk =

is not surjective. First, from the sequence O~ODk_l sequence

~

~O,

ODk ~ODk/ODk_l

we obtain the exact

Now, dimHO(X, ODk/ODk_,) = 1 (the quotient is the sheafwith C at P and 0 elsewhere). Hence, 3f with (j) 2': -Dk but (j) -Dk-l if and only if hO(Dk) > hO(Dk_l), and in this case, hO(Dk) = 1 + hO(Dk-l).

t

Now, by the Riemann-Roch theorem,

m

hO(Dm)

-

hO(Do)

=

I)hO(Dk)

- hO(Dk-l))

=

m - hl(Do)

+ hl(Dm)

Let X be a compact Riemann surface. We denote by C(X) functions on X.

the field of meromorphic

We call X a hyperelliptic Riemann surface, or a hyperelliptic curve if it can be realised as a 2-sheeted (ramified) covering Of]p'l, i.e. if there is f E qX) with either two simple poles or a single pole of order 2. We call such an f a function of degree 2. (Note: if there is f E C(X) with one simple pole, then f : X --+ ]p'l is an isomorphism since 00 is not a critical value and f : X -7 1P'1 is one-sheeted.) If X has genus 9 > 0, X is hyperelliptic if and only if there is an effective divisor D with deg(D) = 2 and hO(D) 2': 2. In fact, if X is hyperelliptic and f is of degree 2, we can take for D the divisor of poles of f [hO(D) 2': 2 since (1) 2': -D, (1) 2': -D]. Conversely, if hO(D) 2': 2, there is a non-constant f with (j) 2': - D.

k=l

and hl(Dm) we have

= 0 for

m 2': 2g - 1, while hO(Do)

=

hO(Dm) = m - 9

1, hl(Do)

= g.

Thus, for m 2': 2g - 1,

+ 1.

Moreover, hO(Dm) -1 = 'L,{, (hO(Dk) - hO(Dk-l)) is the number of k :::;m which occur as the order of pole at P of an f E qX) holomorphic on X - P. Thus, the number "gaps" is m - (hO(Dm) - 1) = g. N ate. We could have applied this argument with an arbitrary sequence H, P2, ... of points and Do = 0, Dk = 'L~ Pi for k > O. We find that there is a meromorphic function f with (j) 2': - D k but (j) D k-l for all k except for 9 exceptional values nl, ... , ng (between 1 and 2g - 1) of k. This is sometimes called the Max Noet7~er gap theorem.

t-

If X has genus 1, it is hyperelliptic; in fact, if D is any divisor of degree 2, we have degD > 2g - 2 = 0, and hO(D) = 1 - g + degD = 2. If X has genus 2, it is also always hyperelliptic; in fact, if P is a Weierstrass point, there is a fuaction f E qX), holomorphic on X - P, with a pole of order 2 at P. Let X 'be hyperelliptic and f : X -7 1P'1 a function of degree 2. We can arrange that f-l(oo) consists of two distinct points by composing f with an automorphism of pl. If C is the set of critical points of f, and B = f(C), we have B C 1P'1 - {oo} = C Moreover, orda(j) = 2 if a E C, so that f-l(b) consists of exactly one point if bE B. The map l' : X - C -7 X - C which takes x to the point x' # x with f(x) = f(x') extends to a holomorphic map l' : X --+ X (by setting 1'(0.) = a for a E C) and we have 1'2 = identity. It is called the hyperelliptic involution of X. By the Riemann-Hurwitz

formula, if X has genus g, we have since 1P'1 has genus 0, 2g-2=-4+

L(orda(f)-l), (~EC

so that (since orda(j) = 2 for a E C), the number of branch points of each of these is a Weierstrass point of X).

f

is 2g

+2

(and

Let u be a merolllorphic function on X such that u(x) # 1!(1'(X)) for some x E X - C, Then, for all z near fCr), 'U,(Xl) # U(X2) if (Xl,X2) = f-l There exist rational functions 0.1,0.2 E iC(z) so that (writing z for f(x)) we have

u being holomorphic at both :r and r(x).

ai -

(proof of Theorem 4 in §7). Then ('11. + a1)2 = a2 = plq, with p, q E iC[z]; if we write p . q = P ..Q2 with P, Q E iC[z] and with P having no multiple roots, we have w2 = P(z), where w = q(u + a1)/Q. Moreover, there is an open set of values of z such that W(X1) 1=W(X2) if (X1,X2) = 1-1(z). Let Y be the Riemann surface (compact) of the algebraic function w2 - P(z). We obtain a map 7f : X -> Y by setting 7f(x) = (j(x),w(x)) outside a finite set onX (corresponding to the poles of I, wand the branch points of Y over jp'1, 'liz, the zeros of w); this extends to a holomorphic map X -> Y. Now, for an open set of x EX, we have w(x) 1= W(T(X)); since w(x)2 = W(T(X))2, we have w(x) = -W(T(X)), and, by the principle of analytic continuation, we have w( x) = -w (T( x)) for all x EX. It follows easily that 11" is an analytic isomorphism commuting with the projections I : X-> 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

<

2) Given P, Q E X,

P

1=

Q, 3w E HO(X,O)

with w(P)

= 0, ecJ(Q)

1=

O. If not,

the map HO(X,O_P_Q) ~ HO(X,ll_p) is an isomorphism. We have hO(O_p_Q) = 1 - 9 + (2g - 4) + h1(ll_p_Q) = 9 - :3 + hO(Op+Q)' If hO(Op+Q) > 1, there is a non-constant meromorphic function f with (f) ? - P - Q, so that I is of degree 2 and X is hyperelliptic. Hence hoe OP+Q) = 1, and hO(ll_p_Q) = 9 - 2 < hO(fLp). 3) Given P E X, 3w E HOp::, 0) with ordp(w) = 1. If not, we have w(P) = 0 ==? ordp(w)? 2, i.e. hO(O_p) = hO(fL2P)' As in 2) above, this implies the existence of a non-constant I with (f) ? -2P and X would be hyperelliptic. The theorem follows from these three statements

as in the imbedding theorem in §10.

The image of any compact Riemann surface X under 'PJ<x is called the canonical curve of X. If X is hyperelliptic, the canonical curve is isomorphic to jp'l; (jthervvise, it is isomorphic to X.

with 'P'.(t) =f. 0 on D. for some j and 'Pk == 1. Let K cUbe compact. For simplicity of not~tion, we suppose that k = O. If cozo + ... + CnZn = 0 is a hyperplane H, then H n X is not transversal at some point of K if and only if:L~=l cv'Pv(t) = -co and :L~=l cv'P~(t) = 0 have a common solution. Since 'P~(t) =J. 0 for some //, we may assume . that :L~=l cv'P~(t) =J. 0; the set S = {t E U :L~=l cv'P~(t) = O} is discrete, and we can choose A arbitrarily close to 0 so that :L~ cv'Pv(t) =f. -co - A if t E S; then (co + A)ZO + ... +cnzn = 0 meets X transversally on K; thus the set Wg = {HI X nH is transversal at points of K} is dense. It is clearly open, and the set {H E (jp'n)* meeting X transversally} is a finite intersection of sets W g.

I

We begin with some general remarks. If M is a complex manifold of dimension nand

A c M is a submanifold of dimension n - 1 (codimension 1), A defines a holomorphic line bundle as on a Riemann surface: if {U;} is an open covering of M, Ii E O(Ui) is such that Ui n A = {x E Uilfi(X) = 0, dfi =f. 0 at any point of Ui}, then gij = fi/fj

is holomorphic, nowhere zero on Ui n Uj and form the transition functions for a line bundle L(A). The family {f;} define the standard section SA of L(A) (whose divisor is A). Consider now M = IP'n, with homogeneous coordinates (zo, ... , zn). A hyperplane H (linear subspace of codimension 1) is given by {£(z) = OJ, where £ is a non-zero linear form in Zo,··· , Zn; we shall denote the corresponding line bundle also by H [or OlfDn (1) or 0(1)]; two hyperplanes define isomorphic bundles. If Uv = {(zo : ... : zn) I Zv =f. O},

(31., ... ,~, ... , '""-) form local coordinates on Uv. in fact an isomorphism onto en (th~"hat o;er a te~m means it is omitted). If H i~ defined by {£(z) = O}, the functions fj = e~z) define H n Uj (j = 0, ... , n) and the transition // = 0, ... , n, the functions

¥,.

functions for H on Ui n Uj are given'ty gij = The set of all hyperplanes forms the "dual" projective space (IP'n)*, the coefficients of £ forming homogeneous coordinates for (IP'n)*. Let X c Ipm be a connected complex submanifold of dimension 1 (= smooth imbedded projective algebraic curve). We set deg(X) = deg(OlfDn(I)IX); it is called the degree of the curve X. If SH is the standard section with divisor the hyperplane H, then DIP'" (1)IX = I'lnpP is the divisor of sHIX and deg(X) = ~np. If X n H is transverse at every point, then np = IVP E X n H, and deg(X) is the number of points in X n H. A well-known theorem of Bertini implies that the "general" hyperplane transversally; we shall prove it in the special case we need. Proposition

1.* (Special case of Bertini's theorem).

The set of HE

meets X

(]pm)"such that

In what follows, we shall assume that our curve X C jp'n is non-degenemte in the sense that it is not contained in any hyperplane. One can always consider X imbedded in the smallest dimensional linear subspace jp'k of jp'n containing X.

Proof. Let H be a generic hyperplane and XnH = {Xl"", Xd}, d = deg(X). If d < n, choose points Yl, ... , Yn-d on X. Now, any n points of jp'n lie on a hyperplane. Let H' be a hyperplane containing Xl, ... , Xd, Yl,"" Yn-d. Then, if SH' (standard section of 0(1) with divisor H') is such that SH' IX =J. 0, we would have deg(sH,IX) :2: n > d, contradicting the definition of deg(X). Hence SH' IX == 0 and X is degenerate. GEOMETRIC FORM OF THE RIEMANN-ROCH THEOREM. Let X be non-hyperelliptic and let X C jp'y-l be the canonical imbedding. Let D :2: 0 be an effective divisor on X. Assume that D =f. O. If e is a linear form on jp'g-l, then £IX is just a holomorphic I-form on X and all such I-forms are obtained in this way. Given a hyperplane H in jp'y-l, we shall say that H contains D if the divisor of fiX, (fiX) :2: D, £(z) = 0 being an equation of H. We denote by [D] the linear subspace jp'g-l generated by D, which is, by definition, the intersection of all hyperplanes containing D. If D is of the form ~Pi with the Pi distinct, a hyperplane containing D is just one containing the points Pi and [D] is the subspace spanned by the Pi· The geometric form of the Riemann-Roch

theorem states simply that

H meets X transversally is an open dense set in (IP'n)*. Proof.

Let a E X and U be a small open set given by a biholomorphic map 'P : D. -+ U,

'P = ('po, ... , 1, ... ,'Pn), D. = {t E q ItI < I} and 'Po, ... ,'Pn are holomorphic functions

*

If X is allowed to have singular points, the condition means that H avoids the singularities and is transverse elsewhere. The proposition holds also for such curves; if Ho avoides the singularities of X, then all H near Ho avoid an open set containing the singularities, and the proof applies.

In fact hO(ll_D) is the ma.."imum number of linearly independent I-forms w on X with (w) :2: D, i.e. the maximum number of linearly independent linear forms f'.on jp'g-l with (f'.IX) :2: D, hence 9 _. 1 ~ hOell_D) is the dimension of the intersection of all hyperplanes containing D, i.e. 9 - 1- hOell_D) = dim[D]' and the result follows from the R.iem.anll··Roch theorem.

Remark that any k + 1points (k + 1 ::;n) in lP'nlie on a k-dimensionallinear subspace of IP'n. We say that PI, ... , Pk+l E lP'nare (linearly) independent if they are not contained in a plane of dimension < k, i.e. if their linear span has the maximal possible dimension. CASTELNUOVO'S degenerate algebraic property: if X n II independent (i.e. do

GENERAL POSITION THEOREM. Let X C ll:'n be a noncurve of degree d. Then, the set of hyperplanes with the following = {Xl, ... ,Xd}, then any set of n points Xi,,· .. , Xin are linearly not lie in a plane of dimension n - 2) is dense in (IP'n)*.

In proving this result, we shall assume familiarity with basic algebraic geometry. We begin with the following. Lemma 2. Sllppose that n ?: 3. Let U C (ll:'n)* be the open set of hyperplanes meeting X transversal/yo (The condition that HE (IP'n)* is not transverse to X is algebraic and U is the complement of a proper algebraic set in (ll:'n)*.) There is a proper algebraic subset A of U such that, if H E U - A, then no three pO'ints of X n Hare colin ear (lie on a line). [We suppose X is irreducible, but not necessar-ily smooth.] Proof. We shall prove that the (algebraic) family of lines in ll:'nwhich meet X in three (or more) points has dimension 1; these lines are called trisecants. Since the family of hyperplanes containing a given line has dimension n - 2, this will show that hyperplanes containing some trisecant form a family of dimension n - 1. Suppose that trisecants form a family S of dimension?: 2. Since the family of all secants to X (family of lines joining two points on X) has dimension 2 and is irreducible (it is the closure of the image of X x X - 6x, 6x = diagonal, under the map sending P, Q to the line joining them), it would follow that every secant is a trisecant. We first show that this implies that the tangent lines Tp and TQ to X at any two smooth points P, Q must intersect. Fix a point Po E X and consider the map 710: X - Po --;- lP'n-l induced by projection of lpm to IP'n-l from Po. Let Y be the image curve and y E Y a smooth point such that 710is of maximal rank (= 1) at points of 7Iol(y). If P, Q E 7IOl(y), then Tp, TQ map onto the tangent line L of Y at y, so are contained in the plane generated by L and Po and so must intersect. Thus for fixed Po and an open set of points P, Tp, TQ intersect if KO(P) = KO(Q). If we vary Po in an open set, the corresponding points Q also fill out an open set. Thus there is an open set of pairs (P, Q) such that Tp, TQ meet; this must then hold for all pairs (P, Q). Choose P, Q with distinct tangenflines, and let B be the 2-plane spanned by Tp, TQ. Now B n X is finite (since X is non-degenerate). and let a E X, a tf. B. Then, the tangent line Ta meets both Tp and TQ, and since, Ta rt B and two lines meet in at most one point, Ta must contain the point Po = Tp n TQ. Thus To contains Po for all but finitely many points a and hence Po E Ta Va. But this is absurd, since then projection

from Po to lP'n-l restricted to X, would be of rank 0 everywhere, its image would be a point, and X would actually reduce to a line. This proves the lemma. . Proof of Castelnuovo's general position theorem. Let U C (lP'n)* be as in the lemma above and let I C X x U be the socalled incidence correspondence: 1= (P, II) with P E X n II. Then I is irreducible of dimension n. [It is irreducible since projection on X has as fibre over P E X the irreducible family of hyperplanes through P; it is of dimension n because projection 011 U has finite fibres.] Consider the subvariety 10 C I of pairs (P, II) such that there are points P = PI,' .. , Pn E X n H which are dependent. If dim 10 < n, then its projection on U would be proper, and the theorem, proved. Suppose dim 10 = n. Then, we have 10 = I. If P E X is a general point and 7Ip : X --+ lP'n-l the projection from P, the general position theorem must be false for the image X' C lP'n-l of X if it is false for X because if P = H, P2, ...• Pn E Hn ...'I. lie in a plane B of dimension n-2, then 7Ip(P2),· .. ,7Ip(Pn) lie in 7Ip(B) which has dimension n - 3. 'vVecan iterate this process as long as n > 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.

<

We have the following exact sequence of sheaves:

where a(h) = (h, -h) and (3(1, g) = f + g. The exactness is seen as follows. If (h) 2: -Do, we have both (h) 2: -D and (h) 2: -D'. If (I) 2: -D, (g) 2: -D', we have orda(l+g) 2: -max(D(a),D'(a)) = -(D(a)+D'(a)-min(D(a),D'(a))), so that a,(3 are maps between the sheaves in question. If orda(l) 2: -max(D(a), D'(a)) (I a germ of meromorphic function at a), then either (I) 2: -D(a) or (I) 2: -D'(a); in the first case, f = (3(f, 0), in the second, f = .8(0, f). If ,6(1, g) = 0, then f = -g and (f) 2: -D, (I) = (g) 2: -D' so that (f) 2: --Do and (j,g) = a(f). Thus (*) is exact.

+ hO(D')

Since D' ~

J{ -

9

::; hO(Do)

+ hO(D + D'

- Do) = hO(Do)

+ hO(K

- Do) .

D, this gives

+ 1 = harD) + hO(K

- D) ::; hO(Do)

+ hO(K

- Do) ::; 9

+ 1,

the last inequality following from the remark at the beginning of the proof. This proves the existence of Do, and hence, the proposition. Corollary degree d genus of system.

<

to Clifford's theorem. Let X C p" be a compact Riemann surface of 2n and suppose that X is non-degenerate. Tben 9 ::; d - n (g being the and if equality holds, then the hyperplane sections form a complete linear

Proof. Let H be a hyperplane ili'P7l and D = X n H. Then harD) 2: n + 1 linear form on p" gives a section of OD, and no linear form vanishes on X unless it is zero since X is non-degenerate). Since d < 2n. we have 1

dimlDI 2: n

n

> "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)

<0

//=1

1m L

CvCkBvk

<

0 ,

CvCkBvk

>

0 .

v,k

Let

WI, .. '

,wg be a basis of HO(X,fJ). Ajk =

r

lak

Wj'

Let 1m L v,k

= fJvj .

cg E JR:., not all 0, and let W =

9

1m

Corollary.

Av(wj)

.

I:%=l

CkWk· By

Given two distinct points P, Q, P of Q, on X, there is a meromorphic I-form 'P on X with simple poles at P and Q and resp('P) = +1, resQ('P) = -1 (by the Mittag-Leffler theorem for I-forms given in §1O). Because of the corollary to Prop. 1 above, we can add to 'P a holomorphic I-form 'P' on X such that the a-periods of wPQ = 'P + 'P' are zero (we assume the ai, bj so chosen as not to contain P or Q); the form wPQ is then uniquely determined. It is called a normalized abelian differential of the third kind. Given P E X an integer n 2: 1 and a coordinate system (U, z), at P with z(P) = 0, there is a unique meromorphic I-form w~nl on X, holomorphic on X - {P} and such that (i) w~'l - Z~~, is holomorphic at D and (ii) the a-periods of w};) are 0. This is called a normalized abelian differential of the second kind. (Abelian differentials of the first kind are simply holomorphic I-forms. Any meromorphic I-form on X is a linear combination of these three kinds of I-form.) RECIPROCITY

THEOREM. Let

Wj,

j = 1, ... , n be a normalized basis of HO(X,

fl),

and let w~n), wPQ be normalized abelian differentials of the 2nd and 3rd kind respectively. We have

Jbk

J;

1) wPQ = 271i X-Uai-Ubj)

Wk (the integral being taken along a Curve joining P to Q in

1

(n) -- ')_712 '. ~f(n-l)(p) ,k n.

Wp

bk

.

Proof. We identify X - U ai - U bj with a convex polygon 6. as before, and set Uk(X) = Wk (Po a fixed point in 6., the integral being along any path. in 6.).

.r;o

By Lemma 1, we have

r

.J 86.

UkWPQ = "L{AvCWk)Bv(wPQ)

- Bv(wklAv(wPQ)}

1/

= Bk(WPQ) =

I'

Jbk

[since A.,,(Wk) = bvk and Av(wpQ)

= 0]

,upO . •

On the other hand, since L2..is siniply connected andleJPQ has residue the residue theorem gives

+1

at P, -1 at Q,

This proves 1). 271iresP(lLkW~n)) d:lzk = !k near P).

The proof of 2) is similar:

=

271iresP(uk(Z)z~~')

=

fbk w}:'l

21ri~(ttl!k(Z)

=

JM

=

1!kW<;l (as above) = 21ri~f~n-l\z)

(since

.The results in the reciprocity theorem are sometimes also referred to as bilinear relations for periods of differentials of the second and third kind. This aspect becomes clearer if we drop the normalization conditions we imposed above.

We shall denote both these maps again by the same symbol A. The theorem that is truly central in the study of the relationship between X and J( X) is usually known as Abel's theorem. Abel's formulation of the theorem was rather different (and, in some ways, even more general). The theorem, as it is usually formulated today . was first given by Riemann in his fundamental paper on abelian functions [2]. Let X be a compact Riemann surface of genus g 2: 1. We use the description of X in terms of a convex 4g - gon as in §14, and the corresponding basis ai, bj of HI (X, Z). Let WI, ... , wg be a normalized basis of HO(X,

fl): Ia Wk = Djk. Let A be the subgroup

I/'

of reg consisting of the vectors A/' = (I/, WI,.", wg)' as I runs over HI (X, Z). We have Aak = (0, ... , 1, 0) = ek, the vector in reg with 1 in the k-th place and 0 elsewhere, and Ab> = (Ibk WI, , Ibk wg) (= Bk say) consists of the columns of the matrix B = (Bjk), Bjk = Ibjwk. Since Im(B) is positive definite, the vectors {el, ... ,eg,Bl, ... ,Bg} are linearly independent over JR. Since {ai, bj} generate HI (X, Z), we also have A = Zel + ... + Zeg + ZBl + ... + ZBg. These remarks imply that A is a lattice in lC9 with a compact quotient

ABEL'S THEOREM. Let D be a divisor to 0 if and only if A(D) = 0 in J(X).

Intrinsically, J(X) can be described as follows. Let V be the dual of HO(X,fl) (canonically isomorphic to H1(X,0) by the Serre duality theorem). We obtain a map of Hl(X,Z) into V as follows if I E H1(X,Z), let its image in V be the linear form W f-7 I/'w on HO(X,fl). Then, the choices made above (of ai,bj and Wk) identify HO(X, fl)* with reg and the image of H1(X, Z) with A. [The remarks made above also show that HI (X, Z) maps isomorphically onto a lattice in H°(.X, O)*.J We have

Thus, the theorem asserts thefollowing. Let PI," ., Pr; Ql, ... , Q" be points on X (with Qj of Pi Vi,j). The necessary and sufficient condition that there exist a meromorphic function with L:.Pi as its divisor of zeros and L:.Qj as its divisor of poles (i.e. (f) = PI + ... + Pr - Ql _ ... - Qr) is that k

".£,

r w=".£, r Pv

k

,,=1 }Po

wmod

(wg)

}Po

map A : X

---t

J(X)

as follows:

mod A

Ie

Using the fact that J(X) is an abelian group, we can define a map XN ---t J(X) by (PI, ... , PN) f-7 Lf=l A( Pj). Let"Div(X) be the set of all divisors on X. We can also define a map Div(X) ---t J(X) by "

i=l

l'

I--->

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(
If now PI, , Pn E X, and we number the P's so that PI = ... == Pr, (= Q1 say), PT1+1 = = PTl+T2 (= Q2 say), ... ,PTI+··+Tp_l+1 = ... = PTI+'.+rp (= Qp say), 1'1 + ." + 1'p = n, and Q1,"" Qp are distinct, the group of elements of Sn which fix (PI, ... , Pn) is SrI X ... X Srp (ST' acting by permutation on the kth block of rk .points), so that a neighbourhood of the image of (PI, ... , Pn) in sn(x) is isomorphic to a neighbourhood of (0, ... ,0) in
Thus, Av (
Theorem 1. A: sg(X) -+ J(X) is a birational map; there is an analytic setY C S9(X) oj dimension < 9 such that A: S9(X) - Y -+ J(X) - A(Y) is an analytic isomorphism. We start with Lemma 1. The set oj distinct points PI, ... , P9 E X such that the rank at D = 'E,Pi oj the differential oj A: S9(X) -+ J(X) is maximal, = g, is open and dense in xg. Proof. Let (Uj, Zj) be coordinates near Pj (Zj (Pj) = 0). Using the local coordinates on J(X) coming from C9 (J(X) = eN A), the map A can be written A(Zl,' .. ' Z9) = ~jrW (w = (W1, ... ,W9))· Ifw = f~dzj matrix of A at D = 'E,P, is given by

on Uj

(J~

=

(j~" .. ,h)) the Jacobian

el" is the vector with 1 in the ~L-th place, 0 elsewhere, and BI" is the vector (B1"1, ... , B1"9); since the el"' BI" form a Z-basis of A, we conclude that (*) holds if and only if

This proves the theorem. We now study the relationship between J(X) and X9. Let Sn be the symmetric group on n letters [= group of permutations of (1, ... ,n)]. Sn acts on the cartesian product xn =~; the quotient sn(x) = XnlSn is called the nth symmetric power of

The existence of (PI, ... , Pg) such that this matrix has rank 9 follows from §13, Lemma 4 (If dim HO(X, L) = k, :3k points Xj such that any s E HO(X, L) vanishing at the Xj is identically 0) since hoeD) = g. The fact that this set is dense follows from the proof of §13, Lemma 4. Lemma 2. Ij D = ojf'T withr=dimIDj.

~i Pi

E S9(X), then A-1A(D) is the bijective In particular, A-1A(D) is connectedVD.

holomorphic

image

n-times

X. sn(x) is a complex manifold of dimension n. To introduce coordinates (especially at a point fixed by a non trivial element of Sn), we proceed as follows.

c

Consider the action of Sr on r, and consider a neighbourhood of 0 in CT. Any germ of holomorphic function F at a in Cr invariant under Sr is a holomorphic function of the elementary symmetric functions in the coordinates Zl, ... ,ZT of CT (Newton's theorem); equivalently, it is a holomorphic function of WI = Zl + ... + Zr, W2 = f,(Zf + ... + z;), ... , WT = ~(zr + ... + zn and we can take WI, ... , Wr as coordinates for I Sr.

e

Proof. If Dl, D2 E S9(X) and A(DIl = A(D2), then Dl is linearly equivalent to D2 by Abel's theorem (since D1 - D2 has degree 0). Let f'T = f'(HO(X, 00)) (D E S9 (X)) be the projective space (HO(X, 00) - {O}IC'. Because of the remark above, the map HO(X, 00) - {O} -+ S9(X), S f--t div(.s) induces a bijection of f'r onto A-1A(D). The fact that this map is holomorphic can be seen as follows. Let U c eN be open, and f(x, t) a function holomorphic on .6." x U [tie = {x E
I

(counted with multiplicity) in Ixl mEZ,

~(

b x,

<

.(t))m

p is constant for t near to, say k, and, for m 2:: 0, = _1

21ri

J

x

m

*(.x, t) f(x,t)

d

x,

Ixl=p

1

so that this sum is holomorphic in t. We have only to apply this to the zeros of a general section toso + ... +trsr E HO(X, (JD) (the Sj forming a basis) in view of the coordinates we are using on sg(X). Note: The above map lP'r Theorem 2 below).

->

A

-1

A(D)

is actually biholomorphic (see the corollary to

I

Proof of Theorem 1. By Lemma 1, the set Y = {D E sg(X) rank(dA) at D < g} is an analytic set of dimension < g. By Lemma 2, A-1 A(D) = {D} if DE sg(X) - Y. The result follows.

fkr,-1)(P1) !k(P2)

iflk =

Remark that if DE sg(X) - Y, then hOeD) = 1. In fact, Lemma 2 implies that if D is an isolated point in A -1 A(D), then dim IDI = O. Since, by the Riemann-Roch theorem

Theorem

2. For any DE

sg(X),

the -rank of the map A : sg(X)

->

J(X)

at D equals

g-dimIDI· Proof. Let D = r1P1 + r2P2 + ... + rnPn with rj > 0, 22rj = 9 and P1,···, Pn distinct. We take as coordinates at D on sg(X) the following functions [X1,···,Xg being coordinates on X at P1,.· . , Ph.'" Pn,·· ., Pn respectively]:

'-----v--" Tl

-times

~

Tn

-times

2:%=1 2:%=1

Now, a linear combination Ckiflk of these column vectors is zero if and only if the holomorphic I-form w = Ckwk has the property that ordpv(w) 2:: rv for v = 1, ... , n, i.e. if and only if (w) 2:: D. Hence the number of linearly independent relations between the columns iflk is hOU:LD); and since hOeD) - hO(fLD) = 1- 9 + 9 = 1, we have hO(fLD) = hOeD) - 1 = dim IDI. Hence the rank of the matrix (ifl1, ... , iflg), i.e. . the rank of dA at D, is 9 - dim IDI· Corollary. For any DE sg(X), map lP'(HO(X, (JD)) -> A-1 A(D)

A-1 A(D) is a smooth submanifold of sg(X) defined earlier is an analytic isomorphism.

and the

In fact, Lemma 2 implies that A -1A(D) is an analytic set of dimension dim IDI· That A-1 A(D) is smooth follows from Theorem 2 and the implicit function theorem. 1 We have seen that the map lP'(HO(X,(JD)) -> 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
The family {V>.(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::;Sj<-r.

nE'7Lg

{},.,s(z) =

"

L--

a

n+rek

e-7rirBki.:.e21ri(n"z)

nE29

It follows at once that if an O. Hence dim Vr::; r9.

f ==

Define

L exp{ 7fi(B(n

+ ;), Tn + s) + 2r.i(z, Tn + s)} .

nEd?

I;ane21ri(n,z+Bk) ==

= f(z

e-rriBkk-21fizk

= I;an+eke-'>riBkk

+ 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
+ ... + A(Q~_l) + "', then 9-1

I:A(Pi)

9

T

I:(Av(Pk)

bv and x:' is the corresponding point of b;/> then

2) If x E av and x' is the corresponding hi

e

L

j mjBjv.

=

e21fi
point of a~, then e21fi
/

Thus F(x)edefines a meromorphic function on X with divisor ~(Pk - Qk) = (I). Thus. we have proved 21fi
Theorem 5. (Riemann's Factorisatio!l Theorem). Let f be a non-constant meromorphic fml.ction on X, with divisor (I) = L~=l Pk - L~=l Q k. Then, there exists W E HO (.X, 0) s'Uch that, 'if ( is a geneTul pO'int on e, we hatle

,

depend only on ( and not on P.

Consider now a non-constant merom orphic function f on X, and let (I) = L~=l Pk L~=l Qk· We choose ( E e, (rf. UdY' + '" - A(Pk») u Uk(Y' + K, - A(Qk»), and write ( = A(Do) + "', Do = A(Q~) + ... + A(Q~_l)' We transform X into a simply

_ . J:o IT' uJ

f(x)

-- c e

iI(A(x) k=lO(A(x)

- A(Pd - () _ A(Qk) _ ()

This theorem provides the analogue of the factorisation factors.

of rational functions· into linear

(we have]"v

r.p = 0 since the w~)

Let

()

F x = exp

19(A(x) - A(P) - () dx log -19(~A-(x-) --A.(-Q-) _-(-) is a meromorphic 1-form on X, holomorphic on X - {P,Q}, and with simple poles at P, Q, with residue + 1 at P and -1 at Q . Moreover, its a-periods are 0 since 19 is periodic. One can also construct the same way.

B.A. Dubrovin: Theta functions and non-linear equations Russian Math. Surveys (Uspekhi) 36 (1981), 11 - 92. I.M.Krichever and S.P. Novikov: Holomorphic bundles over algebraic curves and nonlinear equations, Russian Math. Surveys (Uspekhi) 35 (1980), 53 - 79. Tata LectuTes on Theta, 2 vols., Birkhauser,

T. Shiota: Characterization of Jacobian tiones Math. 83 (1986), 333 - 382.

1983, 1984.

varieties in terms of soliton equations,

The literature surrounding this relationship PDE has become enormous.

between

algebraic

Inven-

curves and non-linear

Let P be a fixed point on the compact Riemann surface X, let (U,.z) be a local coordinate at P, z(P) = 0, and let u be a polynomial in I-variable over C. Let D be a non-special effective divisor of degree 9 on X. We assume that F rf- suppeD). Theorem

7. TheTe exists a function

(F) 2': -D

(i)

(ii)

F meromorph'ic

on X - P such that

on X - P;

exp(-27fiu(~))

is holomorphic

at P.

Proof. Let u(t) = Co + C1t + ... + cTtT (cT =I- 0). Let w~) be the normalized abelian differential of the second kind with pole 7~~' (n 2': 1) at F (holomorphic on X - F). Then d1L(~)

(3

=

+ ~~=1

(,],X)

19 (A(x) - A(D) + (3 - Ii) 19(A(x) - A(D) - K)

r.p ~--------~

27f~

Po

First, F is single valued on X: consider it as a function on .6. as in Theorem 5. If x E bv and x' is the corresponding point of b~, then A(x') = A(x) + ev and F(x) = F(x') since x = = O. r a","

J

Xl

If)

1

(j)

forms with higher order poles at one point, having residue 0, in

The idea behind Riemann's factorisation theorem can be used to construct functions with certain essential singularities on X. These functions are of great importance in the study of certain non-linear partial differential equations which have turned out to be very closely connected with the geometry of algebraic curves. For an introduction to this circle of ideas, one may consult

D. Mumford:

are normalized).

nCnw~n)

is holomor;hic

((31, ... , (69) be the vector of b-periods of

at F. r.p:

Let. r.p = -

~~=1

nCnw~n)

Let

while A(x) - A(x') = -Ev, so that 19(A(x) - A(D) + /3 - K) = v(A(x') - A(D) (3 - K) X exp(27fi(Av(x') - /clAD) + (3v - K,,) + 7fiEw), and v(A(x) - A(D) - K) 19(A(x') - A(D) - K) exp(21fi(A(x') - Av(D) -Kv) + 1fiEvv). Hence

F(x) F(x')

= e-27ri(3v

.

exp(21fi(Av(x') - Av(D) + (3v - Kv) + 1f'iEvv) exp(21fi(Av(x') - Av(D) - liv) + 1fiEw)

Since '-P ~ due ~) is holomorphic

at P, Fe-27riu(

~)

is holomorphic

Finally, the poles of F are at the zeros of 19(A(x) ~ A(D) K)) = D since D is non-speciaL

K)

+

= 1.

at P.

and div( v(A(x)

- A(D)-

It is not hard to see that if D and 1, are generic, _this function is uniquely determined up to a constant multiple. In fact, one sees that the divisor D' of zeros of the function constructed above is non-special; if Fo is a function satisfying the conditions of Theorem 7, then Fo/F' is meroHlOrphic on X, and (Fo/F) 2': -D'; since D' is nOIFspecial, F'o/F is constant.

Torelli's theorem asserts that the pair (J(X), El) determines the Riemann surface X. There are several proofs available of this theorem. The one we shall give here is due to Henrik Martens [12]. There are more "geometric" proofs, some of which will be found in Griffiths-Harris [9] or Arbarello-Cornalba-Griffiths-Harris [10]. We begin with a general fact about complex tori. Let T1 = em/AI, T2 = in en).

en /A2

We shall also denote an effective divisor of degree k on X by D k, Die ,/::;.k, ... (In other words, subscripts on a divisor wilLind.icate its degree in the proof of the above theorem).

map and F : cern

---t

en

Lemma 1. Let 0 a E Wg-I-r,b· Proof.

g~

ce

TORELLI'S THEOREM. Let X, Y be compact Riemann Elx, Ell' the theta divisors on J (X), J (Y) respectively. Suppose that there exists an analytic isomorphism 'P : J (X) analytically 'isomorphic.

surfaces of gen'us 9

---t

?::

1, and

.I (Y) s'uch that 'P* (8y)

=

Elx. Then X and Yare

We shall assume that (l(X),

El x) and (l(Y),

We denote by Ax : X ---t .1(X), the Abel-Jacobi .1(X) of sr(x), 1 :s; r :s; 9

:s;

r

:s;

g-

be a lifting of f.

Proof. Since F lifts f, if A E AI, then F(z +.\) - F(z) E A2 for all z E em, and so is constant. Hence, for 1 :s; v :s; m, is invariant under translation by A, so defines a holomorphic map T1 ---t n, which is constant since T1 is compact. This proves the lemma.

If a = Ax(Dg-1-r)

+b,th.~nA(l)r)

To prove the converse, we may assume that b = O. By assumption, V Dr (D,. ?:: 0 of degree r), there is /::;'g-l such that Ax(D,.) + a = Ax (/::;.g-Il· Now, if Po is the base point in X defining the Abel-Jacobi map, we have Ax(rPo) = 0, so that a = Ax(8), where 8 ?:: 0 has degree 9 -1. We now have Ax(Dr + 8) = Ax (/::;.g-l+r Po), so that, by Abel's theorem Dr + 8 ~ /::;'g-l + rPo (linear equivalence). Hence Dr + J(x - /::;'g--l~ (Kx - 8) + rPo; moreover KX-/:),.g-l and J(x ~ 8 are linearly equivalent to effective divisors [since hO(D~_l) = h°(1f.xT D~"l)l. Thus, Kx ~ 8 + rpp is linearly eqmvalent to a divisor of the form Dr + D~_l VDr; hence dim IKx ~ 8 + rP61 ?:: r. Hence, by the Riemann-Roch theorem hO(8-,- rPo) = h°(IG - 8 + 'rPo) + 1- 9 + (g --1- r) ?:: 1,so that 8 - rPo ~ D~-l-r> and we have Ax(D~_l_r)= Ax(8 - rPo) = Ax(8) = a, and a E \;Vg-1-r.

map of X, and by Wr the image in in

'vVeshall actually prove the following theorem, which clearly implies Torelli's theorem

TF* i'~'g-l-r

=

n

TF n' g-l,a --

aEM~

zs a translate

of

+ a =A(Dr + Dg-1-r) + b E Wg-1,b.

Ell') have been identified by 'P.

We let Ay : Y ---t .1(Y) be the Abel-Jacobi map of Y, and set 1,~= image of S"(Y) .1(Y), 1 :s; r :s; g.

Theorem. Assume that Wg-1 WI of -WI'

of E by a:

be two complex tori (AI is a lattice in em, A2, a lattice

Lemma 1. Let f : TI ---t T2 be a holomorphic Then F is a polynomial of degree :s; 1.

Proof.

.For any set E c J(X) and a E. J(X), we denote by Ea the translate Ea = E + a. With this notation, we have

\lg-1.

Then

VI

If E is a subset of J(X), we denote by E* the set E* = A(Kx) canonical divisor on X. We shall call E* the dual of E.

is

(I

n

(TV· , g-1,-a )*

.

aEM~

translate of either Proof. \;Vg-1-r

- E, where Kx is a

If a E W,., we have \;Vg-1,-a .

==

Ax(Dr)

(Dr)

so that

C naEW

T

Let now ( E naEWT,W9-l,-a, soth(j,t (+ Wr C Wg.~l' By Ler::mla 1, this implies that ( E Wg-I-r, and the first state.tl1entis proved. The second follows from the first by taking duals.

Lemma 3. Let 0 ::; r ::; 9 - 2, let a E J(X), x E WI, Y E Wg-1-r. Then we have: Either Wr+1,a C llVg-1,b

Wg-1,b where S = Wr+1,a

n

- y.

= x ' and DOg-l-r

there is P E X with Ax(P) Then x - y = -Ax(D')

+ (Ax(D')

+ Wr+1)

2) Suppose that P f:- suPP(D~_l_r)'

such that

where deg D' = 9 - 2 - r

.

C b + Wg-1,

= (Wg_2,y_a)*;of C

course,u E Wr+1,a

Wr,a+x U S.

We have still to check the opposite inclusion. We have Wr +a+x == Wr +a+Ax(P) C Wr+1,a' Since a+x = b+y E b+Wg-1-ro we have Wr,a+x C b+ Wg-1-r+l-Vr = b+liVg_1. Finally, (Wg-2,y-a)* = W;_2 +b-x = Ax(Kx) - Wg-2 -x+b C Ax(Kx) - Wg-1 +b =

Wr+1,a = Wr,a+x uS

a = b + Ax(D') =b

(by Lemma 1); thus u E (W;_2h-x = (W;_2)a-y by assumptIon. Thus, we have Wr+1,a n Wg-1,b

Wg-1

1) Suppose that P E suPP(D~_l_r)' (and D' :2:0), and we have

+ Wr+1

+x

n (Wg-2,y-a)*.

Proof. o By definition of Wk, Ax(Dg-1-r) = y.

so that a

Set b = a

which is our first alternative.

and let

+ b.

This proves that Wr,a+x C Wr+1,a is proved.

n Wg-1,b,

and that S C Wr+1,a

n Wg-1,b;

Proof of Torelli's theorem. Recall that we have identified J(X) say) and that Vr is the image of sr(y) in J under Ay.

the lemma

with J(Y)

(= J

Let r be the smallest integer :2:0 such that VI is contained in some translate of either 17Vr+1or 17'11:+1;since VI C Vg-1, and Vg-1 is a translate of Wg-1 by hypothesis, there is such an integer (e.g. 9 - 2). The theorem asserts that r = O. Assume that r :2:1, and that VI C W-r+1,a' Let x E WI, Y E Wg-1-," Remark that if x is fixed, there is an analytic set Z(x) C Wg-1-r, such that if y f:- Z(x), then VI Wg-1,b where b = a + x - y.

et

In fact, suppose that VI C Wg-1,b contradicting the definition of T. so that, since Dr+1 +D~-l-r that

and t.g-1 +P both have degree g, Abel's theorem implies

Case

(i). D,.+1

+ D~-l-r

Then V1,-x-a

C nyEW9_l_,'

of

Wg-1-ro

17Vg_1,_y = Wr,

Now, if VI C l-Vg-1,a+x-y = Vg-1,a-y, 0' = Co + x, Co fixed, then VI + y C Vg-1,,,· By Lemma 1, this means that y E Vg--2,a- Hence, if Z(x) = Vg-2,,, n TVg-1-r of 17Vg-1-ro then, for y E Wg-1-r - Z(x), we have VI Wg-1,b, b = a + x - y.

et

Consider now (with b Since P if suPP(D~_l_r),

Vy.

Z(x)

=

t.g-1

=

a

+x

- y, y if Z(x))

+ P.

we have P E supp(Dr+1),

and we have since vi C Wr+1,a and VI

et

Wg-1,b,

we have VI n Wg-1,b =

we have Wr+1,a

eVl

et

17Vg-1,b, so that by Lemma 3,

rl Wr,a+x) U (Ill n S) ,

et

where S = VVr+1,an(Wg-2,y-a)*' Since 1'1 Wg-1,b, (and Wg-1 is a translate of1~-ll, there is a divisor D(b) of degree 9 on Y such that (if AV1; sg(y) -. J is the natural map) Case

(ii). Dr+1

+ D~-l-r

of

Ay(D(b))=1l1,Wg-l,b,

t.g-=-l + P.

In this case, the complete linear system It.g-1 + PI contains two distinct effective divisors, so that dim It.g-1 + PI :2:1. Hence, for any Q E X, we can find a t.' -1 > 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;

'-<:"v

are linearly independent,

( -

and

tzO (z) = - tz: ( -z), v

we have

g~(() =

t; =

A(D2

+ t..o)

,deg(D2

+ 60)

= 9- r

+r

- 1= 9 - 1.

Fu~ther 60 could be an arbitrary divisor in a non-empty open set in sr-1 (X) [since Do = P 60 could be anywhere in a non-empty open set in sr(X)l· Hence dim ID2 + 601 ;::: r-1.

+

,g.

Suppose conversely that x f-+ 19(A(x) - A(P) - () is "t 0; let div( 19(A.(x ) - A(P) - ()) = P + Do((), degDo(() = 9 - 1. By Lemma 3, there exists Q ~ suppDo(() such that 19(A(x) - A(Q) - () -=j. O. By Lemma 2, div(19 (A(x) - A(Q) - ()) = Q +Do((), so that

Conversely, suppose that (-K, = A(D), where D ;:::0, deg D = g-l

and dim jDj ;:::s-1.

Given any effective divisor D1 of degree s - 1, we may assume that

Q is a s'imple zero of 19(A(x) -- A(Q) - (), so that

D ;::: D1 (since

dim!DI;::: s -I).

L azEMy (A(x) 9

- A(Q) - () Ix=Q wy(Q)

1= 0,

v=l

and

t~(-()1= 0 for some

1I.

This theorem can be formulated as follows. Recall that if E, E' are subsets of leX) and ( E leX), then E -- E' - ( is the set {x - y - (I:r E E, y E E'}; we use similar notation E + E' and so on. Theorem

1'. Let ( E

e. Then ( -is a s'ingular point of e if and only 'ij-{)(W1 - W1 - () ==

o. Suppose now that the genus 9 ;:::2. Given ( E e, there is an integer r < 9 such that 19(Wr - Wr - () -=j. 0; in fact, since W9-1 = "4(Kx) - \-,119-1 (see proof of §17, Theorem 3), we see that W9-1 - W9-1 is a translate of VV9-1 + W9-1 = leX).

Let Eo, E1 be effective divisors of degree s - 1. Choose D ;:::0, deg D = 9 - 1, with (K = A(D) and D ;:::Eo· We have

A(Eo) - .4.(E1)

-

(= A(Eo - D) - .4.(E1)

-

K,

= -(I~+ --4(E1

+ (D - Eo»)) ;

since D - Eo ;:::0 and has degree 9 - 1- (s - 1), E1 + D - Eo ;:::0 and has degree 9 - 1; hence K + A(E1 + D - Eo) E W9-1 + K, = e, so that A(Eo) - A(Ed - ( E -8 = e, and 19(.4.(Eo) - A(Ed - () = 0, i.e. 19(Ws-1 - VVs-1 - () == o. Theorem

3. (Riemann's

vanishing of 19 at(,

singularity

i.e. ifa=(a1,

al0'.11I

&0'.19

-a -;;-(() Z

Let ( E e and let Tn be the order of andlal=a1+···+a9 <m,then

theorem).

... ,ag) =

a.0'.1 41

...

8-O'.g (() 49

= 0,

while there is 13 with !131 = m so that

fz1(() 1= O.

Equivalently, the expansion of 1J in a series of homogeneous wzth a homogeneous polynom'ial of degree m.

polynomials

in z - ( starts

point in aj, we have A(xd

the corresponding theorem in §14,

- A(xD

=

-Bj

and, by the reciprocity

Then m = r(, so that m = 1 + dim IDI, where D ~ 0 is a divisor of degree g - 1 such that (- n; = A(D). so that Proof. Suppose ~~~t k ~ 1is an integer such that 1J(Wk - Wk - () == O. We claim la! ~ k, we have ozQ Wk-[<>I - ( == O. We prove tlllS by induction on lal = n. If ~hlS ISJust our hypothesis. Assume that the assertion is proved for lal = n-l < k. If u, v E ~~~-n and x, y E X, we have A(x)+u(A(y)+v) -( E Wk-(n-l)-Wk-(n-l)-(, so t~1at ozQ (A(x) + u - A(y) - 1l - () = O. Differentiating this with respect to settmg x = y, we obtain

I

2: 9

& &<>1J -& & <> (u - v - ()w,,(x)

1.1=1

Zu

~:! (-()

n =

0,

Then,

F(Xl) F(x~)

=

e2rri(A;(D1)-A;(D))

Thus F( Xl) is single-valued x and

= 0 for lal

+ 1= n

"Ix EX,

r(.

Let D = L:=l Pv, Dl = L:=l Qv and let wp"Q" be the normalized abelian differential of the . third kind (residue + 1at P v,. -1 at Q v, holomonhic on X - {P Q } a n d 1lavIng .' . 1 a-penods 0). Let 'P = L:=l wp"Q"' ].1)

(F is holomorphic

= div(Xl

f-+

E

= div(Xl

f-+

1= 0

+ 1J(A(Xl) + 1J(A(Xl)

+ A(xr) + A(xr)

,Pn)

1= 0). -

Again, fix X2,···,

Xn

en '

-

A(Dl)

-

A(D) - ()) .

V

- Wr-l

'vVenow prove (using the same reasoning as in Lemma 2) that Di = D'. In fact, we have dim [EI= 0 (since 19 (.4.(Xl) + ... + A(xr) - A(D) - () '1= 0 in Xl) so that dimlD'1 = O. Moreover, A(D') = A(E) - A(D) = (( + A(D) - L; A(x,,) - n;) - A(D) = (- n;L; A(x,,); in the same way, A(Di) = (- n; - L; A(x,,), so that, since deg D' = deg Di, Abel's theorem implies that Di ~ D'. Since dim ID'I = 0, we have D' = D~.

I,::

Consider now the function exp (L~=l 'P) with X2, ... ,x,- fixed. If z", 7J)" are local coordinates at P", Q" respectively (with z,,(P,,) = 0,1.0,,( Q,,) = 0), then if Xl is near P", we have 'P = L~=l WPkQk = !if:: + h, h holomorphic at Pv while, if :r;1 is

I;;

set

and

= 1.

on X.

El

near Q", we have ... ,Xr EX,

+ Jri Bjj) + JriBjj)

we have L~ A(xv) - A(Dl) - ( = L; A(x,,) - A(Dl - Xl) - ( E - (, so that 1J vanishes at this point. Hence El = Dl + Di, where D~ ~ 0 and has degree 9 - r. Similarly, E = D + D', where D' ~ 0 and deg D' = deg Di = g - r.

Wr-l

The inequality m :S. r( is harder to prove. Let r = r' so that 'i)(VV _ - HT () = 0 1 {Ie " r 1 vv r-l . , ane Wr - W,. - () '1= O. We can find three effective divisors D, Do, Dl of degree r such that suppeD + Do + Dl) consists of 3r distinct points and such that

Forxl,

- Aj(Dl) - (j) - Aj(D) - (j)

If Xl E supp(Dd,

and all u,v E Wk-n.

:S k, so that we have m ~

exp(27ri(Lv Aj(x,,) exp (2Jri (L" Aj(xv)

We now claim that F is constant (1= 0 since F(Pl,'" generically in X and consider the divisors

Z

so that ~:~~(u - v - () = 0 if la'i = lal This implies that

= 0

that if

I;: I;: 'P = -

I;:

1;0'

d:::; + hi,

hi holomorphic

at Q". Hence near P",

at (Pl, ... ,Pr) if 2',p" = Do.)

We claim that F defines a meron1orphic function on X x ... x X (r times). To see this, we fix X2,· .. ,Xr and consider F as a function F(Xl) defined on the simply connected polygon 6. obtamed by slitting X along the homology basis ai, bj. Ifx~l E bj and x~ is the corresponding point of bj, we have A(xd = A.(xU + ej and Ix; 'P = 0 smce the a-periods of'P are O. Hence F(Xl) = F(xU. If Xl E aj and x~ is

Consequently,

the divisor of exp (L~=l

D - Dl- Hence, the divisor of Xl

f-+

I;: 'P),

as a function is

F(Xl, X2, ... ,

of Xl, is 2',P" - 2',Q"

Thus, this function is holomorphic and non-zero on X, and so is a constant # 0 (for fixed generic X2, .. ' ,xr). From the symmetry of the function in Xl, .. ' ,X" it follows that F == Co E
#

0 so that

.+A(xr)-A(Dl)-()

=

We have proved: There is a constant

jXV

expct 1'=1

'P )19(A(Xl)+"

Co

co19(A(Xl)+"

.+A(xr)-A(D)-()

.

Po

We differentiate with respect to when Xl = Fl (see above)

Xl

and set

Xl = Fl·

This gives, since exp

(1;: 'P) = 0

One of the applications of the singularity theorem is to the proof that the set 8sing of singular points of 8 has pure dimension 9 - 4 if X is not hyperelliptic (and dimension 9 - 3 if X is hyperelliptic). This is an important property, and shows that Jacobians are special among the so called principally polarised abelian varieties (complex tori defined . by a lattice having a basis (1, B) where I is the identity, and B a complex symmetric matrix with positive definite imaginary part). The theta divisor in the generic case (defined as on the Jacobian) is smooth. We shall not prove this theorem, but shall say a few words about why 8sing # 0 for 9 2 4. The theorem is discussed in [8], [10], as well as a partial converse studied by A. Andreotti and A. Mayer [On period relations for abelian integrals on algebraic curves, Annali Sc. Norm. Pisa 21 (1967), 189 - 238.]. Because of §19, Theorem 1', 8sing is a translate of the set W;_l = { Divisors D 2 0 of degree 9 - 1 with dim IDI > O}. Thus, the statement 8sing # 0 is equivalent to the statement that there is a non-constant-meromorphic function on X whose divisor of poles has degree::; 9 - 1. If X is hyperelliptic, this is obvious when 9 2 3. One way to prove this when X is not hyperelliptic (and 9 2 4) is to prove the following: Let X

C jpg-l

be the canonical imbedding. Then, there is a quadric Q =

~

aijZiZj

l~i,jS.g

here Zl (zv) is a local coordinate at Fl (Fv) and Cl # 0; in fact, we saw above that exp 'P) = zveh, h holomorphic at Fl, in a neighbourhood of Fl. Iterating this argument, we find that

(1;:

Cl ...

=Co

Cr

19(A(Fl)

+ ... + A(Pr) - A(D]) - ()

~ l:Sk" ...,kr:Sg

or19 OZk,,,.OZk

dzl(Fl) ... dz,.(Fr) I

(A(Fl)+

... +A(Fr)-A(D)-()Wk,(Fl),,,Wkr\Fr)'

(with (aij)

symmetric) such that QIX

==

0 and 0

< rank(a.ij)

::; 4.

[This can actually be done by an easy dimension count, using Noethcr's theorem given in §13. Of course, the result is trivial if 9 = 4.] Given this theorem, one can proceed as follows. diagonalised, we can make a linear transformation form

Since a symmetric matrix can be of (Zl,' .. , Zg) to put (aij) in the

r

Since the term on the left is # 0 and A( Fl) exist kl, ... , kr (between 1 and g) so that

+. . + A( Fr)

= D, it follows that there

In presenting essentially this argument in [3], Riemann does not use the normalized abelian differentials wPvQv explicitly, but rather, their expression in terms of theJ3function given by §17, Theorem 6. Thus, we could have worked directly with the funct'ion

and the rank condition means that there are at most four l's on the diagonal, so that = r ::; 4. If r ::; 2, Q is a product of linear forms, so cannot vanish on X since X is non-degenerate (not contained in any hyperplane). Thus Q = + z? + z~ or Since 0.2 + b2 = (0 + ib)( a. - ib), we can, by another linear transorn;atioll, assume that Q = + ZlZ2 or Q = Z122 + Z3Z4. If , ... , wg) is the basis of HO(X, fl) corresponding to this form of Q, the relation becomes

where (0 is a general point of 8. The argument, using this function, remains essentially the same.

(as section of KJi2). In either case, if div(w3) = I:~~~2 PI" Wl or iU2 wIl must vanish on at least 9 - 1 of the points Fv, and WdW3 then has a polar divisor of degree ::;g-1.

This proves that m ::; T' =

1'(,

and with it, Riemann's theorem.

Q

I:~

I:;

zr

The study of quadrics containing the canonical curve is a very rich and beautiful See, in particular

One.

B. Saint-Donat. On Petri's analysis of quadrics through a canonical curve, Math. nalen 206 (1973), 157 - 175.

An-

M. Green. Quadrics of rank four in the ideal of the canonical curve. (1984), 85 - 104.

Inv.

Math,

75 The literature on Riemann surfaces and the topics dealt with in these notes is vast, and we content ourselves with just a few references.

Riemann's two papers, which form the basis of much of the material in these notes are: [2] B. Riemann. Theorie der Abel'schen Functionen. 54 (1857). Collected Works: pp. 88 - 144.

J. fur die reine und angew. Math.

[3] B. Riemann. tiber das Verschwinden der Theta-Functionen. angew. JIIlath. 65 (1865). Collected Works: 212 - 224.

J. fur die reine und

Two texts on Riemann surfaces which are accessible and have much in common with the first part of these notes are: [4] O. Forster. Riemannsche tiOli available. [5] R.C. Gunning. 1966.

Fliichen, Springer 1977. There is also an english transla-

Lectu7'es on R'iemann

surfaces.

For the topology of orient able surfaces, in particular consult: [6] W.S. Massey. Algebraic 1967.

Princeton

Mathematical

Notes,

the classification theorem, one can

Topology: An Introduction.

Harcourt

Brace, NeVi York,

For the standard material from complex analysis (in particular properties of 8) used in the first part of these notes, as well as a different arrangement of the proof of the finiteness theorem, see: [7] R. Narasimhan.

Complex Analysis

in one Variable, Birkhauser,

1985.

As a quick introduction to the many aspects of the geometry of curves and .Jacobialls. one cannot recommend too strongly the following beautiful book: [8] D. Mumford.

Curves and their Jacobians.

University of Michigan Press, 197.5.

Two other indispensable books dealing not only with the material of these notes, but with very much more are:

[9] P.A. Griffiths and J. Harris. Pr'inciples of Algebraic Geometry. 1978. [10] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris. Curves, Vol. I, Springer, 1985.

Wiley, New York,

Geometry

of Algebraic

Serre's paper on the duality theorem is: [11] J.-P. Serre. Un theoreme de dualite.

Comm.

Math. Helv. 29 (1955), 2 - 26.

Martens' proof of Torelli's theorem is in: [12] H. Martens. 111.

A new proof of Torelli's theorem.

Annals

of Math.

78 (1963), 107 -

C. de Boor Splinefunktionen 1990. ISBN 3-7643-2514-3

G. Baumslag Topics in Combinatorial Group Theory 1993. ISBN 3-7643-2921-1

J.D. Monk Cardinal Functions on Boolean Algebras 1990. ISBN 3-7643-2495-3

M. Giaquinta Introduction to Regularity Theory for Nonlinear Elliptic Systems 1993. ISBN 3-7643-2879-7

D. Battig/H. Knbrrer Singularitaten 1991. ISBN 3-7643-2616-6

R.J LeVeque Numerical Methods

O. Nevanlinna Convergence of Iterations for Linear Equations 1993. ISBN 3-7643-2865-7

for Conservation

Laws

2nd Edition, 3rd Printing 1994. 1992 ISBN 3-7643-2723-5

R. -Po Holzapfel The Ball and Some Hilbert 1995. ISBN 3-7643-2835-5

J.F. Carlson Modules and Group Algebras

R. Narasimhan Compact Riemann Surfaces 1992. ISBN 3-7643-2742-1

Notes by Ruedi Suter 1996

A.J. Tromba Teichmi.iller Theory in Riemannian 1992. ISBN 3-7643-2735-9

M. YOI' Some Aspects of Brownian 1992. ISBN 3-7643-2807-X

Problems

Motion

Geometry

ISBN 3-7643-5389-9

DMV1

M. Knebusch / W Scharlau: Algebraic Theory of Quadratic Forms (ISBN3~7643-1206-8)

DMV2

K. Diederich / I. Lieb: Konvexitat in der komplexen Analysis (ISBN 3-7643-1207-6)

DMV3

S. Kobayashi / H.-H. Wu: Complex Differential Geometry (ISBN 3-7643-1494-X)

DMV4

R. Lazarsfeld / A. van de Ven: Topics in the Geometry of Projective Space (ISBN3-7643-1660-8) WM. Schmidt: Analytische Methoden fUr diophantische Gleichungen (ISBN3-7643-1661-6) A. Delgado / D. Goldschmidt / 8. Stellmacher: Groups and Graphs: New Resultsand Methods (ISBN3-7643-1736-1)

DMV7

R. Hardt / L. Simon: Seminar on Geometric Measure Theory (ISBN 3-7643-1815-5)

DMV8

Y.- T Siu: Lectures on Hermitian-Einstein Metric for Stable Bundles and Kahler-Einstein Metrics (ISBN3-7643-1931-3)

DMV9

P. Gaenssler / W Stute: Seminar on Empirical Processes(ISBN3-7643-1921-6)

DMV 10

l. lost: Nonlinear Methods in Riemannian and Kahlerian Geometry Delivered at the German Mathematical Society Seminar in DUsseldorf in June, 1986 (ISBN3-7643-2685-9) T tom Dieck / I. Hambleton: Surgery Theory and Geometry of Representations (ISBN3-7643-2204-7) J.H. van Lint / G. van der Geer: Introduction to Coding Theory and Algebraic Geometry (ISBN3-7643-2230-6) H. Kraft / P. Siodowy / TA. Springer (Eds): Algebraische Transformationsgruppen Invariantentheorie / Algebraic Transformation Groups and Invariant Theory (ISBN3-7643-2284 ..5)

und

DMV14

R. Bhattacharya / M. Denker: Asymptotic Statistics (ISBN3-7643-2282-9)

DMV 15

A.H. Schatz / V. Thomee / WL. Wendland: Mathematical Theory of Finite and Boundary Element Methods (ISBN3·7643-2211-X) l. Mecke / R.G. Schneider / D. Stoyan / WR.R. Weil. Stochastische Geometrie (ISBN3-7643-2543-7) L. LJung / G. Pflug / 1-1.Walk: Stochastic Approximation and Optimization of Random Systems (ISBN3-7643-2733-2)

DMV18

K.W Roggenkamp / M.J. Taylor. Group Rings and Class Groups (ISBN3-7643-2734-0)

DMV19

P. Groeneboom / l.A. Wellner: Information Bounds and Nonparametric Maximum Likelihood Estimation (ISBN3-7643-2794-4)

DMV 20

H. Esnault / E Viehweg: Lectures on Vanishing Theorems (ISBN3-7643-2822-3)

DMV 21

M.E. Pohst: Computational Algebraic Number Theory (ISBN3-7643-2913-0)

DMV22

H. W Knobloch / A. Isidori / D. Flockerzi: Topics in Control Theory (ISBN3-7643-2953-X) M. Falk /1. Husler / R.-D. Reiss: Laws of Small Numbers: Extremes and Rare Events 3 1/2" Diskette included (ISBN3-7643-5071-7) M. Aubry: Homotopy Theory and Models. Based on lectures held at a DMV Seminar in Blaubeuren by H.J Baues, S. Halperin and J-M. Lemaire (ISBN7643-5185-3) W 8allmann: Lectures on Spaces of Nonpositive Curvature (ISBN3-7643-5242-6)