Compact Riemann Surfaces 9780817627423, 0817627421, 9783764327422, 3764327421

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Lectures in Mathematics ETH Ziirich Department of Mathematics Research Institute of Mathematics Managing Editor: Oscar E. Lanford

Raghavan Narasimhan Compact Riemann Surfaces

Springer Basel AG

Author's address: Raghavan Narasimhan Department of Mathematics University of Chicago Chicago, IL 60637 USA

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data Narasimhan, Raghavan: Compact Riemann surfaces / Raghavan Narasimhan. Springer Base! AG, 1992 (Lectures in mathematics) ISBN 978-3-7643-2742-2 ISBN 978-3-0348-8617-8 (eBook) DOI 10.1007/978-3-0348-8617-8

This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, 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, perrnission of the copyright owner must be obtained. First reprint !996 © 1992 Springer Base! AG Originally published by Birkhauser Verlag in 1992 produced from chlorine-free pulp. TCF 00

ISBN 978-3-7643-2742-2 98765432

Preface 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. 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. It is a great pleasure to express my thanks to the ETH for its hospitality, to Prof. J. Moser for his encouragement, and to Dr. A. Stadler for the enormous amount of work he undertook which made these notes easier to write. But special thanks are due to Prof. K. Chandrasekharan. But for him, I would not have been at the ETH, nor would these notes have been written without his advice and encouragement.

Chicago, August 1991

R. Narasimhan

Contents 1. Algebraic functions

3

2. Riemann surfaces.. .... ............ ........ ..... .......... ............. .. ...

8

3. The sheaf of germs of holomorphic functions ................................

12

4. The Riemann surface of an algebraic function ............................... 15 5. Sheaves ....................................................................

17

6. Vector bundles, line bundles and divisors ...................................

27

7. Finiteness theorems ........................................................

32

8. The Dolbeault isomorphism ................................................

38

9. Weyl's lemma and the Serre duality theorem...... .................. ........ 43 10. The Riemann-Roch theorem and some applications.. . .......... ............ 49 11. Further properties of compact Riemann surfaces ............................

58

12. Hyperelliptic curves and the canonical map ................................. 63 13. Some geometry of curves in projective space ................................

66

14. Bilinear relations ........................................................... 77 15. The Jacobian and Abel's theorem.... ..................... ... ............... 84 16. The Riemann theta function .............................. '" .. .. ........... 91 17. The theta divisor ........................................................... 97 18. Torelli's theorem ........................................................... 106 19. Riemann's theorem on the singularities of

e ................................

111

References ................................................................. 119

1. Algebraic Functions Let FE ([[x, y] be an irreducible polynomial in two variables (with complex coefficients). We assume that its degree in y is ;::: 1. Recall that by the so-called Gauss lemma, if we identify ([[x, y] with ([[x][y], and if F is irreducible, it is also irreducible in C(x)[y], the polynomial ring over the field ofrational functions in x. Moreover, ([[x, y] is a factorial ring (i.e. a unique factorisation domain). An algebraic function is, intuitively, "defined" by an equation F(x, y) = 0 (where F is irreducible in ([[x, y]). To make this statement more precise, we begin with the following. The implicit function theorem. Let f be a holomorphic function of two complex variables X,y defined on {(x,y) E C 2 11xl < rl, Iyl < r2}, rl,r2 > O. Assume that

f(O,O) = 0,

of

oy (0, 0) # 0 .

Then, there exist positive numbers E, 0 > 0 such that for any x E Dc = {z E C Ilzl < 1o}, there is a unique solution y(x) of the equation f(x, y) = 0 with ly(x)1 < O. The function x f--+ y(x) is holomorphic on Dc.

:s

Proof. Since %(0,0) # 0, we can choose 0 > 0 such that f(O,y) # 0 for 0 < Iyl o. Choose now 10 > 0 such that f(x, y) # 0 for Ixl 10, Iyl = 0 (possible since f is non-zero on the compact set {O} x {y Ilyl = o}).

:s

By the argument principle, if

Ixl < E,

2~i

J{~~(x,Y)/f(x,Y)}dY

iYi=b

is an integer n(x) equal to the number of zeros of the function y f--+ f(x,y) in Iyl < 0; by our choice of 0, nCO) = 1. On the other hand, since f(x, y) # 0 for Ixl 10, Iyl = 0, the integrand, and thus also the integral, is a continuous function of x for Ixl < E. Thus n(x) = 1 for Ixl < 10, which means precisely that there is a unique zero y(x) of f(x,y) with ly(x)1 < O. That x f--+ y( x) is holomorphic follows from the formula

:s

1

y(x) = 2Jri

J

iyi=b

%(x,y) y f(x, y) dy

1. Algebraic Functions

4

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

Lemma 1. Let a E C be such that ao (a) =F 0 and such that there is no bEe with F( a, b) = 0 = ~~ (a, b). Then, there is f > 0 and n holomorphic functions Yl (x), ... ,Yn(x) in the disc {x E C Ilx - al < f} with the following properties: (i) Yi(X) =F Yj(x') if i =F j, Ix - al

< f, lx' - al < f; moreover

F(X,Yi(X))==O (ii) if'T) E C and F(x, 'T)) and n.

= 0,

Ix-al 0 would divide bk- 1Qk-2, hence Qk-2 (since bk- 1 E qx] and deg y P > 0). From the equation

5

1. Algebraic Functions

bk- 2Qk-3 = A k- 1 Qk-2 + Qk-l, it would follow that P divides bk- 2Qk-3 and hence Qk-3. Repeating this argument, P would divide all the Qj (j ~ 1), hence also ~; and F, contradicting the irreducibility of F. Thus Qk = Qk(X) E IC[x] is "t o. If now a, bEe and F(a, b) = 0 = ~~ (a, b), we see from the above equations that Ql(a, b) = 0, then that Q2(a, b) = 0, ... , Qk(a, b) = Qk(a) = O. Since Qk "t 0, the set

{x ECI:Jy E C with

F(x,y) = 0 =

~: (x,y)}

C

{x Eel Qk(X)

= o}

is finite. Before proceeding further, we insert some toplogical preliminaries. All topological spaces we consider will be Hausdorff.

Definition. A continuous map p : X ----+ 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. takes closed sets in X to closed sets in Y.

----+

Y is necessarily closed,

Proof. Let A C X be closed, and Yo E Y. Let K be a compact neighbourhood of Yo in Y. Then p(A) n K = p(A n p-l(K)) is compact (since A is closed and p-l(K) is compact), hence closed in K. Remark. A continuous map p : X ----+ Y between locally compact spaces X, Y is proper, if and only if, for any locally compact topological space Z, the product

p x id z : X x Z

---+

Y x Z, (x,z)

f------+

(p(x),z)

is closed. If X, Y have countable bases, this can be seen by using the following remark: if {Xl, ... ,X n , ... } is a sequence of points in X, without limit points and such that {P(Xn)} n>l converges in Y, then the image of the closed set {(x n , ~) I n ~ I} in X x lR is not closed in Y x R The property in this remark can be used to define proper mappings between spaces which are not locally compact.

Remark. Let p : X ----+ Y be a proper map between locally compact spaces. Let Z C Y be a locally compact space (with the induced topology). Then p-l(Z) : p-l(Z) ----+ Z is again proper.

pi

In fact, a compact subset of Z is a compact subset of Y.

6

1. Algebraic Functions

Lemma 3. Let Cl, ...

Then

,C

n E!C. Let w E C and suppose that w n + Cl w n - l + ... + Cn = O. Iwl < 2maxlcvll / v v

(unless

Cl

= ... =

n = 0).

C

Proof. Let C = max v Icvl l / v that, since Icv I :::; cV ,

> O.

If z =!!!. c' we have zn

Izl n If Izl 2: 2, we would have 1 :::; 1;1 Izl < 2, i.e. Iwl < 2c.

:::;

Izl n - l

+ .£l.zn-l + ... + £n. = 0, c en

so

+ ... + 1 .

+ ... + Izln :::; ! + ... + 2~ < 1, a contradiction.

qx, y], F(x, y) = ao(x)yn + ... + an(x), ao :t. {(x,y) E C 2 I F(x,y) = O} and So = {x E C I ao(x) = O}. Let 7r : V projection (x, y) ....... x. Then 7r 17r- l (C - So) --> C - So is proper.

Proposition 2. Let F E

Thus

o. Let V -->

=

C be the

Proof. Let K C C - So be compact. Then there is {j > 0 so that lao(x)1 2: {j and lav(x)l:::; for x E K. If (x, y) E V, x E 7r- l (K), we have

i

y

n

al(x) n-l an(x) +-y + ... + - =0, ao(x)

ao(x)

so that, by (1.8), Iyl :::; 2max v (j-2/v. Thus 7r- l (K) is bounded. Since clearly (K x q n V is closed in C 2 , 7r- l (K) is compact.

7r- 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: \:Iyo 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 \:Ij E J. The triple (X, Y,p) is then called an (unramified) covering. We also say that X is a covering of Y.

An open set V C Y with the property in the definition is said to be evenly covered by p.

It follows from the definition that the cardinality of p-l(y) is a locally constant function on Y. (With the notation in the definition, the cardinality Ofp-l(y) is that of J\:Iy E V.) Thus, if Y is connected, "the number of points" in p-l(y) is independent of y E Y. 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 said to be isomorphic if there exists a homeomorphism r.p : X' --> X such that po r.p = p'.

Examples. 1) Let .6. = {z E C Ilzl < I} and .6.* = .6. - {O}. Then, if n 2: 1, the map --> .6.* given by Pn (z) = zn is an n-sheeted covering.

Pn : .6.*

7

1. Algebraic Functions

It is a standard fact in the theory of covering spaces that any connected n-sheeted covering of ~ * is isomorphic to Pn. 2) p: C

---+

C*, p(z)

=e

Z

is an infinite covering of C*.

3) Let X, Y be locally compact, let p : X ---+ Y be a local homeomorphism (i.e. \fa EX, :3 an open neighbourhood U of a such that V = p(U) is open in Y and p U is a homeomorphism onto V). 1

Then, p is a finite covering if and only if it is proper.

Proof. If p is a finite covering, if Yo E Y and V is an open neighbourhood of Yo which is evenly covered by p, then p 1 p-1 (V) ---+ V is clearly proper. It follows easily that p is proper. Conversely, let p be a proper local homeomorphism, let Yo E Y and let p -1 (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 Vj = 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 rf- E. Let V = Y - E. Then p-1(V) C U{ u··· u U~, and we have V C V1 n··· n Vn . If we set Uj = Uj np-1(V), then p-1(V) = U~ Uj and Uj is a homeomorphism onto V. 1

pi

Let F E qx,y] be irreducible, F(x,y) = ao(x)yn + ... + an(x). Let 50 = {x E C 1 ao(x) = o} and 51 = {x E C :3y E C with F(x,y) = 0 = ~;(x,y)}. Then, if V = {(x,y) E C 2 1 F(x,y) = o} and 7r: V ---+ C the projection (x,y) f--+ x, then 1

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 50 U 51 and the point at 00 in C to define the algebraic function completely, we shall introduce the notion of a Riemann surface and some related topics.

2. Riemann Surfaces Let X be a 2-dimensional manifold (i.e. X is a Hausdorff space and any point in X has a neighbourhood homeomorphic to an open set in lR?). Consider pairs (U, :p) where U is open in X and :P : U onto an open set in C.

-7

:p(U)

C C

is a homeomorphism

Two such pairs (UI , :pd, (U2, :P2) are said to be (holomorphically) compatible if the map :P2 0 :PIl : :Pl(Ul n U2) -7 :P2(Ul ,nU2) is holomorphic; its inverse is also holomorphic by a standard result in complex analysis. A complex structure on X is a family S of pairs {(U,:p)} which are pairwise compatible and such that UU = X; there is then a unique maximal family of pairs with these two properties and containing S; we shall usually assume that the complex structure is maximal. The elements (U,:p) of this (maximal) complex structure are called charts or coordinate neighbourhoods. In a coordinate neighbourhood, we usually identify U with :p(U) and write z for :P as one does with the usual complex variable in C. A Riemann surface is a connected 2-dimensional manifold X with a complex structure S. We shall 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 n c X is open (X is a Riemann surface) and I : n -7 C is continuous, we say that f is holomorphic if for any chart (U,:p) of X, the function 10 :p-l : :p(n n U) -7 IC is holomorphic. If X, Yare Riemann surfaces, f : X -7 Y a continuous map, f is called holomorphic if, for any chart (V,1/') of Y, the function 1/' 0 f: I-l(V) -71/'(V) C C is holomorphic. Non-constant holomorphic maps between Riemann surfaces are open. Also, a bijective holomorphic map f : X -7 Y has a holomorphic inverse 1- 1 : Y -7 X. Such bijective holomorphic maps are called analytic isomorphisms (or biholomorphic maps). Examples

1. The complex projective line = Riemann sphere. Let pl be the one-point compactification C U {oo} of C. We set Ul = pI - {oo} = C, :Pl : U l -7 C being the identity;

u.2

= pl _ {O}

,

:P2

(z) = { 1/ z if z E C - {O} = C* O'f 1 z=oo.

The map :P2 0 :PIl is the map z f---* 1/ z of C* into itself, so that these two charts define a complex structure on pl. This Riemann surface is called the projective line or the Riemann sphere.

2. Riemann Surfaces

9

2. Tori. Let T E C, Im( T) > O. Let A = {m + m I m, n E Z}. A is an additive subgroup of C. Consider the quotient group X = C/ A and let 7r : C ---7 X be the canonical projection. With the quotient topology, X is a compact Hausdorff space, and C ---7 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+A+JLT A,JL E JR., -~ < A,JL < 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

+H,

As charts, we use pairs (U, cp) obtained as follows: let V be any open set in C such that 7r1V is a homeomorphism onto an open set U in X; set cP = (7r1V)-1 : U ---7 V C C. Two such charts (U1, CPl), (U2 , CP2) are holomorphically compatible: we clearly have 7r(cp2 0 cp;-l(z)) = 7r(z) for z E CPl(U1 n U2 ) thus CP2 0 cp;-l(Z) - z E A Vz E CPl(U1 n U2 ), so must be constant on connected components (because CP2 0 cp;-l is continuous and A is discrete). The Riemann surfaces X constructed above are called tori or elliptic curves. 3. Surfaces of "higher genus". Let g be an integer> 1, and let 0 < r < 1. Let ~ = {z E C II z I < I}. There is a unique bijective holomorphic (= biholomorphic) map T : ~ ---7 ~ such that T(r) = re37ri/2g and T( re 7ri / 2g ) = re27ri/2g. Let (J : ~ ---7 ~ be the rotation z f-+ ze27ri/4g.

For k E Z, we set and denote by

r

the group of biholomorphic maps of

~

generated by A k, Bk(Vk 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: Sur les polyg6nes generateurs de groupes Fuchsiens, 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 ~, and the quotient ~/r is compact. One sees that the canonical projection 7r : ~ ---7 ~/r is a covering map, and obtains a complex structure on ~/r 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 ---7 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 pi U is a homeomorphism onto an open set V in Y such that V C Vj for some j, where {(Vj,1)Jj)jEJ} is the given complex structure on Y. Let CPu : U ---7 C be the map CPu = 1)Jj 0 p. It is easily checked that two such pairs (U, cpu), (U', cpu') 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 plU, a homeomorphism onto V C Y. Then, if pis holomorphic, the map (pIU)-l : V ---7 U is again holomorphic.

2. Riemann Surfaces

10

Consider now a Riemann surface X and a holomorphic map p : X ----> C which is also a local homeomorphism. We consider C as the complement of 00 E pI, and p as a local homeomorphism X ----> pl. We shall define boundary points of X. Let the following properties:

{Xv}./~1

1)

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

2)

{p(xv)} converges to a point a E pI;

be a sequence of points in X with

n

Let DE = {z E C liz - al < c} if a E C, and let DE = {z E C Ilzl > U {oo} if a = 00. Then, for all sufficiently small c > 0, all but finitely many of the {xv} lie in the same connected component of p-I (DE)' 3)

Two such sequences {Xv}, {Yv} are called equivalent if the sequence

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

Zv -

for £or

1/ 1/

odd even

again has the three properties above [i.e. limp(xv) = limp(yv) = a say, and the connected components of p-I (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 an equivalence class of sequences {xv }v~1 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>I' We define neighbourhoods of P in X as follows. Let c > 0 be small and Dc: = {z liz - al < c} (a E q or DE = {z Ilzl > U {oo} (a = 00), where a = limp(xv). Let fiE be the connected component of p-I(Dc:) containing all but finitely many of the Xv, and let Of be the union of fiE with those boundary points Q with the following property: if {Yv}v>1 defines Q, then {1/ I Yv tJ. flf} is finite (this is independent of the sequence {Yv} defi;;ing Q). The OE (c > 0 small) form a fundamental system of neighbourhood of P EX - X.

n

This topology is Hausdorff: if P, Q are boundary points defined by {xv}, {yv} respectively, and P i=- Q, then, by the definition of the equivalence relation, there is c > 0 such that the components flc:,lflE,2 of p-I(D E) containing all but finitely many of the Xv, Yv respectively are distinct, and OE,I n Oc:,2 = 0. Moreover, p clearly extends to a continuous map f5: X ----> pI: f5(P) = a = limp(xv). A boundary point P of X is said to the algebraic if the following holds: let Dc: be a small disc around a = f5(P) and let fI be the connected component of p-I(Dc:) containing all but finitely many points of a sequence defining P; then p(fI) c DE - {a} and the map p : fI ----> DE - {a} is a finite covering. If we set b. R = {z E C Ilzl < R} and b.'R = b. R - {O}, then there is n 2: 1 such that the map p : fI ----> Dc: - {a} is isomorphic to the map Pn : b.;l/n ----> Dc: - {a} given by Pn(z) = a + zn if a E C, Pn(z) = z-n if a = 00 (see Example 1 after Definition (1.10)).

2. Riemann Surfaces

11

n

In this case, 0 = u {P} is a neighbourhood of P in X containing no other boundary points of X. Since pin ~ Dc - {a} is isomorphic to the map Pn defined above, there is a homeomorphism ip : 0 ~ ~El/n with ip(P) = 0 and P 0 ip-l = Pn on ~;'/n' Clearly, ipln is holomorphic. 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 (0, ip) constructed above. Let p = pIX. The pair (X,p) will be called the (algebraic) completion of (X,p). The map p : X ~ jp'l is holomorphic, but does not have to be a local homeomorphism. With the notation above, if P is an algebraic boundary point and n is an n-sheeted covering of Dc - {a} with n > 1, then p is not a local homeomorphism at P. 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.

3. The Sheaf of Germs of Holomorphic Functions Let X be a Riemann surface, and let a E X. We consider pairs (U, f), where U is an open neighbourhood of a and I is a holomorphic function 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 IIW = glW. An equivalence class is called a germ of holomorphic function at a; the class of a pair (U, I) is called the germ of I at a and denoted by -a I . The value at a of -a I is defined by -a I (a) = I(a) for any pair (U, f) defining -a I . If we choose a chart (U, cp) with a E U and cp( a) = 0, and identify functions on subsets

of U with functions on subsets of cp(U), then we can also speak of the values of the derivatives of a germ -a I at a:

I(kl(a) = (.!i)k 1 0 cp-l I _ dz z-O

-a

defining

if (V, f)

is a pair

I .

-a

Let Oa be the set of all germs at a. Oa is clearly a ring, even a I(>algebra. The set ma of germs -a I with -a I (a) = 0 is an ideal; the complement Oa - ma consists of the units of 0 a (f has an inverse in 0 a ¢:::::} -a I (a) 1= 0) so that ma is the unique maximal ideal -a in Oa. If we choose a chart as above, then the map La

f----> L~=o -!trL~nl(a)zn is an isomorphism (as 1(> algebras ) of 0 a with the ring C{ 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 I E Ox· Let now La E Ox, and (U, f) a pair defining the germ La' We set N(U, f) = {,Lx I x E U}, the set of all germs defined by I at the different points of U. We define a topology on Ox by the condition that the sets {N(U, I)} form a fundamental system of neighbourhoods of -a I when (U, f) runs over all pairs defining -a I .

Lemma. With this topology, Ox is Hausdorff and the map p : Ox homeomorphism.

L,

---+

X is a local

L

Proof. Let flb E Ox, and suppose that 1= fl b' We must show that they can be separated by open neighbourhoods, and consider two cases.

L,

1) a 1= b. Let (U,/), (V,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(V, g) = 0.

13

3. The Sheaf of Germs of Holomorphic Functions

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, 1), (U, g) define -a f ,g-a respectively. We claim that N(U, 1)nN(U, g) = in fact, if flx (x E U) is a germ in the intersection, then both f and g induce the 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 g , a contradiction.

o

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

U and

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]).

We can now construct the "Riemann surface of an analytic function" . Let X = C and consider 01(. Let M be a connected component of 01(, and p : M -+ C C WI the restriction to M of the map -a f 1-+ a constructed earlier; p : M -+ C is a local homeomorphism, so there is a unique structure of a Riemann surface on M for which p is holomorphic; p is then a local analytic isomorphism, i.e any a E M 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 M by h(J ) = -a f (a) [if (U, 1) is a pair defining -a f-a with U connected, then N(U,1) c M and h(J ) = f(x) \/x E U, so that h is -x holomorphic]. Intuitively, this "universal" function h describes all the germs that can be obtained by "analytic continuation" of a fixed germ -a f E M. Let now M = M U {algebraic boundary points of M} and phic map extending p : M -+ C we constructed before.

p : M -+ WI

be the holomor-

Let U be a connected open set in C and j, a holomorphic function on U. We assume that (U,1) defines a germ -a j E M for some a E U (and hence for all a E U, since U is connected) . The set E = M - M is a discrete set in M, and we have our universal function h on M. Let Xj be the union of M with those points PEE which are not essential singularities of h (i.e. points where h has either a holomorphic extension or a pole). Thus, there is a meromorphic function h j on Xj with hjlM = h. Let Pj : Xj -+ WI be the restriction of p. The triple (X j, p j, h j) is the Riemann surface of the function j on U. If P E Xj - M, and pj(P) = a, then near P, the map Pj is equivalent to the map z 1-+ a + zn (n 2: 1) or z 1-+ z-n. Thus, if z is a local coordinate at P on X j and w

14

3. The Sheaf of Germs of Holomorphic Functions

denotes a local coordinate at a on pI, the local description of the maps PI, hi can be written:

4. The Riemann Surface of an Algebraic Function Let F(x, y) = ao(x)yn + a1(X)yn-l + ... + an(x) E qx, y] be an irreducible polynomial, let V = {(x, y) E C 2 F (x, y) = o}. Let 50 = {x E Ciao (x) = O}, 51 = {x E C 3y E C with F(x,y) = 0 = ~~(x,y)} and let 5 = 50 U 51 U {oo} C pl. Let 7r: V --+ C be the projection (x, y) f--+ x, V' = V - 7r-l(5) = V - 7r-l(50 U 5d and 7r' = 7rIV'. We have seen that if DE is a small disc around a E 5 (DE = {Izl > :} U {oo} if a = 00), then 7r'I7r -1 ( DE - {a}) --+ DE - {a} is a finite covering (of n-sheets). In particular, 7r- 1 (Dc - {an has only finitely many connected components. Moreover, if W is a connected component of V', then 7r'IW is again a covering, and so maps W onto pI - 5. 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 = 7rIWj --+ pI - 5 is a finite covering, hence every boundary point of Wj is algebraic. Let irj : Wj --+ pI be the algebraic completion of 7rj : Wj --+ pI - 5. If P E Wj - Wj and a = irj(P), there is a neighbourhood U of P and c > 0 such that irjlU --+ DE is isomorphic to the map z f--+ a + zm (or z f--+ z-m) for some m > 0, so that, in particular, irjlU --+ DE is proper. It follows that for any a E S, there is c > 0 so that irj 1ir;- 1 (D E) --+ DE is proper. Since irjlWj --+ pI - 5 is proper, irj : Wj --+ pI is proper, so that Wj is compact.

I

I

Let P2 : V --+ C be the second projection (x, y) f--+ y. Then the function TJ holomorphic on V', so that TJj = TJIWj is holomorphic.

=

P2IV' is

We claim that TJj extends to a meromorphic function on Wj. To see this, let a E 5. Let P E Wj ' irj(P) = a, and choose local coordinates z at P and w at a so that irj becomes the map z f--+ zm = w. If U is a small neighbourhood of P, by the definition of V and TJ, we have, if z f:. 0, n a1(w) n-l() TJ) + aow () TJj z

+

...

an(w) - 0

+ aow () -

,

= 0, so that there exist constants C > near w = O. By Lemma 3, we have ITJj(z)1
0 so that :~~:\

e 'N//v 2 max,., Iwl v

::;

I ::; I,;fN

e £or some constants C1, k. tit

Hence TJj has a meromorphic extension

to Wj. We now claim that V'is connected. If this were not the case, 7r1 : WI r-sheeted covering with 1 ::; r < n.

--+

pI - 5 is an

For x E pI - 5, let b,., (x) (1/ = 1, ... , r) be the 1/ th -elementary symmetric function of Yl, . .. , yr, where the Yj are the values taken by TJ1 at the points of 7fl1(x); by the

16

4. The Riemann Surface of an Algebraic Function

definition of 1'/1, the Yj are values of the second projection P2 at points (x, y) E V, so that F(x,Yj) = 0, j = 1, ... ,T. We claim that bv extend to meromorphic functions on pl. In fact, since the Yj are values of the function 1'/1 which is meromorphic on W1 , in the neighbourhood of a E 5 we have an estimate of the form

::; C1 1x [(P1 , ... , Pr ) =

11 (X )]

1f

al- f '

(resp. C1

Ixl-R'

if

a = (0)

.

Thus, the bv are meromorphic on p1, and are therefore rational functions of x. Let G(x, y) = yr + b1(x)y r- 1 + ... + br(x). Then, if x E p1 - 5, the roots of G(x, y) (viz Y1,'" ,Yr), are also roots of F(x,y). Hence G divides F in C(x)[y], and, since deg y G ~ I, F is not irreducible (Gauss' lemma). Thus V'is connected, and W, the algebraic completion of V', is a compact Riemann surface. W carries a meromorphic function 1'/, and if ft : W -+ p1 is the extension of 1f' : V' -+ p1 - 5, we have

F(ft(P),1'/(P))=O

on

W.

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 V'. This statement is equivalent to the following: Let a E Pl-5, and let Y1,"" Yn be the germs at a satisfying the equation F(x, Yj(x)) = O. Then, for any j, there is a closed curve "y in p1 - 5 starting at a such that analytic continuation of Y1 along "y leads to Yj.

5. Sheaves Let X be a topological space. A presheaf of abelian groups on X consists of the following data. 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) 2) A family (p~)vcu of group homomorphisms p~ : F(U) -+ F(V) whenever U, V are open sets with V c U (called restriction maps) having the following properties: (a) (b)

pg = identity on F(U) V open U; if W eVe U are three open sets, then U v Pw = Pw

0

U

Pv .

If the groups F(U) have additional structure (rings, vector spaces, ([>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) -+ O(V) will be restriction: f f---* f IV.

A presheaf (F(U), p~) will be called a sheaf if the following two conditions are satisfied. Let U C X be open, U = UiEl Ui where the {Ui } are again open (i) If f, 9 E F(U) and pgi (f) = pgi (g) Vi, then

f

= g.

(ii) Given, for each i, an element fi E F(Ui ), if P~:nUj(fi) = P~~nuj(1i) Vi,j E I, then -::Jf E F(U) with pgi (f) = fNi. We shall usually denote p~(f) by of a mapping).

flV

(as if we were actually dealing with restriction

We can associate a sheaf to any presheaf (by the construction used to define germs of holomorphic functions). Let F = (F(U), p~) be a presheaf on X, and let a E X. On the disjoint union U aEU F(U) we introduce an equivalence relation as follows: f E F(U), 9 E F(V) are equivalent if -::J an open set W, a EWe Un V such that flW = glW. The set Fa of equivalence classes is called the stalk of the presheaf F at a. [It is also the direct limit of the directed system (F(U),p~).l We can introduce a topology on IFI = UaEX Fa (disjoint union) by taking as a fundamental system of neighbourhoods

5. Sheaves

18

of -a f E Fa the following sets: let f E F(U), a E U, be a representative of the equivalence class -a f , and let N(U, I) == {f Ix E U}, where -x f is the equivalence class in Fx defined -x by (U, I). IFI does not have to be Hausdorff, but the projection map p : IFI -+ X, p(f) == 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~) is a 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 F(U)

9(U)

Ip~

lr~

F(V)

O!v

---+

9(V)

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

--+

9 is called an isomorphism.

If a : F -+ 9 is a morphism of presheaves, we define the kernel, ker a, of a to be the presheaf {ker( au), p~ I ker( au)} . If F and

9 are sheaves, so is ker(a).

If we define the image im( a) to be the presheaf {im(au),r~

I Im(au)}

,

then, even if F and 9 are sheaves, Im( a) does not have to be a sheaf. Example. Let X == C* == C - {O}, let O(U) = {set of functions holomorphic on U}, O*(U) == { set offunctions holomorphic and nowhere zero on U}. If exp : 0 -+ 0* is the morphism defined by expu : f I---' exp(27ril) (J E OW)), then im( exp) fails to satisfy the second condition in the definition of sheaf. Namely, if U1 = C - {x E lRlx ::; O}, U2 = C - {x E lRlx ?: O}, and we set h(z) == z on U1 , h(z) == z on U2 , then fi E im( expui ) since U1 , U2 are simply connected, but there is no f E im( eXPu1 uu2 ) with flUi = fi (i = 1,2) (the function z has no single valued logarithm on C* = U1 UU2 ). 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 I---' -a f == element of Fa induced by (U, I), a E U. It can be checked directly that if F is a sheaf, then a is an isomorphism. If we start with the sheaf U I---' 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

5. Sheaves

19

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

0 to 101 is an isomorphism,

This is often called also the structure sheaf Ox of X.

Definition. If 0: : F --> g is a morphism between the sheaves F and g, we shall denote by Im( 0:) the sheaf associated to the presheaf {im( o:u), r~ I im( o:u) }. Given morphisms 0: : £ the sequence

-->

F and fJ : F

-->

g between sheaves £, F, g on X, we say that

£...:!!.....FLg is exact (at F) if the sheaves ker(fJ) and Im( 0:) (in the sense just defined) are equal. This amounts to saying the following:

(a) fJu 0 O:u = 0 VU and (b) if f E F(U) and fJu (f) = 0, then, there exists an open covering {UdiEI of U such that flUi E im(o:uJ Vi E I. Let now X be a topological space and F a sheaf of abelian groups on X. Let U = {Ui};EI 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

F(UiO n··· n Uiq )

•

We define the coboundary 8 : CO(U, F) --> C 1 (U, F) by 8((fi)iEI) Cij = filUi n Uj - hlUi n Uj . We also set

Zl(U,F) = {(Cij) E C 1 (U,F) I Cij +Cjk = Cik

on

= (Cij)i,jEl, where

Ui nUj nUk Vi,j,k E I}

(strictly speaking, the condition is that

Cij IUijk

+ cjkl Uijk = CiklUijk,

Finally, let Bl(U,F) = Image(8 : CO(U,F)

Zl(U,F).

Uijk = Ui -->

n Uj n Uk

.)

C 1 (U,F)); we have Bl(U,F)

C

We call the quotient group

the first cohomology group of F relative to U.

We also set

HO(U,F) = {(fi)iEI

E

CO(U,F) 18(fi)iEl = o} ;

by the sheaf axioms, the map F(X) --> CO(U, F) defined by f 1--+ (fIUi)iEI induces an isomorphism of F(X) onto HO(U, F) for any opell covering U. Elements of F(X) = HO (U, F) are also called (global) sections of :F.

20

5. Sheaves

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

T*: Hl(U,F)

--+

----+

I such

Hl(V,F)

as follows. If ~ = (cijkjEI E Zl(U,F), we define T*(~) = (ra{3)a,B by 'Ya{3 = r (a)r({3)IVa n V{3. Clearly T*(Bl(U,F)) C Bl(V,F) so that it induces a map (denoted again T*) of Hl(U,F) to Hl(V,F). C

Proposition 1. If T, (5 : A A), then the induced maps

----+

I are two refinement maps (i.e. Va

C

Urea) n Ua(a) Va

E

are equal. Proof. If (iij kj E I E

fr(a)r({3) - fa(a)a(,B)

Zl (U,

F), we have, on Va

= (Jr(a)a(a) + fa(a)r({3»)

n V{3, - (Ja(a)r({3)

+ fr({3)a({3») = ga -

g(3 ,

where ga = fr(a)a(a) I Va· Hence {fr(a)r«(3) - fa(a)a({3)} E Bl(V, F). Proposition 2. If V is a refinement of U, then, the induced map

is injective. Proof. Let T : A ----+ I be a refinement map, and let ~ = {(iij )i,jEI} E Zl (U, F), and suppose that T*(~) E B1(V,F). Thus, 3ga E F(Va ) with fr(a)r({3) I Va n V{3 = ga IVa n V{3 - g{31 Va n V(3. Let i E I and x E Ui. Choose a E A so that x E Va, and define hi(x) = ga(x) + fir(a)(X). If f3 E A is such that x E V{3, we have

because

fir(a) (x) - fir(,B) (x) = -(Jr(a)i(X)

+ fir(,B) (x))

= - fr(a)r({3) (x)

(since ~ E Zl(U, F)) .

Hence, the above formula defines hi E F(Ui)' If x E Ui n Uj , and we choose a with x E V"" we have

+ fir(a)(X) - ga(x) - fjr(a)(x) = fir(",) (x) + fr(a)j(x) = fij(X) .

hi(x) - hj(x) = g",(x)

21

5. Sheaves Thus 8{(h i )} =~, and ~ E BI(U, F). This proves Proposition 2.

We now define the cohomology group HI(X,F). Let U, V be open coverings of X, U = {UiLEI, V = {V}EA, V a refinement of U. Then, there is a map T(U, V) : HI(U,F) -+ HI(V,F) (defined using a refinement map T: A -+ I, but independent of the choice ofthis map). If W is a refinement of V, we have T(U, W) = T(V, W) OT(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 llu HI (U, F) defined by: ~ E HI(U,F) is equivalent to 7] E Hl(V,F) if there is an open covering W which is a refinement of both U and of V and such that T(U, W)~

= T(V, W)7] .

Then, Hl(X,F) = llHl(U,F)jR. For any U, there is a map T(U): Hl(U,F) ~ under R).

-+

Hl(X,F) (T(U)~

= equivalence class of

Proposition 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 = {Ui}iEI be an open covering of X. Suppose that H1(Ui,F) = 0 'Vi. Then, the natural map is an isomorphism. Proof. It is sufficient (because of Proposition 2), to show that for any refinement V = {V}EA of U, the induced map T* : HI(U,F) -+ HI(V,F) is surjective; here T: A -+ I is a map with V c UT(O

hEIIt

1 c

f (z

+ h + w) h

f (z

+ w) -wAw= 1d dw

1

ofz(+ w)-1w d Aw d-

c ox

W

since, f having compact support, limh--->o f(z+h2- f (z) = ¥X(z), uniformly and boundedly on C. We have only to iterate this argument. If E > 0,

-OU = - 1 .hm oz

21fi 10--->0

J

-of (z OZ

+ w) -1 dw A dww

.

'

Iwl~c

now

J

Iwl2:c

J J

0 (f(z + w)) ow w dwAdw

of 1 oz(z+w)wdWAdw=

Iwl2:c = -

Iwl2: c

d( f(z +wW)dW)

23

5. Sheaves and by Stokes' theorem, this

=

J

f(z+w)d w W

IWI=E

= 27rl'f() Z

+

J

f(z

+ W2 -

f(z) dw .

Iwl=E

Since jCz+w2~ jCz) is bounded as a function of w, this last integral the result follows.

---->

0 as

E ---->

0, and

o

If now f E C OO (f1), if we apply this special case to the function ipf where ip E C (f1) and = 1 on a given compact set K C f1, we obtain the following: If f E COO(f1) and K C f1 is compact, there is U E COO(f1) such that ~~ = f on K.

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]. Let f1 be open in 1 be a sequence of compact sets in f1 with Kn C Kn+! (interior of K n+ 1), U K n = f1 a;d such that f1 - K n has no connected component relatively compact in f1. E COO(f1) and let Un E COO(f1) be such that ~ = f on a neighbourhood of Kn. Then Un+! - Un is holomorphic on a neighbourhood of K n , so that there is hn' holomorphic on f1 so that Iu n+! - Un - hnl < 2~n on Kn (n :::: 1). Let

f

Define U = Un + L:m>n(Um+1 - Um - h m ) - hI - ... - hn~1 on Kn; the series converges uniformly on K n' We have U = Un

+ (Un+!

- Un - h n ) +

L

= Un+! +

L

(U m +! - Um - h m ) - hI - ... - hn~1

m:;,n+1

(u m +! - Um - h m ) - hI - ... - h n ,

m:;,n+1

so that this defines a function on f1. Since L:m>n+! (u m +! - Urn - hrn) is holomorphic n ±1 " on Ko n+l:::) K n an d aU0:2 - f on K n, we h ave au 0:2 - f on K n '-" vn, I.e. on H.

Proof of Mittag-LeIDer's theorem. We shall prove that Hl(U, 0) = 0 for any open covering U of f1 (f1 open in .. E O(Ue)' The theorem is proved.

9. Weyl's Lemma and the Serre Duality Theorem

47

Let X be a compact Riemann surface, 7r : E --+ X a holomorphic vector bundle on X. We define a bilinear form ( , ) E : HO(X, E* 12) Kx) X Hl(X, E) --+ C as follows. Consider the bilinear form ( , ) : HO(X, E* 12) Kx) X A~l (X) --+ C defined above. We have seen that (s,8f) = 0 if f E CE(X), s E HO(X, E*I2)K x ), hence this form induces a bilinear form HO(X, E* 12)Kx) x [A~l / 8C E (X)] --+ C, which we denote again by ( , ). If we denote by D the Dolbeault isomorphism D : Hl(X, E) --+ A~l(X) / 8CE (X), we define ( , )E by

This bilinear form induces a map

[V* is the dual of the vector space V], viz 6. E(s)

is the linear map

~

f------+

(S,OE

from

HI(X,E) to C.

SERRE'S DUALITY THEOREM. The map 6. E : HO(X, E* 12) Kx) --+ HI (X, E)* is an isomorphism for any holomorphic vector bundle on the compact Riemann surface X.

e:

01 (X) / Proof. Let Ai fJCE(X) --+ C be any C-linear map. We have seen that 8CE (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) --+ C given by F(cp) = e({cp}) , {cp} = image of cp in A~l(X) / 8CE (X), is continuous for the Coo topology, and is zero on 8CE!(X). By Weyl's lemma, there is s E HO(X, E* 12) Kx) such that

Since D is an isomorphism this proves that 6. E is surjective. To prove that 6. E is injective, given s E HO(X, E* 12) K x ), we must show that if (s,cp) = 0 Vcp E A~l(X), then s == O. This is immediate (the remark starting the proof of Weyl's lemma). In view of the fact that HO(X, E) and Hl(X, E) are finite dimensional, the Serre duality theorem can be stated as follows: Alternate form of the duality theorem. The bilinear form

48

9. Weyl's Lemma and the Serre Duality Theorem

is non-degenerate (or perfect). If F is a holomorphic vector bundle, and we take E = F* Q9 Kx, then E* Q9 Kx F Q9 K'X Q9 Kx = F (since, for any line bundle L, L* Q9 L is canonically trivial). Hence HO(X, F) is isomorphic to the dual of Hl(X, F* Q9 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 Hl(X, E) = Hl(X, 0)]. Proposition. Let 'P be a form of type (0, 1) on the compact Riemann surface X. Then, 3f E COO(X) such that 8f = 'P if and only if, for every holomorphic 1-form w on X, we have

10. The Riemann-Roch Theorem and some Applications Throughout this section, X will be a compact Riemann surface. We begin with some definitions. Let D = 2:~=1 niPi be a divisor on X. The integer d D [and written degD].

= 2:~=1 ni

is called the degree of

Let W be a meromorphic I-form, w "¥=- O. 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 z(a) = 0, and w = f dz, then resa(w) = residue of f at a = the coefficient of ~ in the Laurent expansion 2:~N cvz v of f. It is independent of the coordinate system. In fact, if I is a piecewise differentiable curve in U - {a} whose winding number (= index) at a is +1, then resa(w) = 2;i J-yw. Lemma 1. Let w be a meromorphic i-form, w "¥=- 0, on X. We have 2:aEX resa(w) = O. Proof. Let al, ... , am be the poles of w. Choose coordinate neighbourhoods (Uj , Zj) about aj (zj(aj) = 0) and let 6. j = {x E U I IZj(x)1 < c} (c > 0 small). Let U = X - UT=l 6. j . By Stokes' theorem, we have

l

dw = -

:L Jw = - 2;ri :L resaj (w) . J

aDoj

J

But, a holomorphic I-form on a Riemann surface is closed: if w dz + ~ d:z 1\ dz = 0 . Corollary 1. Let f be a meromorphic function, divisor (J) of f is zero.

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

f

"¥=-

= f dz,

dw

= %f dz 1\

0, on X. Then the degree of the

1df.

Corollary 2. If Dl and D2 are divisors on X and if Dl is linearly equivalent to D 2, then deg(Dd = deg(D2)'

In fact, if Dl - D2

= (J),

we have deg(Dd - deg(D2)

= deg(J).

Let D be a divisor on X; we introduced the sheaf OD: U f-+ OD(U) = {f meromorphic on U I (J) 2': - D on U}. 0 D is isomorphic to the sheaf U f-+ 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 (convergent) Laurent series

f

=

:L

n2-D(a)

cnz n

((U, z) a local coordinate with z(a) = 0) .

50

10. The

Riemann~Roch

Theorem and some Applications

Let D I , D2 be divisors with DI S D2. Clearly OD l (U) C OD 2 (U) for all U open in X and we obtain an injective morphism of sheaves OD l -+ OD2' [In terms of the line bundles L(Dd, L(D 2) and the isomorphism L(D I ) 0 L(D2 - D I ) ~ L(D 2), the map is given by f E HO(U,L(D 1 )), f 1---+ f 0 SDz-Dl' where SD 2 -D l is the standard section of L(D2 - D I ) with (SD 2 -DJ = D2 - Dd Let sE~ be the sheaf associated to the presheaf U 1---+ OD2(U)/ODl(U), We have (SE:t = 0 if DI(a) = D2(a) (in particular if a is not in supp( D I ) U supp( D2)' If DI (a) < D2 (a), then (SE:) a is a finite dimensional vector space, isomorphic to the vector space tL:-D 2 (a):Sn 0 is small, and i is such that ak E Ui , we have

27ri res ak (w) and limc~o

J 0i = 0 (since Oi) is smooth).

&tl.k,e

&tl. k .e

Thus, Lkresak(w) depends only on the cohomology class ~ of {Wij} in HI(X,n) (and not on its representation as a coboundary of nm); we denote this by Res(~). Since HO(X, 0) = HO(X, K'X C?9 Kx) = c., the pairing in the Serre duality theorem for E = K x is given by (A, {Wij }) f--+ A . D ({Wij }). Thus we have

J

Proposition 1. The map Res: HI(X, n)

--+

IC is an isomorphism.

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,n_ D) x HI(X,OD)--+ HI(X, n) defined by (w, {fij}) f--+ {fijw}. [Here, W is a meromorphic 1-form with (w) ::::: D and fij is meromorphic on Ui n Uj with (Iij) ::::: - D, so that J;jW is a holomorphic 1-form on Ui n Uj].

The duality pairing ( , )(D) in the Serre duality theorem equals 27riRes( , )D; in particular, the bilinear form HO(X,n_ D) x HI(X,OD) (w,~)

f---+

--+

Res((w,OD)

IC

54

10. The Riemann-Roch Theorem and some Applications

is non-degenerate.

As a consequence, we have the following analogue of Mittag-LefRer's theorem for compact Riemann surfaces.

Theorem. Let {UdiEI be an open covering of the compact Riemann surface X, and let j; be meromorphic on Ui · Suppose that fi - li is holomorphic on Ui n Uj . There exists a meromorphic function f on X with f - fi holomorphic on Uj Vi if and only if, for any holomorphic 1-form W on X, we have

res(wo) = 0 , where Wo E CO(U, !1 m

)

is the O-cochain (JiW).

[The condition is that if Va E X we choose i(a) with a E Ui , then L,aEX res a (Ji(a)W) =

0.]

Proof. The existence of f is equivalent to saying that the cohomology class ~ = {ji -lilUi n Uj } is = 0 in H 1 (X,O); by the above theorem, this is equivalent to the residue condition. The corresponding theorem for merom orphic forms (rather than functions) is the following. MITTAG-LEFFLER'S THEOREM for forms on a compact Riemann surface. Let U = {Ui};EI be an open covering of X, and let Wi be a merom orphic I-form on Ui such that Wi - Wj is holomorphic on Ui n Uj . Let w = {Wi};E1, 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 Wi - Wj is holomorphic on Ui n Uj.] Then, there exists a meromorphic I-form W on X with W - Wi holomorphic on Ui Vi if and only if a

Proof. The existence of W is equivalent to saying that the cohomology class of {Wij} in HI (X,!1) is zero; since, by Proposition 1, the map Res: HI (X,!1) -> C is injective, this is equivalent to saying that L,'; res a (w) = O. 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 S D be the standard section of L(D) with (SD) = D. Consider the exact sequence of sheaves 0-> !1 ~ !1D -+ CD -+ O. The sheaf CD is zero outside suppeD); if a E suppeD), the

10. The

Riemann~Roch

Theorem and some Applications

55

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

HO(X, 0,D )----4HO(X, CD )----4Hl(X, 0,)----4Hl(X, H D) ; since dim Hl(X, 0,) = 1 and H1(X,DD) ~ HO(X,a~D)* = 0, it follows that image HO(X,DD) has co dimension 1 in HO(X,C D) ~ cdeg(D). But, since if w is a global 1 form, we have La res a(w) = 0, the image lies in the set

such that La ESllpp(D) resa(w a ) = O. Since both spaces have co dimension I, they are equal. The vanishing theorem Hl(X, aD) = 0 = HO(x, D~D) if deg D > 2g - 2 and the Riemann~ Roeh theorem show that h (D) = deg D + 1 - g is determined by the degree of D if it is large.

°

The integer i(D)

= hl(D) = hO(K -

D) is called the index of speciality of D. We give some further applications of these results.

Proposition 2. If D is a divisor on X with degD > 2g - 1, then, ' f @ 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.

Proposition 3. Let L be a holomorphic line bundle on X with degL

(a) if P,Q E X, P f::. Q, ~s E HO(X,L) such that s(P) = 0, s(Q) (b) if P E X, ~s E HO(X, L) such that ordp(s) + 1.

> 2g. Then

f::. 0

Proof. We consider the bundle L @ L( -P); since any line bundle is ~ L(D) for some divisor, Prop. 2 implies that ~s' E HO(X,L@L(-P)) with s'(Q) f::. O. Let s = s'@sp, where Sp is the standard section of L(P). Then, if P f::. Q, we have s(Q) f::. 0, s(P) = O. 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 N = hO(L) - 1 = degD - g. We define a holomorphic map C{JL : X -+ JlDN as follows.

56

10. The Riemann-Roch Theorem and some Applications

°

Let so, ... , SN be a basis of HO(X, L); if a E X choose a neighbourhood U of a and cr E HO(U, L) with cr(x) f. Vx E U. We set i.pdx) = point in projective space JlDN with homogeneous coordinates s::(~x/). Note that ~ is a holomorphic function on U; the point in JlDN is independent of cr for if cr' is another such section and cr' = hcr where h is a holomorphic function on U, h(x) f. OVx, then ~ = h;;; i.pL is, to start with, defined only outside the common zeros of So, ... ,SN, but, by Prop. 2, these sections cannot have common zeros.

(s;g} : ... :

Note. If L is a holomorphic line bundle on X and HO(X, L) f. 0, it defines a holomorphic map i.pL: X --+ JlDN (N = dimHO(X,L) -1) on all of X. If So, ... ,SN is a basis as above and A = {x E X I S j (x) = 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 z( a) = 0, i.p L I U - {a} is the point in JlDN with homogeneous coordinates (10 : ... : fN) where fo, .. ·, fN are holomorphic functions on U and are not outside a. We can write fj = Zk gj where k = minj ord a (1j). i.p L on U is then given by the point in JlDN with homogeneous coordinates (go: ... : gN).

°

°

This construction only works because dim X = 1; in higher dimension, points of indeterminacy of i.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 Theorem. If deg L in JlDN.

> 2g, then i.p L is an imbedding of X

Proof. (i) i.pL is injective. Let P,Q E X, P f. Q. Choose S E HO(X,L) with s(P) = 0, s(Q) f. 0; if S = L~ CvS v , then i.pdP) lies on the hyperplane LCvzv = 0, i.pL(Q) lies outside this hyperplane.

°: :;

(ii) The tangent map of i.pL is injective. Given P E X, choose S E HO(X,L) such that ordp(s) = 1, and let k, k :::; N, be such that Sk(P) f. 0, and let Co, ... , CN E C be such that S = L CvS v . Then

and the functions fv = ~, v f. k, give the inhomogeneous coordinates of i.pL(X) for x near P: i.pdx) = (Jo(x), ... ,l,fk+l(X), ... ,fN(X)). Since ordp(:J = 1, we have dfv(P) f. for at least one v f. k.

°

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

10. The Riemann-Roch Theorem and some Applications

57

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 i.pL®= is an imbedding of X in lP'N, N + 1 = hO(L®m). It is called very ample if i.p L is already an imbedding. We have seen that if deg(L) > 2g, then L is very ample. Hence, if deg(L) > 0, L is ample, since deg(L®m) = mdeg(L). Conversely, if L is ample, then L®m c:= L(D) for an effective divisor D (some m > 0) since L®m must have at least one holomorphic section =I=- O. Moreover, D =I- 0 (since then L®m is trivial, and cannot imbed X). Thus mdeg(L) = deg(L®m) = degD > O. Thus, we have

Proposition 4. A holomorphic line bundle L on X is ample if and only if its degree is > O.

11. Further Properties of Compact Riemann Surfaces Let X, Y be Riemann surfaces and f : X -+ Y a non-constant holomorphic map. If a E X, b = f(a) and w is a local coordinate at b with w(b) = 0, we set orda(f) = ord a (w 0 j). The integer b( a, j) = ord a (f) - 1 is called the ramification index of f at a; f is a local homeomorphism at a if and only if b( a, j) = O. Let now X, Y be compact Riemann surfaces, and let f : X -+ Y be a non-constant holomorphic map. We denote by gx, gy the genera of X, Y respectively. Let b = LaEX b(a, j); b is called the (total) ramification index of f. Let C be the set of critical points of f, i.e. C = {a E Xlb(a,j) > O} and B = f(C) the set of critical values (sometimes called the branching locus). Let w =f:. 0 be a meromorphic I-form on Y and let Wo

= f*(w);

we have deg(wo)

= 2gx-2.

If a EX, b = f(a) and we choose local coordinates z at a and w at b (z(a) = 0 = w(b)) so that near a, the map f is given by z 1-+ zn = w, then n = orda(f). If w = h(w)dw near b, then Wo = f*(w) = h(zn)nzn- 1dz near a, so that orda(wo) = nordb(w)

+n

-

1, where

n =

orda(f) .

If d is the number of sheets of f (= degree of f), this gives (summing first over a E f- 1(b) and then over b)

deg(wo)

=L

( L

orda(f) )ordb(w)

L

+

bEY aEf-l(b)

(orda(f) -

1)

bEY,aEf-1(b)

= ddeg(w) + b . Since deg(w)

= 2gy

- 2 and deg(wo)

= 2g x

- 2, this gives:

The Riemann-Hurwitz Formula. With the above notation, 2g x - 2 = d(2gy - 2)

+b ;

in particular, if there is a non-constant holomorphic map X -+ Y, then gx ::::: gy; if gx = gy ::::: 1, we must have b = 0 and d = 1 unless gx = gy = l. With the same notation as above, let a1,"" aT be the points of C, let bj = f(aj). We denote by X(X), X(Y) the topological Euler characteristic of X, Y, so that, e.g.

x(X) = dime HO(X, q b1(X)

-

dime H1(X, q

+ dime H2(X, q = 2 - b1(X),

= dime H1(X, q = 1st Betti number of X

.

11. Further Properties of Compact Riemann Surfaces

59

We triangulate Y by simplices in which all points of B = f (C) are vertices, and assume that the simplices are sufficiently small. We can then lift the triangulation by f to a triangulation of X. If we denote by eo (X) the number of vertices, by e1 (X) the number of edges (= I-simplices) and by eo(X) the number of faces (= 2-simplices) in the triangulation of X, with similar notation for the triangulation of Y, we have e2(X) = de2(Y), e1(X) = de1(Y), eo(X) = deo(Y) - b [if ai E C, then each edge at bi = f(ai) lifts to b(ai' 1) + 1 edges all ending in the same vertex ai; the cardinality of f-1(B) = d (cardinality of B) -b]. Thus, we have

2-h(X)=d(2-b 1(Y)) -b. If we take Y = p1, there exists a non-constant holomorphic map f : X --+ p1 (= nonconstant meromorphic function on X). Moreover, we have gi¥'" = 0, b1(p1) = O. If we denote by d the degree of f, we have

2gx- 2 =-2d+b and

2 - b1 (X) = 2d - b = -(2gx - 2) .

Thus, we have

2g x =b1 (X); in particular, the genus gx = dim H1 (X, 0) = dim HO(X, 0) is a topological invariant of X. We pass now to a discussion of Weierstrass points. Let X be a compact Riemann surface of genus 9 = dim H1(X, 0). We have seen (Th.3 in §7 and the remark following) that if P EX, there is f meromorphic on X, holomorphic on X - P (but not at P) with a pole of order ::; g + 1 at P. 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 Weierstrass point if there is a meromorphic function f on X and constants co,···, Cg-1, not all zero, such that (i) (ii)

fiX - {a} is holomorphic f - L~:6 Z~+l is holomorphic at P.

According to the analogue of the Mittag-LefRer theorem given in §1O, this is the case if and only if the following holds: There exist Co, ... ,Cg-1 not all zero such that g-l

resp

(2: z~:l w) = 0 1/=0

60

11. Further Properties of Compact Riemann Surfaces

Let WI, ... , w9 be a basis of HO(X, 0), and let

Wk = ikdz on U,ik E O(U).

9- 1

resp

(I: z~:1 Wk) = COik,O + clik,1 + ... + c9-dk,9-1 v=O

Thus we have: P is a Weierstrass point if and only if the system of 9 linear equations 9- 1

I: CvikV)(0) =0,

k=I, ... ,g

v=O

has a solution (co, ... , cg-d i=- (0, ... ,0); this is the case if and only if det [(Jk V)(0)) ISkS9[ =

°SV 2, we have 2g+2 < (g-I)· 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. Lei X be a non-hyperellipiic compact Riemann surface of genus g(? 3). Then, the canonical bundle Kx is very ample, i.e. global sections of Kx have no common zeros and CPK x : X -+ JlDg- I is an imbedding. Proof. 1) Given P E X, :lw E HO(X, 0,) with w(P) #- 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) ~ HO(X, 0,). Now, hO(O,_p) - hI(O,_p) = 1 - 9 + (2g 3) = 9 - 2 and hI(O,_p) = hO(Op) = 1 (since, if there exists a non-constant f with (f) ? -P, f has a single simple pole and f : X -+ JlDI is an isomorphism). Hence hO(O,_p) = 9 - 1 < hO(O,), so that HO(X, O,_p) cannot be isomorphic to HO(X, 0,).

2) Given P, Q E X, P #- Q, :lw E HO(X,o') with w(P) = 0, w(Q) #- O. If not, the map HO(X,o'_P_Q) ~ HO(X,o'_p) is an isomorphism. We have hO(O,_p_Q) = 1 - g + (2g - 4) + hI(o'_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 f is of degree 2 and X is hyperelliptic. Hence hO(Op+Q) = 1, and hO(o'_p_Q) = 9 - 2 < hO(O,_p). 3) Given P E X, :lw E HO(X, 0,) with ordp(w) = 1. If not, we have w(P) = 0 = } ordp(w) ? 2, i.e. hO(O,_p) = hO(0'-2P). As in 2) above, this implies the existence of a non-constant f with (f) ? - 2P and X would be hyperelliptic. The theorem follows from these three statements as in the imbedding theorem in §1O. The image of any compact Riemann surface X under cp Kx is called the canonical curve of X. If X is hyperelliptic, the canonical curve is isomorphic to JlDI; otherwise, it is isomorphic to X.

13. Some Geometry of Curves in Projective Space 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 (co dimension 1), A defines a holomorphic line bundle as on a Riemann surface: if {Ud is an open covering of M, fi E O(Ui ) is such that Ui n A = {x E Uil!i(X) = 0, dfi =I- 0 at any point of Ud, then gij = fdiJ is holomorphic, nowhere zero on Ui n Uj and form the transition functions for a line bundle L(A). The family {Jd define the standard section SA of L(A) (whose divisor is A). Consider now M = JP'n, with homogeneous coordinates (zo, ... , zn). A hyperplane H (linear subspace of co dimension 1) is given by {e(z) = O}, where e is a non-zero linear form in zo, ... , Zn; we shall denote the corresponding line bundle also by H [or OlP'n (1) or O( 1)]; two hyperplanes define isom~rphic bundles. If Uv = {( Zo : .•. : zn) I Zv =I- O}, 1/ = O, ... ,n, the functions (Bl., ... ,.£.c, ... ,"""-) form local coordinates on Uv , in fact, Zv Zv Zv an isomorphism onto (the hat over a term means it is omitted). If H is defined by {e( z) = O}, the functions fj = define H n Uj (j = 0, ... , n) and the transition J functions for H on Ui n Uj are given by gij = The set of all hyperplanes forms the "dual" projective space (JP'n)*, the coefficients of eforming homogeneous coordinates for

en

¥.

¥..

(JP'n)*.

Let X c JlDn be a connected complex sub manifold of dimension 1 (= smooth imbedded projective algebraic curve). We set deg(X) = deg(OlP'n(l)IX); it is called the degree of the curve X. If SH is the standard section with divisor the hyperplane H, then OlP'n(l)IX = l:.npP is the divisor of sHIX and deg(X) = l:.np. If X n H is transverse at every point, then n p = 1't:/P 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 meets X transversally; we shall prove it in the special case we need. Proposition 1.* (Special case of Bertini's theorem). The set of H E (JP'n)* such that H meets X transversally is an open dense set in (JP'n)*. Proof. Let a E X and U be a small open set given by a biholomorphic map cP : Ll ---+ U, = (CPo,···, 1, ... , CPn), Ll = {t E q It I < I} and CPo,···, CPn are holomorphic functions

cP

* 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.

67

13. Some Geometry of Curves in Projective Space

with 0, i = 1,2). Then dim IDII

+ dim ID21

::; dim IDI

+ D21

Moreover, if equality holds, then any DE IDI + D21 (i.e. D 2': 0, D '" Dl written D = D~ + D~ with D: E IDil, i = 1,2,.

+ D 2)

can be

Proof. If ri = dimlDil and Pl, ... ,Pr1 , Ql, ... ,Q r 2 are any points in X, there is D: '" Di , D: 2': 0 with Pi E supp(Di), Qj E supp(D~). Then D~ + D~ E IDI + D21 and contains all (rl + r2) points Pi, Qj in its support; hence the inequality. The divisors D~ + D~ with D; 2': 0, D; '" Di form an (rl + r2)-dimensional subvariety of the projective space IDI +D21 = JFD(HO(X,L(DI + D2)))' If equality holds, this subvariety must be the whole projective space. We come now to an important theorem. We denote by K a canonical divisor on X. CLIFFORD'S THEOREM. Let D be an effective special divisor on X (so that hO(K - D) > 0). Let d be the degree of D. Then dim IDI ::;

~d = ~ degD.

Moreover, if equality holds, then D must be 0, or D '" K, or X must be hyperelliptic.

Proof. Since K - D is linearly equivalent to an effective divisor, we have dim IDI

+ dim IK - DI ::; dim IKI

,

13. Some Geometry of Curves in Projective Space

71

i.e. Moreover, by the Riemann-Roch theorem

Adding, we have 2hO(D) :::; d + 2, hO(D) :::; ~d + 1. Moreover, if equality holds, then dim IDI + dim IK - DI = dim WI so that any divisor K' :2: 0, K ~ K', can be written K' = Dl + D 2, Di :2: 0 with Dl ~ D, D2 ~ K - D. Assume that X is not hyperelliptic, and consider X C lP'g-l imbedded by the canonical map. If H is any hyperplane transverse to X, then the points of H n X give us a divisor K' ~ K, K' :2: 0, and we can write K' = Dl + D2 with Dl ~ D, D2 ~ K - D, D 1 , D2 :2: o. Assume also that D 1 , D2 -j. O. If [Dil is the linear subspace of lP'g-l generated by Di , we have (by the geometric form of the Riemann-Roch theorem) dim[Dll = degDl - hO(D 1 ) = d - hO(D) dim[D2l = degD2 - hO(D 2) = 2g - 2 - d - hO(K - D) . Since the assumption of equality dim IDI = ~d implies that hO(D) + hO(K - D) this gives dim[Dll + dim[D2l = 2g - 2 - (g + 1) = g - 3.

= g + 1,

Hence both Dl and D2 span linear subspaces of dimension:::; g - 3. If d :2: g - 1, the points of Dl are linearly dependent, if d < g-l, 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 Dl or D2 = 0, and the theorem is proved. We now give another proof of Chifford's theorem not using Castelnuovo's general position theorem. We have only to prove the statement about equality. Proposition 3. Let D :2: 0 be an effective divisor of degree d. Assume that 0 :::; d :::; 2g - 2. Then dim IDI :::; ~d. If D -j. 0 and D f K, and if equality holds, then X is hyperelliptic. Proof. We have hO(D) - hO(K - D) = 1- g + d :::; -~d + d (since g -1 :2: ~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 consequence 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) :::; g + 1.

Assume that D is special and that hO(D)+hO(K -D) = g+l (i.e. that hO(D) = ~d+1).

13. Some Geometry of Curves in Projective Space

72

= 2, then hOeD) = 2 and there is a non-constant meromorphic function f with (1) 2: - D and f is of degree 2 so that X is hyperelliptic. We shall show that if deg D > 2 and K f D, then there is a divisor Do 2: 0 with deg Do < d such that hO(Do)+hO(K -Do) = g+l. Since degDo < d ~ 2g-2 = degK, we have K -Do f 0 and we can continue till we obtain a divisor D' with deg D' = 2, hOeD') = 2, so that X is hyperelliptic.

If d

Let D' 2: 0, D' '" K -D. Then D' i:. O. Choose points P E suppeD'), and Q tf. suppeD'). 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 = La D(a)a, D' = La D'(a)a, then Do = Lamin(D(a),D'(a)). a. Clearly, Do(P) > 0, Do(Q) = 0 so that degDo < deg D and Do i:. O. We have the following exact sequence of sheaves:

where a(h) = (h, -h) and (3(j,g) = f + g. The exactness is seen as follows. If (h) 2: -Do, we have both (h) 2: -D and (h) 2: -D'. If (j) 2: -D, (g) 2: -D', we have orda(j + 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(j) 2: - max( D(a), D' (a)) (j a germ of meromorphic function at a), then either (j) 2: -D(a) or (1) 2: -D'(a); in the first case, f = (3(j, 0), in the second, f = (3(0, f)· If (3(j, g) = 0, then f = -g and (j) 2: -D, (j) = (g) 2: -D' so that (j) 2: -Do and (j,g) = a(j). Thus (*) is exact. It follows from the exact cohomology sequence that

hOeD)

+ hOeD')

~ hO(D o) + hoeD

+ D' - Do)

=

hO(D o) + hOCK - Do) .

Since D' '" K - D, this gives

g + 1 = hOeD)

+ hOCK - D)

~ hO(D o) + hOCK - Do) ~ g + 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 to Clifford's theorem. Let X c lpm be a compact Riemann surface of degree d < 2n and suppose that X is non-degenerate. Then g ~ d - n (g being the genus of X), and if equality holds, then the hyperplane sections form a complete linear system. Proof. Let H be a hyperplane in lpm and D = X n H. Then hOeD) 2: n + 1 (any linear form on lPn gives a section of 0 D, and no linear form vanishes on X unless it is zero since X is non-degenerate). Since d < 2n, we have dim IDI 2: n

>

~d ,

13. Some Geometry of Curves in Projective Space so that D cannot be special, i.e. hOCK - D)

n

+ 1:::; hOeD) =

= O.

1- 9

73

Hence

+ d, g:::;

d- n .

Equality implies that hOeD) = n + 1, i.e. restriction to X of linear forms on lP'n 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 lP'n is rational, i.e. 9

= O.

In fact, it can be shown that the only such curve is the closure of the image of C under the map z f---+ (1 : z : z2 : ... : zn) 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 d > > n in lP'n. Let X c lP'n be a non-degenerate imbedding of a compact Riemann surface in lP'n; let d be the degree of X. Then, as we have seen d :::: n. Let N = [~=il (integral part). Let D be the divisor on X given by a general hyperplane section X n H. Lemma 4. 1) Let 1 :::; k :::; N. Then hO(kD) - hO ((k -1)D) :::: 1 + ken -1). Moreover, if equality holds for a certain value of k, then HO(X, OkD)/ HO(X, O(k-l)D) is generated

by HO(X,D), i.e., the natural map SymkHO(X,OD)

->

HO(X,OkD)/HO(X,O(k_l)D)

is s'urjeciive.

2) If k > N, we have hO(kD) - hO((k - I)D) = d, and lIO(X,OD) generates HO(X, OkD)/ HO(X, O(k-l)D). Proof. We suppose 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 ken - 1) :::; d - 1, 1 + ken - 1) :::; d. Choose a set E of 1 + ken - 1) points of D. If PEE, write E - {P} = El U ... U Ek where each E j has n - 1 points. By Castelnuovo's general position theorem, the points of E j (j = 1, ... , k) generate a plane B j of dimension n - 2 which does not contain P. Hence there is a hyperplane H j with P ¢ H j , E j C H j , so that there is a linear form Aj on lP'n with Aj(P) #- 0, Aj(Ej ) = O. Let Ap = AI .. . Ak; then Ap is a homogeneous polynomial of degree k with Ap(P) #- 0, Ap(E - {p}) = O. Let s(P) 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(k_l)D) are linearly independent. In fact, If S D is the standard section of D with divisor D, if {cp }PEE are complex numbers such that

°

L PEE

cps(P) E

SD .

HO (X, (k - I)D) ,

13. Some Geometry of Curves in Projective Space

74

then LPEE cps(P) = 0 on D, hence on E; but the value of the sum LCpS(P) on Q E E is cQs(Q)(Q) [since s(P)(Q) = 0 if Q =J P], so that, since s(Q)(Q) =J 0, we have cQ = 0 (VQ E E). Hence dimHO(X, QkD)/ HO(X, O(k-1)D) ::::: cardinality of E = 1 + k(n - 1). Since the sections s(P) are clearly E Sym k HO(X, 0) (since Ap is a product of linear forms), we have shown that the image of Sym k HO (X, D) 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.

°

To prove part 2) we remark that if k > ~=i, and P E supp(D) we can write supp(D)- P = El U· . ·UEk where each E j has at most n-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) =J 0, Aj(Ej) = O. 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, O(k-l)D), and we find that hO(kD) - hO((k -l)D) ::::: d. On the other hand, the exact sequence

(where SD is the standard section, and CD,x = OkD,x/O(k-l)D,x = C if xED, = 0 otherwise) implies that

hO(kD) - hO((k -l)D) :::; dimHO(X, CD) = d.

.)

ThIs proves 2 and also that for k

°

() > N, H O X, D generates

From this, we obtain

H

°HO(X,OkD) (X,O(k-l)D ).

Castelnuovo's genus estimate. Let X be a (smooth) nondegeneratc curve in lpm. Let d = deg(X), and set N = [~=i]. Define E (0:::; E < n -1) by

d - 1 = N(n - 1)

+E .

Then, the genus 9 of X is bounded as follows: 1

g:::; 2N(N - l)(n - 1) + NE. Further, if equality holds, then, for k ::::: 2,

is surjective; i.e. HO(X,OD) generates HO(X,OkD) for all k::::: 2. Proof. Let r be a large positive integer. Then hI ((r+N)D) = 0, and by the RiemannRoch theorem hO((r+N)D) = (r+N)d+1-g.

13. Some Geometry of Curves in Projective Space

75

On the other hand N

hO((r + N)D) = I:(hO(kD) - hO((k -1)D))

+ hO(O.D)

k=l

+

N+r

I:

(ho(kD) - hO((k -1)D)

k=N+l N

2':

I: (1 + k(n -

1))

+ 1 + rd

(by Lemma 4)

k=l

= 1 + rd + N

1

+ 2N(N + l)(n -

1) .

It follows that

g:::; (r

1

+ N)d - rd - N - 2N(N + l)(n - 1) 1 2

= N(d-1) - -N(N + 1)(n-1) = N 2 (n - 1)

1

1

+ EN - 2N (N + 1) (n - 1) = 2N (N - 1) (n - 1) + EN.

Further, equality implies that hO(kD) - hO ((k -l)D) = 1 + k(n -1) for all k :::; N, and the fact that HO(X,OD) generates HO(X,OkD) for all k 2': 2 follows by induction on k from the lemma. [Note that the function 1 E HO(X, OD), so that Sym k - 1 HO(X, OD) c Sym k (HO(X, OD))'] 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 Curves, I. (Springer-Verlag). We mention only one consequence, a famous theorem of Max Noether. NOETHER'S THEOREM. Let X be a compact Riemann surface of genus g 2': 3. Suppose that X is not hyperelliptic. Then, if Kx is the canonical line bundle of X and m 2': 2, the natural map

is surjective.

Proof. We consider X C jp'9- 1 as the canonical curve. Then the hyperplane section D is a canonical divisor, hence deg D = deg K = 2g - 2. The integer N above is N = [299~23] = 2 if g > 3, = 3 if g = 3. If g > 3, E = 2g - 3 - 2(g - 2) = 1 and ~N(N - l)(n - 1) + NE = g - 2 + 2 = g. If g = 3, E = 0, N = 3 and

76

13. Some Geometry of Curves in Projective Space

!N(N - l)(n - 1) + NE: = 3(g - 2) theorem follows from Ca.'ltclnuovo's.

=

3

= g.

Thus we have equality, and Noether's

It should be added that if (g ~ 3 and) X is hyperelliptic, the above result definitely fails. This follows, e.g. from the fact that K~m is very ample for large m, but the mapping tpKx induced by Kx is not injective.

14. Bilinear Relations 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 orient able surfaces is the following: A compact orient able Coo surface X without boundary is diffeomorphic to a sphere with a finite number of handles attached. Attaching a handle is illustrated below.

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 g handles can be described, up to diffeomorphism, as follows. Start with a convex polygon ~ with 4g sides aI, bl , a~, b~, ... , ag, bga~, b~ in C, oriented, as usual, "counter clockwise". If aI, a~ are the directed segments pq, p' q', we identify aI, a~ by a linear map of pq onto q'p' (i.e. one taking p to q' and q to p'). Thus, a~ 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.

78

14. Bilinear Relations

q'

I

I

p

I

I

(

I

I

r

,r ,-

r

~

r

'"

q

Under this identification, ~ becomes a compact surface X diffeomorphic to a sphere with 9 handles. All the vertices of ~ map onto the same point Xo EX, and aj, bj map onto closed curves at Xo in X; we shall call these curves again aj, bj . The segments l I b'j map ont 0 a -1 'baj' j respec t'lveIy. j 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 (8 ij is the Kronecker 8; 8ij = 1 if i = j, otherwise).

°

These curves are indicated schematically in the figure below.

Xo

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 ~ 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 - gon ~ by

14. Bilinear Relations

79

the above identification process. This gives us (piecewise differentiable) curves ai, bj on X. If ip is a Coo I-form defined in a neighbourhood of these curves, and is closed, we set

and call these the a- and the b-periods of ip. Let 0: be a Coo closed I-form on X, ip a Coo closed I-form defined in a neighbourhood of Uai U Ubj. We identify them with I-forms on ~(= ~) and on a neighbourhood of a~ respectively. Fix Po E A and, for P E ~, set u(P) = I~ 0: (~ is simply connected). We then have Lemma 1.

Proof. Let PEak and let P' be the corresponding point of ak. Let "I be a curve joining P' to P as shown.

If

Then u(P) - u(P') = 0:; now, the image of "I in X is a closed curve homologous to b;,l, so that, since 0: is closed,

u(P) - u(P') =

1

1 bk

0:

= -Bk(O:) .

Similarly, if Q E bk and Q' is the corresponding point on bk, we have

u(Q) - u(Q') =

1

0:

ak

= Ak(O:) .

80

14. Bilinear Relations

Now

iat1

ucp =

=

L (J + Ja~ + 9

k=l

tJ

k=l

ak

1+ 1~ )

ucp

bk

(u(p) - u(P'))cp(P)

+

t1

(U(Q) - u(Q'))cp(Q)

k=l

ak

bk

(with the notation above: PEak, Q E bk and P', Q' are the corresponding points on respectively)

a~, b~

which proves the lemma. 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 I-form on X,

> O. We use the

=I- 0, we have

W

L Ak(W)Bk(W) < 0 . 9

1m

k=l

In particular, if wE HO(X,!1) and all its a-periods are 0, then Proof. We apply the lemma with a

{

J8t1

uw

= w,

cp

so that ~ Jx

W

1\

Corollary. Let

w 1\ w =

= { du 1\ W = {

Jt1

i 1112

w < 0 unless w == O.

Wl, ... ,Wg

dz

Jx

1\

= O.

= w. Now, by Stokes' theorem

If (U, z) is a local coordinate on X, and z = x 1 E O(U),

i

W

dz

W

+ iy,

= -2i

1\

w.

we have setting w =

J1112

By the lemma,

be a basis of HO(X,!1). Let

dx 1\ dy ,

1 dz

on U,

14. Bilinear Relations Then, the matrix

(Ajkh:Sj,k:Sg

81

is invertible.

In fact if Aj is the g-vector (fa, Wj, ... , Jag Wj), then, if ~Cj Aj ~CjWj are zero, so that ~CjWj = 0, Cj = 0 Vj. In view of this corollary, we can choose a basis

1

Wj

=

8kj

WI, ... , W 9

= 0, the a-periods of

of HO (X, r!) such that

(Kronecker 8) .

ak

We shall call this a normalized basis of HO(X, r!) [relative to the choice ai, bj of basis of H 1 (X, Z)].

Theorem. (Riemann's bilinear relations). Let X be a compact Riemann surface of genus g > O. Let WI,"" Wg be a normalized basis of HO(X, r!). Set

= (B jk ) is symmetric, and its imaginary part is positive

Then, the complex matrix B definite.

Proof. Let aj, bk, ~ have the same meaning as before, and let Uj(P) = J~ Wj. We have

r

JM

UjWk

=

r

Jx

Wj

1\ Wk

(Stokes' theorem)

=0;

on the other hand, (by Lemma 1),

r

Jot::.

ujWk

=

t(Av(wj)Bv(Wk) - Bv(wj)Av(Wk)) v=l

= Bj(Wk) Thus, B is symmetric. Now, let Prop. 1,

- Bk(Wj)

C1, ... , c g E

since

L Av(~CkWdBv(~CkWk) < 0 . v=l

1m

L

CvCkBvk

0 .

v,k

i.e. 1m

L v,k

= 8 vj

JR, not all 0, and let

9

1m

Av(wj)

W

.

=

2:%=1 Ckwk.

By

82

14. Bilinear Relations

Given two distinct points P,Q, P of. Q, on X, there is a meromorphic I-form cp on X with simple poles at P and Q and resp(cp) = +1, resdcp) = -1 (by the Mittag-Leffler theorem for I-forms given in §1O). Because of the corollary to Prop. 1 above, we can add to cp a holomorphic I-form cp' on X such that the a-periods of wPQ = cp + cp' 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 11 2 1 and a coordinate system (U, z), at P with z(P) = 0, there is a unique meromorphic I-form w~) on X, holomorphic on X - {P} and such that (i) w~) - z~!l is holomorphic at D and (ii) the a-periods of w~n) are O. This is called a normalized abelian differential of the second kind. (Abelian differentials of the first kind are simply holomorphic I-forms. Any merom orphic I-form on X is a linear combination of these three kinds of I-form.) RECIPROCITY THEOREM. Let Wj, j = 1, ... ,11 be a normalized basis of HO(X, fl), and let w~), wPQ be normalized abelian differentials of the 2nd and 3rd kind respectively. We have 1) fbk wPQ = 27ri f~ Wk (the integral being taken along a curve joining P to Q in X - Uai - Ubj) 2) If (U, z) is the coordinate system at P used to define w~), and

1 b.

(n) -- 27rZ'.

Wp

~f(n-l)(p) ,k 11.

Wk

= hdz, we have

.

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

Proof. Uk(X)

By Lemma I, we have

On the other hand, since 6. is simply connected and the residue theorem gives

wPQ

has residue +1 at P, -1 at Q,

14. Bilinear Relations

83

fbk w~) = fM UkW~) (as above) = 2Jl'iresP(ukw~)) = 2Jl'iresp(uk(Z)z~~1) = 2Jl'i;bUzrUk(Z) = 2Jl'i;bf~n-l)(z) (since ~ = fk near P).

This proves 1).

The proof of 2) is similar:

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.

15. The Jacobian and Abel's Theorem Let X be a compact Riemann surface of genus 9 ;.:::: 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, 0):

fa Wk = bjk . J

Let A be the subgroup

of([:::g consisting of the vectors A'Y = (f'Y WI,"" f'Y Wg) as 'Y runs over HI (X, Z). We have Aak = (0, ... , 1, ... 0) = ek, the vector in C g with 1 in the k-th place and 0 elsewhere, and Ah = (fbk WI, ... , fbk Wg) (= B k say) consists of the columns of the matrix B = (B j k), Bjk = kWk. Since Im(B) is positive definite, the vectors {el, ... ,eg,B1 , ... ,Bg} J are linearly independent over R Since {ai, bj } generate HI (X, Z), we also have A = Zel + ... + Ze g + ZB 1 + ... + ZBg. These remarks imply that A is a lattice in C g with a compact quotient J(X) = O/A, called the Jacobian of the Riemann surface X. Intrinsically, J(X) can be described as follows. Let V be the dual of HO(X, 0) (canonically isomorphic to Hl(X,O) by the Serre duality theorem). We obtain a map of HI (X, Z) into V as follows if 'Y E HI (X, Z), let its image in V be the linear form W f-+ f'Y W on HO (X, 0). Then, the choices made above (of ai, bj and Wk) identify HO(X,O)* with C g and the image of H 1 (X,Z) with A. [The remarks made above also show that H 1 (X,Z) maps isomorphically onto a lattice in HO(X,O)*.] We have

Fix a base point Po E X. We define the Abel-Jacobi map A : X choose a curve c from Po to P and set

A( P) = ( {p WI, ... , (

}Po

}Po

wg)

----+

J(X) as follows:

mod A

(where all integrals are along c). If c' is another curve from Po to P, there is an element 'Y E HI (X, Z) with fe Wk = fel Wk + f'Y Wk for all k, so that the map is well-defined. [In the intrinsic description, A(P) = class of the linear form W f-+ fe won HO(X, 0).] Using the fact that J(X) is an abelian group, we can define a map X N ----+ J(X) by (PI, .. ' ,PN ) f-+ 2:f=1 A(Pj ). Let Div(X) be the set of all divisors on X. We can also define a map Div(X) ----+ J(X) by

L niPi T

i=1

L niA(Pi ) . T

f---+

i=1

15. The Jacobian and Abel's Theorem

85

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]. ABEL'S THEOREM. Let D be a divisor on X of degree O. Then D is linearly equivalent to 0 if and only if A(D) = 0 in J(X). Thus, the theorem asserts the following. Let P1, ... , Pr ; Q1, ... , Q r be points on X (with f= 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. (1) = P1 + ... + Pr - Q1 - ... - Qr) is that Qj

r LJ

P"

k

1/=1

f

Po

W ==

rQ" L J Wmod k

1/=1

f

Po

A,

the integration being along some curve from Po to PI/, and from Po to QI/ respectively (the curves being, for each v, the same for all Wk).

Proof. Since D has degree 0, we can write D = 2:~=1 (Pk - Qk), the Pk , Qk being points of X (and no Qk being one of the P's). Suppose there is a meromorphic function 2:~=1 CI/WI/ with any of Pk , Qk).

Conversely, if

CI/

CI/

E

E

C. Further, J'Y

c:

f

with (1) = D. Then

!!f =

!!f E 27fiZ V closed curves 'Y on X

2:~=1 WPkQk

+

(not containing

are such that J'Y 'P E 27fiZ V closed curves 'Y, where 'P =

2:~=1 WPkQk + 2:~=1 CI/WI/, then (1) = D where f(P) defined because of the condition on J'Y 'P).

= exp(J~ 'P) (exp(J~ 'P) is well-

We assume X identified with a convex polygon ~ as described earlier by slitting X along curves ai, bj which do not pass through Pk, Qk. Let r

'P

9

= LWPkQk + LCI/wi/' k=l

1/=1

1.

Then, J"I 'P E 27fiZ for all closed curves in X - U{ Pk , Qk} if and only if AI/ ('P) = a" 'P E 27fiZ and BI/( 'P) = Jb" 'P E 27fiZ for all v = 1, ... , g; in fact, if C k , q denote small circles around Pk , Qk respectively (with respect to coordinate neighbourhoods around these points), then'Y is homologous to an integral linear combination of al/, bl/, C k , CL and JCk 'P = +1, Jc~ 'P = -1 Vk since wPQ has residue +1 at P and -1 at Q. Thus, there exists a meromorphic function

f with (1) = D if and only if:

86

15. The Jacobian and Abel's Theorem

Av(cp), Bv(cp)E27riZ, v=I, ... ,g. Now, Av(cp) = Cv because WPkQk are normalized to have a-periods 0, and Av(wJL) Moreover, by the reciprocity theorem

= Dvw

Thus, Av (cp), Bv (cp) lie in 27riZ if and only if there exist integers (nl,' .. , n g ), (ml," . . . . , mg) such that and

(v = 1, ... ,g). This last condition can be written

eJL is the vector with 1 in the Jl-th place, 0 elsewhere, and BJL is the vector (BJLI' ... ,BJL9); since the eJL , BJL 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 xg. Let Sn be the symmetric group on n letters [= group of permutations of (1, ... ,n)]. Sn acts on the cartesian product xn =]( X • ~. x X; the quotient sn(x) = xn / Sn is called the nth symmetric power of 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. Consider the action of Sr on Cr , and consider a neighbourhood of 0 in cr. Any germ of holomorphic function F at 0 in C r invariant under Sr is a holomorphic function of the elementary symmetric functions in the coordinates Zl, . . . ,Zr of C r (Newton's theorem); equivalently, it is a holomorphic function of WI = Zl + ... + Zr, W2 = :h(zf + ... + z;), ... ,Wr = ~(zr + ... + z;) and we can take WI, ... ,Wr as coordinates for (7/ Sr.

15. The Jacobian and Abel's Theorem

87

If now PI"", Pn E X, and we number the P's so that PI = ... = PTI (= Ql say), P r, + l = ... = Pr, +r2 (= Q2 say), ""PTl+"+Tp_l+l = ... = PT1 +"+Tp (= Qp say), rl + ... + rp = n, and Ql" .. ' Qp are distinct, the group of elements of Sn which fix (PI' ... ' F,,) is STI x ... X ST p (STk 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 CTI / STI X ... X CTp / ST p ' and we can use the coordinates mentioned above for each factor. Consider now the map A: xg -7 J(X), (PI" .. , Pg) 1-+ 2::%=1 A(Pk ). Clearly, A induces a map (which we shall again call A) A: sg(X) -7 J(X). We shall identify sg(X) with the set of divisors D ~ 0 of degree g. Theorem 1. A: sg(X) -7 J(X) is a birational map; there is an analytic set Y c sg(X) of dimension < 9 such that A: sg(X) - Y -7 J(X) - A(Y) is an analytic isomorphism.

We start with

= !:'Pi of J(X) is maximal, = g, is open and dense in xg.

Lemma 1. The set of distinct points PI, ... ,Pg E X such that the rank at D

the differential of A: sg(X)

-7

Proof. Let (Uj,Zj) be coordinates near Pj (Zj(Pj) = 0). Using the local coordinates on J(X) coming from C9 (J(X) = C9/A), the map A can be written A(Zl, ... ,Zg) =

2::jrjw (w = (Wl,".'Wg)), Ifw = f~dzj on Uj (J~ = (f~" .. ·,h)) the Jacobian = !:'Pi is given by

matrix of A at D

The existence of (PI, ... , Pg) such that this matrix has rank 9 follows from §13, Lemma 4 (If dimHO(X, L) = k, 3k points Xj such that any s E HO(X, L) vanishing at the Xj is identically 0) since hO(O) = g. The fact that this set is dense follows from the proof of §13, Lemma 1. Lemma 2. If D = 2::i Pi E sg(X), then A-I A(D) is the bijective holomorphic image ofJPT with r = dimlDI. In particular, A-lA(D) is connectedVD. Proof. If D l , D2 E sg(X) and A(Dl) = A(D 2), then Dl is linearly equivalent to D2 by Abel's theorem (since Dl - D2 has degree 0). Let JPT = JP(HO(X, OD)) (D E sg(X)) be the projective space (HO(X,OD) - {O}/C*. Because of the remark above, the map HO(X, OD) - {O} -7 S9(X), S 1-+ div(s) induces a bijection of JPT onto A-I A(D). The fact that this map is holomorphic can be seen as follows. Let U C C N be open, and f(x,t) a function holomorphic on ~E x U [~E = {x E C Ilxl < E}l. Suppose that f(x, t) =f. 0 for Ixi = p( < E), t E U. Then, if to E U, the number of zeros Xi(t) of f(x, t)

88

15. The Jacobian and Abel's Theorem

(counted with multiplicity) in Ixi mEZ,


0 such that

The convergence follows. Clearly '!9(z + ek) = '!9(z): '!9 is clearly periodic of period 1 in each variable, being a standard Fourier series. We have

'!9(z

+ B k) =

L e7ri (n,Bn)+27ri(n,z)+27ri(n,Bk)( Be k = Bk) nEZg

- 7ri(ek, Bek) - 27ri(ek, z))

= e-27rizk-7riBkk'!9(z) (since n + ek runs over zg when n does) . That '!9 t= 0 follows from the fact that a Fourier series whose coefficients are not all zero cannot vanish identically. That '!9(z) = '!9( -z) is obvious if we replace n by -n in the series defining '!9. Lemma 2. Any theta function of order 1 is a constant multiple of the Riemann theta

function. Proof. Let f(z) be a theta function of order 1. Since f is periodic, with period I, in each variable, it has a Fourier expansion

f(z) =

L

an e21fi (n,z) .

nEZg

Now,

= f(z + Bk) = e-1fiBkk-21fizk f(z) = e-1fiBkk~ane21fi(n-ek'Z)

~ane21fi(n,z+Bk)

--

~a

U

n+ek e-1fiBkk e

21fi (n,z)

.

93

16. The Riemann Theta Function Hence an+ek = e7riBkk+27ri(n,Bk)an' It follows that if an = 0 for some n, then a n+ek \:Ik, and hence that an = 0 for all n. In particular, J == 0 if and only if ao = O.

=

0

Applying this to J - aotJ, we conclude that J == aotJ. 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. Let L be the line bundle on J(X) defined by the factor of automorphy 'fiek == 1, 'fiBk = e-27rizk - 7riB kk. Lemma 2 asserts that H O(J(X), L) has dimension 1, and tJ defines a non-zero section of L. Let e be the divisor on J(X) defined by this section: e = div(tJ). Locally on J(X), e is defined by the equation tJ(z) = 0; more precisely, if a E J(X) and Zo E ([:g maps under the projection 1f : ([:g ----+ J(X) onto a, if V is a small neighbourhood of Zo and 1f(V) = U, en U is defined by (U,tJ 0 (1f1V)-1). Set theoretically, e is the image in J(X) of {z E ([:g 18(z) = O}. It is called the theta-divisor oj J(X). 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 r g ; in particular it is finite dimensional. Note: The finite dimensionality of HOU~I, E), where M is a compact complex manifold and E, a holomorphic vector bundle on M, can be proved exactly as in the proof of §7, Theorem 1.

Proof of Lemma 3. Let J E Vr ; then, J is periodic of period 1 in each variable, and can be expanded in a Fourier series: J(z) = LnEZg a n e 27ri (n,z). we have

-- '~ " a n e-7rirBkke27ri(n-rek,z) -- '~ " a n+rek e-7rirBkke27ri(n,z) nEZg nEZg

so that a n+rck = e7rirBkk+27ri(n,Bk) an. It follows at once that if an n = (nl,"" ng) with 0 :::; nj < r, then J == O. Hence dim Vr :::; r g. Let s = (SI,"" Sg) E '!lg, tJr,s(z)

=

o for

those

0:::; Sj < r. Define

L

nEZg

exp{ 1fi(B(n + ;), rn

+ s) + 21fi(z, rn + s)} .

The series converges uniformly for z in any compact set in ([:g as in Lemma 1, and one verifies, as in Lemma 1, that tJr,s E Vr \:Is. Since the non-zero Fourier coefficients of tJr,s are at the lattice points {s + rnln E '!l9} and these sets are pairwise disjoint for

94

16. The Riemann Theta Function

O:S Sj < r, it follows that {'!9 r ,s hence form a basis of Vr .

IS=

(SI, ... ,Sg) E zg,O:S Sj < r} are independent,

Consider now a basis 0 = (0 0 , ... , ON), N + 1 = 3g of the space of theta functions of order 3; as we shall see, the functions OJ do not have common zeros; moreover, if), E A, O(z +).) = eWA(z)O(z) where w.\ is a polynomial of degree :S 1 (as follows immediately from the definition of theta functions and the fact that ek, Bk generate A over Z). Hence, 0 defines a holomorphic map, which we denote again by O.

THE LEFSCHETZ IMBEDDING THEOREM. The map 0 : J(X) theta functions of order 3 is an imbedding.

----7

]l'N

defined by

Proof. We start by showing that V2 has no base points, i.e. V Zo E (:g, :3 a theta function f of order 2 with f(zo) i=- O. This follows from the following remark: if '!9 is Riemann's theta function and a E (:q, then f(z) = '!9(z + a)'!9(z - a) E V2. Also, if a, bE (:g, then '!9(z + a)'!9(z + b)'!9(z - a - b) is a theta function of order 3. It follows that V3 has no base points either (i.e. the basis functions 00 , ... , () N have no common zeros). We now show that () : J(X) ----7 ]l'N separates points. Suppose that WI, W2 E that ()(Wl) = t()(W2), t i=- O. Then

'!9(Wl

+ a)'!9(wl + b)'!9(Wl -

a - b) = t'!9(W2

+ a)'!9(w2 + b)'!9(W2 -

(:g

and

a - b) Va, bE C9 .

We claim that this implies the following: the function z 1-+ ~i:~!~j is holomorphic and nowhere 0 on (:g. In fact, given Zo E (:g, we can choose b E (:g so that '!9( Wj + b) i=- 0 and '!9(Wj - z - b) i=- 0 j = 1,2 for all z E U, where U is a small neighbourhood of Zo; we then have '!9(Wl + z) _ t'!9(W2 + b)'!9(W2 - Z - b) U '!9( W2 + z) - '!9( WI + b)'!9( WI - z _ b) ,z E , and so is holomorphic and non-zero on U. We now use the following lemma. Lemma 4. If W E (:g is such that {)~(;)z) is holomorphic and nowhere 0 on (:g, then W

EA.

Equivalently, if ( E J(X) and the theta divisor = 0 in J(X).

( : e = e + (, then (

e

is left invariant by translation by

Of course, this lemma and our remark above imply that WI -W2 E A, so that () : J(X) is injective.

]l'N

----7

16. The Riemann Theta Function Proof of Lemma 4. There exists a holomorphic function g on

95

0; in other words D is a special divisor

of degree g. Proof. A(X) c 8( if and only if A(P) - ( E 8 VP E X, i.e. ( - A(P) E 8 = W g- 1 + K VP E X. Thus, the condition is that ( - K = 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) = ( - K, 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. 8(, then there is a unique divisor D ~ 0 of degree 9 such that A(D) + K = (. D is given by the divisor of zeros of 1J(A(P) - ().

This follows from Theorem 4 and §15, 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 : sg(X) -7 J(X). We shall give another application of these results. Consider the map A : S9(X) -7 J(X), (PI' ... ' Pg) f-7 ~A(Pi)' and let Y C S9(X) be the set of critical points, i.e. Y = {D E sg(X) I rankD(dA) < g}. Y is an analytic set of dimension::; 9 - 1. Further, if DEY, then A-I A( 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::; 9 - 2. In particular, no finite union of translates of Y' can contain 8. Let now P E X and let x be a variable point on X. By Theorem 4, if A( P) + ( - K rt. Y', then the function x f-7 1J (A( x) - A( P) - () has exactly g zeJ;os PI, ... , Pg, and ~A( Pi) = ( + A(P) - K; further ~Pi is the only divisor ~ 0 of degree 9 satisfying this equation. If we assume, in addition that ( E 8, (

=

A(Q~)

+ ... + A(Q~_I) + K,

then

g-1 LA(Pi ) = A(P) + LA(Q5) , 9

1

so that ~Pi

=P

j=1

+ ~Q5.

Thus, if ( E 8 and (

rt. -A(P) + K + Y', then the zeros of x f-71J(A(x) - A(P) - () are

given by (P, Q~, ... , Q~-I) , where Q~, ... , Q~-1 depend only on ( and not on P. Consider now a non-constant merom orphic function f on X, and let (J) = 2::~=1 Pk 2::~=1 Qk. We choose ( E 8, ( rt. Uk(Y' + K - A(Pk)) u Uk(Y' + K - A(Qk)), and write ( = A(Do) + K, Do = A(Q~) + ... + A(Q~_I). We transform X into a simply

103

17. The Theta Divisor

connected polygon ~ as in §14, with the a v , bv avoiding a suitable finite set of points in X. Consider the function on ~ defined by

F(x) =

IT

'!9(A(x) - A(Pk) - () k=1 '!9(A(x) - A(Qk) - ()

Its divisor = (L,Pk + rD o) - (L,Qk function defined on X.

+ rD o) = L,(Pk - Qk) = (I).

If x E bv and x' is the corresponding point of F(x) = F(X').

b~,

then A( x')

It is not, however, a

= A( x) + ev , and we have

If x E av and x' is the corresponding point of a~, we have A(x' ) = A(x)

+ Bv

and

By Abel's theorem, there exist integers nl, ... ,ng, ml, ... ,mg so that 9

T

2:)A v (Pk) - AAQk))

= vth

component of

1

L njej j=1

9

+ L mjBj j=1

9

= nv + LmjBjv. 1

Now if WI, ... ,Wg is the normalized basis of HO(X, n), and W = LJ=1 mjwj, and we set ip(x) = J;o W (Po fixed), we find that e 27ri