From Number Theory to Physics

--- ---->
----. (y)

;'

,/

Fig. 25. Proof of equations (A.2.1) and (A.2.2).

and according to (A.2.1) and (A.2.2), we have

and

hi-B; f1]' = h; (f - f

0

¢i)1]'

=

1;

17

·l1]',

which proves the required identity.

Appendix B. Holomorphic line bundles on compact Riemann surfaces B.1. COO and holomorphic line bundles This first Section presents a series of definitions, without motivations. These should be provided by the next two Sections. B.l.l. Let X be a Coo manifold. Recall that a Coo (complex) line bundle £ on X consists of a family {£x} xEX of one-dimensional complex vector spaces parametrized by X, 'which depend on x in a Coo way'. A section s of £ over a

110

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

subset E c X is a map which attaches a vector sex) E Cx to any point x E E. When s( x) =J. 0 for any x E E, the section s is said to be a frame over E. A rigorous way to define the COO-structure of the line bundle C is to give a covering U of X by open sets and to give, for any U E U, a frame Su over U in such a way that the transition functions

'Pvu : Un V

--4

C* ,

defined on the non-empty pairwise intersections of open sets in U by the relation

(B.l.l)

sv(x) = 'Pvu(x) su(x),

are Coo functions. Then an arbitrary section s of C over an open subset D of X is Coo (resp. continuous) if, for any U E U, the function

lu : U n D

--4


defined by

sex) = lu(x) su(x) is Coo (resp. continuous). The set of Coo (resp. continuous) sections of Cover D will be denoted Coo(D; C) (resp. CD(S?; C)). It is clearly a vector space. Moreover the pointwise product 1· s of a function 1 E COO( D;
B.l.2. Observe that the transition functions satisfy the following cocycle condition: if U, V and Ware open sets in V with a non-empty intersection, then

(B.l.2)

'Pwu = 'Pwv . 'Pvu

Un V

on

n W.

Conversely, any family ('Puv) offunctions, 'Puv E COO(UnV; C*), parametrized by the pairs (U, V) of open sets in U with a non-empty intersection, arises from a Coo line bundle C on X, provided these functions satisfy the cocycle condition. Indeed, starting from such a family ('Puv) we may define a line bundle C on X and sections Su on U, U E U, as follows: for any x E X, let

Ux = {U

E

U, x

E

U};

define C x as the one-dimensional complex vector space obtained as the quotient of U x x C by the equivalence relation rv such that

(U,>.)

rv

(V,{t) {:} {t = 'Puv(x)>',

J. -B. Bost

111

and define sv by sv(x) = [(U,l)]. Then the sv's satisfy the relations (B.1.1) by construction. Therefore £ is a Coo line bundle and ('-Pvv) is the associated family of transition functions. B.1.3. A Coo morphism £ from a Coo line bundle £ on X to another line bundle £1 on X is the data, for every x EX, of a linear map

Rex) : £x

-7

£~

such that for any open subset U of X and any s E Coo(U; C), the section

£. s : x

I--t

£( x) . s( x)

of £1 on U is Coo. If U is an open covering of X and if (SV)VEU and (s\; )VEU are families of Coo frames of £ and £/ over the open sets in U, then we may define functions Lv E Coo(U; q, U E U, by the relations (B.1.3)

£. Sv

= Lv . s\;.

Moreover, if ('-P v v) and ('-Pu v) are the transition functions at tached to (s V ) and (s\;), we have

(B.1.4)

i.pvv Lv = Lv i.pvv

on

Un V.

Conversely, for any family (Lv)vEu, of functions Lv E Coo(U; q, satisfying (B.1.4), there exists a unique Coo morphism £ from £ to £1 such that (B.1.3) holds. Morphisms of line bundles clearly may be composed. A Coo morphism £ : £ -7 £1 is called an isomorphism if, for any x EX, the map €( x) is an isomorphism; this is equivalent to the non-vanishing of the Lv's defined in (B.1.3) or to the existence of a Coo morphism of line bundles from £f to £ inverse of £. Continuous morphisms and isomorphisms between line bundles are similarly defined. B.1.4. The trivial line bundle over X is the line bundle such that £x = C for any x E X and whose Coo structure is defined by U = {X}, Sx = 1. Its (COO) sections may be identified with (COO) C-valued functions. There is an obvious notion of restriction of a Coo line bundle on X to an open subset [l of X. A Coo line bundle £ on X is said to be trivial on [l if its restriction to [l is isomorphic to the trivial line bundle over fl. This is easily seen to be equivalent to the existence of a trivialization of £ on fl. B.1.5. If L and V are two one-dimensional complex vector spaces and if vEL and VI E V, we will write v . VI instead of v 0 VI for their tensor product, which is an element of the one-dimensional vector space L 0 V. If vEL - {O}, we will denote v-I the linear form on L taking the value 1 on v; it is an element of the one-dimensional vector space L * . The tensor product £ 0 £/ of two Coo line bundles £ and £1 over X is the Coo line bundle over X such that (£ 0 £/)x = £x 0 £~ and whose Coo

112

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

structure is such that, for any open subset E Coo ( Q; £'), the section

Q

of X, any

8

E

Coo(Q; £) and any

8'

8 . 8' :

x

I-t 8(

x) . 8' (x)

of £ ® £1 over Q is Coo. The external tensor product £ 0 £1 is the Coo line bundle over X x X such that (£ 0 £/)(x,y) = £x ®£~ and whose Coo structure is such that, for any two open subsets Q and QI of X, any 8 E COO(Q; £) and 8' E Coo(Q';£'), the section 8

[ill

8' :

(x, y)

I-t

8(X)· 8'(y)

of £ 0 £' over Q x Q' is Coo. The dual line bundle £* is the Coo line bundle over X such that (£*)x = (£x)* and whose Coo structure is such that, for any open subset Q of X and any Coo frame 8 E Coo(Q; C), the section 8- 1 :

x

I-t

8(X)-1

of £* over Q is Coo.

B.1.6. When X is a Riemann surface (or more generally, a complex manifold of any dimension), we can define holomorphic line bundles over X and extend to them all the preceding statements by simply replacing 'Coo, by 'holomorphic' whenever it occurs. Clearly, if £ is a holomorphic bundle over a Riemann surface X, it is automatically a Coo line bundle over X considered as a Coo surface. Conversely, if M is a Coo line bundle on X, the data of a structure of holomorphic line bundle on M compatible with its Coo structure is essentially equivalent to the data of an open covering U of X and of Coo frames 8U over U for every U E U such that, for any two intersecting U and V in U, 'f!uv defined by (B.1.I) is holomorphic on U n V. Observe also that the notion of meromorphic function extends easily to sections of holomorphic line bundles: a section of a holomorphic line bundle £ is called meromorphic if, locally, it can be written as the product of a holomorphic frame of £ by a meromorphic function. B.1.7. A major difference between the Coo and the holomorphic situations regards the existence of global sections. For any Coo line bundle £ on a Coo manifold X, the space Coo(X; £) of its global Coo sections is 'very large'. Indeed, if 8 is a Coo frame of £ on an open set U in X and if p is any Coo function on X which vanishes outside a compact subset of U, then we define a global section p. 8 E Coo(X; £) by setting

(p. 8)(X) =

p(x)· 8(X)

o

if if

x EU x 1. U.

On the contrary, if £ is a holomorphic line bundle over a Riemann surface X, the space of holomorphic sections of £ over X, usually denoted HO(X; C), may

J. -B. Bast

113

e

be 'very small'. For instance, suppose that is the trivial holomorphic bundle 0 over a compact connected Riemann surface X. Then the space HO(X; 0) is the space of constant C-valued functions on X, hence isomorphic to C Indeed, an element 1 of HO(X; 0) is nothing else than a holomorphic function from X to C By compactness of X, 111 assumes a maximum value at some point P of X. The maximum modulus principle shows that 1 is constant on a neighbourhood of P, hence, by analytic continuation, on X which is connected. B.loS. A C= Hermitian metric II lion a C= line bundle e over a C= manifold X is the data for any x E X of a Hermitian norm II lion ex such that, for any open subset U of X and any s E C=(U; e), the function

II s II: x f-tll

s(x)

11

2

,

from U to IR.+, is C=. A C= Hermitian metric on a holomorphic line bundle is a C= Hermitian metric on the underlying C= line bundle. B.lo9. Let X be a C= manifold of dimension 1? We will denote /\ the C= line bundle of complex differential forms of degree n over X. By definition, for any x EX

x

/\ x,x = /\ Tx,x ®lR C n

and the C= sections of /\ X are the C= complex differential forms of degree The line bundle /\ X may also be defined as follows: let {(U,,,pu)} UEU be a family of C= coordinate charts on X such that U is a covering of X; for any intersecting U and U' in U, denote 1?

"pu = (xih:S;i:s;n

and "pu' = (X:h:S;i:s;n

and (B.1.5)

'f'U'U = det

( aax~X ' ) l:S;i,j:S;n ; J

the function 'f'U'U belongs to c=(Unu*; IR.*) and the 'f'U'u'S satisfy the cocycle condition (B.1.2) and are easily checked to define the line bundle Finally, we will denote 1/\ Ix the C= line bundle of complex densities over X. It is defined as the C= line bundle associated to the transition functions l'f'uu,l, which are the absolute values of the transition functions (B.1.5) defining If X is oriented, then 1/\ Ix may be identified with (indeed, we may assume that the (U,,,pu),s are oriented charts; then I'f'u'ul = 'f'u'u), More generally, a local choice of orientation on X determines locally an idenand 1/\ Ix and two different choices of orientation give tification between rise to opposite identifications. Recall that, if X is oriented, for any compactly supported continuous section w of /\ the integral X w makes sense. These observations show that for any compactly supported continuous section w of 1/\ lx, the integral makes sense, without any orientation assumption on X.

/\x.

/\x.

/\x

/\x

x'

Jxw

J

114

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

B.2. Holomorphic and meromorphic differential forms B.2.1. Let X be a Riemann surface. Let a be a complex Coo I-form defined on an open subset U of X. For any holomorphic coordinate chart (D, z), D c U there exist Coo functions

on D. The functions

= --; dzv

it is a ~on-vanishing holomorphic function on Un V. The
o oz

and is Coo or holomorphic iff J is. Let wx be the dual line bundle Tx of Tx. It is a holomorphic line bundle on X which can be defined by the transition functions

dzv dzu.


Using a local holomorphic coordinate z, a section s of w x may be written locally s=J(z)dz

and is Coo or holomorphic iff J is. The Coo sections of w x may be identified with the Coo forms of type (1,0). A holomorphic (resp. meromorphic) section of Wx on an open subset D of X is called a holomorphic (resp. meromorphic)

J. -B. Bost

115

differential on X. The vector space HO (X; w x) of holomorphic differentials on X is denoted ill (X). Let wx be the 'complex conjugate' of the line bundle wx. It is the Ceo line bundle on X defined by the transition functions

dz V ) 'Pvu = ( dzu . Using a local holomorphic coordinate z, a section s of Wx may be written locally s = J(z) d:Z and is Ceo iff J is. The Ceo sections of wX may be identified with the Ceo forms of type (0,1). Complex conjugation transforms Ceo sections of w X into Ceo sections of wx. The sections of wx which are complex conjugate of holomorphic differentials on X are called antiholomorphic differentials. Their vector space will be denoted ill (X). The following statement may be proved by a simple local computation using the Cauchy-Riemann equations and will be left as an exercise to the reader (see also inJra (B.7.I)).

Proposition B.2.1. For any Riemann surface X, a Ceo section of Wx (resp. of wx) over X belongs to ill(X) (resp. ill(X)) iff, considered as a Coco complex i-form over X, it is closed. Observe that there exists a canonical isomorphism of Coo line bundles (B.2.I) which, locally, sends dz ® d-z to dz 1\ d-z. Finally, the data of a Hermitian metric II " on wX is equivalent to the data of a Riemannian metric ds 2 on X compatible with its holomorphic structure (cf. §I.2.I). Namely, for any local holomorphic coordinate z = x + iy on X, ds 2 and " " are determined by each other via the relation

The volume form I-l associated to ds 2 is then given by (B.2.2)

I-l =

~ " dz ,,2 . dz 1\ d-z .

B.2.2. From now on, we restrict our attention to holomorphic and meromorphic differentials over compact connected Riemann surfaces. We start with a few examples. Examples. i) There is no non-zero element in ill (JP'IC). Indeed let There is an entire function J(z)

Cl'

E ill(JP'lC).

116

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties 00

such that, over C = fez) dz.

Q

Since

Q

is holomorphic on a neighbourhood of

f

(~ )

d(

00

~) = - ~ an

in jplC, the differential

z-n-2 dz

is holomorphic near O. This clearly implies that all the coefficients an vanish. ii) Let X = Cj(Z + TZ) be an elliptic curve. Then dz is a non-vanishing holomorphic differential on X. Therefore W X is a trivial holomorphic line bundle on X, and Ql (X) is a one-dimensional vector space generated by dz (cf. §B.l. 7). iii) Let C be the hyperelliptic Riemann surface defined by the polynomial

2g+2 P(X, Y) = y2 -

II (X -

Qi)

i=l

where the Qi'S are pairwise distinct complex numbers (cf. §I.4.2). The coordinates X and Y defines meromorphic functions x and y on C. The vector

space Ql (C) has dimension g, and the following holomorphic differentials build a basis of Ql(C); Wi=---

Y

i = 1, ... ,g.

Indeed, .near the points at infinity of C, 1x may be used as local coordinate and

± (~)9+1;

near the points (Qi'O), y may be used as local coordinate and x - Qi '" Ay2; near any other point of C, x may be used as local coordinate and y does not vanish. This immediately implies that the Wi, 1 ::; i ::; g, are holomorphic differential forms on C and that WI = dx has a zero of order 9 - 1 at the two points at infinity and does not vanish else~here on C. Therefore any Q E Ql (C) may be written }

r-.J

Q=fWl

where f is a meromorphic function on C which is holomorphic everywhere except at the points at infinity where the orders of its poles are at most 9 - l. On the other hand, the function f may be written

f

= R(x) + S(x)y

where Rand 5 belong to qX) (cf. §I.4.2, Exercise). Since f is holomorphic at finite distance, Rand 5 must be polynomials. It is now easy to deduce that 5 = a and that R is a polynomial of degree at most 9 - 1 by examining the local behaviour of Q near the points at infinity. iv) Let F be an irreducible homogeneous polynomial of degree d ;::: 3 in o , Xl, X 2 ] such that the algebraic curve X in jp 2C of equation

crX

J. -B. Bost

11 7

is smooth (e.g. F(X O ,X1 ,X2 ) = xt + xt - xg; cf. §I.4.2). The coordinates (X O ,X1 ,X2 ) define meromorphic functions x = and y = on X. Elementary computations show that, for any polynomial P in qx1 , X 2 ] of degree ::; d - 3, the meromorphic differential form on X

¥o

a(P):=

aF ) -1 P(x,y)dx ( aX2 (l,x,y)

= -

(

t

aF (l,x,y) ) -1 P(x,y)dy aX 1

is holomorphic. Moreover, it may be proved that the map P 1--4 a(P) is an isomorphism from the space of polynomials of degree::; d - 3 in qX l ,X2 ] onto Q1(X). More generally, if X is the normalization of an algebraic curve, it is always possible to describe Ql (X) in terms of the algebraic data defining this curve. be a cocompact discrete group in PSL(2, JR), acting freely on S) v) Let and let X = S)jr. Then Q1(X) may be identified with the modular forms of weight two with respect to r (cf. [Z]).

r

B.2.3. Let (U, z) be a holomorphic coordinate chart on a Riemann surface and let P E U. If w = f dz is a holomorphic differential on U - {P}, its residue at P is defined as the coefficient of (z - z(P))-l in the Laurent series expansion of'ljJ in term of (z - z(P)). It is noted Resp wand is also the integral

1

w 1 27ri aD

-

where D is a small disc in U which contains P (this expression shows that Resp w is independent on the choice of the local coordinate z).

Proposition B.2.2. For any meromorphic differential w on a compact Riemann surface X, we have

L

Res x w = O.

xEX

(This sum is finite since the set of poles of w is finite.) Indeed, let {Pjh::;j::;N be the poles of wand let {Djh::;j::;N be a family N

of disjoint discs on X such that D j :7 Pj. Consider U = X - U D j. It is a J=1

surface with boundary, bounded by

N

.u

J=1

aDj. Moreover, restricted to U, w is a

holomorphic differential, hence a closed Coo I-form. Therefore we have: N

L xEX

Res x w = LResPj w j=1

118

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

t_1 laD'{

. 21l'i )=1 = -

w

J

2~i!au w

__ 1 { dw 21l'i lu

=

(Stokes formula)

o.

Consider now a meromorphic function f on a Riemann surface X and a point P of X such that f is not constant on a neighbourhood of X (if X is connected and non-constant, this holds for any P E X). Let (U, z) be a holornorphic chart on X such that P E U and z(P) = O. The multiplicity with which f takes the value f( P) at P is the positive integer n defined by the following conditions: . if f(P) i= 00, there exists a E C* such that, for x E U f(x) = f(P)

. if f(P) =

00,

+ az(xt + 0

(z(xt+ 1 )

there exists a E C* such that, for x E U f(x) = az(x)-n

+0

(z(x)-n+l) .

For any A E C, the meromorphic differential form (f -A)-ldf is holomorphic at P iff f(P) 1. p, oo} and has a simple pole with residue n (resp. -n) if f(P) = A (resp. f(P) = (0). Therefore Proposition B.2.2 applied to (f - A)-ldf implies: Proposition B.2.3. For any non-constant meromorphic f1lnction f on a compact connected Riemann surface X, the number (c01lnted with multiplicities) of preimages under f of an element a E pIC is independent on a.

This number is easily seen to be the degree of f (cf. [Mil]). In the sequel, we will use the following consequence of Proposition B.2.3: Corollary B.2.4. If on a connected compact Riemann surface X there exists a meromorphic function f which has exactly one pole and if this pole is simple, then f establishes an isomorphism from X onto pIc.

B.2.4. We end this Section with a remarkable result - a special case of Hodge decomposition: Theorem B.2.5. For any (connected) compact Riemann surface X, the map t:

Ql(X)EJ1Ql(X)->HbR(X;
0/. EJ1 f3 is an isomorphism.

1-+

[0/. + f3l

J. -B. Bost

119

(Observe that a + j3 is a closed complex I-form on X by Proposition B.2.1 and has a well defined class in the de Rham cohomology group HI>R(X; q; cj.

§A.l.) We will prove Theorem B.2.5 in §C.5. It has the following consequence:

Corollary B.2.6. If X is a connected compact Riemann surface of genus g, then the complex vector space QI (X) has dimension g. When X is the Riemann sphere or a hyperelliptic Riemann surface, the corollary follows from the explicit description of QI (X) given above combined with the determination of their genus (cj. Figure 3 and Examples in §I.2.2 and §I.4.2). In general, it shows that dim QI (X) which a priori depends on the holomorphic structure of X is in fact a topological invariant of X. Conversely, when X is realized as an algebraic curve, Corollary B.2.6 provides an algebraic interpretation of the genus of X.

B.3. Divisors and holomorphic line bundles on Riemann surfaces One of the most important questions in the theory of Riemann surfaces is the existence of meromorphic functions on a given Riemann surface with prescribed zeros and poles. For instance, consider a compact connected Riemann surface X. We have seen that there exist non-constant meromorphic functions on X (cj. Theorem I.4.2); more precisely, according to Theorem 1.4.3. 2), there are enough meromorphic functions on X to separate the points of X and even to provide a projective embedding of X. On the other hand, any holomorphic function on X is constant (see Section B.1.7). A natural question is then the following: Let PI, ... , P k be distinct points on X and let ml, ... ,mk be positive integers. Is there any non-constant meromorphic function on X which is holomorphic on X - {PI, ... ,Pk } and whose pole at Pi has order at most mi 18 for any i = 1, ... ,k? A variant of this question is to consider some other points Q1, . .. ,Qf on X and positive integers nl, ... ,nf and to ask for meromorphic functions on X which satisfy the preceding conditions and moreover have a zero of order at least ni at Qi. Observe that if X is a plane algebraic curve and the Pi'S are points at infinity on X, by Theorem I.4.5, this essentially amounts to finding a polynomial P(X, Y) which satisfies some growth conditions at 00 and with zeroes of order ni at Qi (1 :::; i :::; E). Geometrically, it is equivalent to find another plane algebraic curve of bounded degree, which meets X at Qi with a multiplicity at least ni (1 :::; i :::; E) and has a suitable behaviour at infinity. 18

By definition, this condition is satisfied by any meromorphic function hoJomorphic at Pi.

120

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

In this Section, we explain how such questions may be translated into questions concerning the existence of holomorphic sections of some holomorphic line bundles on X. Let X be a compact Riemann surface. A divisor on X is an element of the free Abelian group whose generators are the points of X. In other words, a divisor on X is a finite formal sum k

D= LniPi i=1

of (distinct) points Pi of X, affected with multiplicities ni E Z. Such a divisor is called effective if all the ni's are non-negative. More generally, for any two divisors D1 and D2 on X, one writes D1 ~ D2 when D1 - D2 is effective. The multiplicity np(D) of a point P of X in D is ni if P := Pi and is 0 if P is not one of the P;'s. The finite subset IDI of X formed by the points P E X such that np(D) '" 0 is called the support of D. The group of divisors on X will be denoted by Div(X). Let us now explain how a divisor is attached to any non-zero meromorphic section of a holomorphic line bundle on X. Let X be any Riemann surface. For any P E X and any non-zero meromorphic function defined on an open neighbourhood of P in X, the valuation of f at P is the integer v p(f) defined as follows: . if f is holomorphic at P and has a zero of order n at P, then v p(f) = n; . if f has a pole of order n > 0 at P, then vp(f) = -no Clearly, we have: i) vp(f) = 0 iff f is holomorphic and does not vanish at P ii) if f and g are non-zero meromorphic functions on a neighbourhood of P, then

(B.3.1)

vp(fg) = vp(f)

+ vp(g).

and

(B.3.2) Let C be a holomorphic line bundle on X. Consider a point P E X and a holomorphic trivialization t of C over an open neighbourhood of P. The preceding properties show that, if s is any non-zero meromorphic section of I: on an open neighbourhood V of P and if we define a meromorphic function f on Un V by s = ft, then vp(f) does not depend on t. This integer is called the valuation of s at P, and is denoted by v p( s). When I: is the trivial line bundle, this definition is clearly consistent with the preceding one. Moreover, the properties i) and ii) are still true if now f and g are non-zero meromorphic sections of two holomorphic

J. -B. Bost

line bundles £ and M; then fg is a section on £ @ M and f- 1 is a £*. From now on, suppose that X is compact and connected. Then, non-zero meromorphic section of £ on X, the points of X such that are isolated, hence form a finite set. Thus, we can define the divisor divs =

L

121

section of if s is any p( s) #- 0 of,9 as

11

vp(s)P;

PEX

this sum is indeed a finite sum. Observe that divs is effective iff sis holomorphic. Consider now the 'set' E of pairs (£,s) where £ is a holomorphic line bundle on X and s is a non-zero meromorphic section of £ on X, and let us say that two such pairs (£,8) and (M, t) are isomorphic iff there exists a holomorphic isomorphism r.p : £ ..:::+ M such that r.p·s =

t.

This isomorphism relation, which we will denote by"', clearly is an equivalence relation on E. The following Proposition makes precise the link between holomorphicline bundles and divisors on X provided by divisors of meromorphic sections.

Proposition B.3.1. If two pairs (£, s) and (M, t) in E are isomorphic, then div8 = divt. Moreover the map div: Ej

-+

Div(X)

1-+

divs

'V

[(£,s)] is a bijection.

Clearly, (£, s) '" (M, t) iff t· S-1 is a non-vanishing holomorphic section of £*. This holds iff dive t . 8- 1 ) = 0 and, according to (B.3.1) and (B.3.2), this is equivalent to divt = divs. This proves the first assertion of proposition (B.3.1) and the injectivity of div. Let us now prove its surjectivity. Let

M

@

k

D= LniP; i=1

be a divisor on X, where the Pi'S are distinct. For.each i, let us choose a local variable Zi at Pi and a small enough Ci > 0, such that the discs

Di = {M EX; zi(M) is defined and IZi(M)1 <

cd

are pairwise disjoint subsets of X. Let £ be the line bundle defined by the covering {U;}o:Si9' where

122

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

Uo = X - {PI, ... ,Pd

Ui = Di if i = 1, ... , k, and by the transition functions 'PUoU;

n'

= zi'

defined for i = 1, ... , k on the intersections 19

Uo n Ui = {M E X; zi(M) is defined and 0 < IZi(M)1 < cd. Then £, is trivial on X - {PI, ... , Pk} by construction. Moreover, the constant function 1, seen as an element of HO (X - {PI, ... , Pk}; £,), defines a meromorphic section s of £, over X, holomorphic and non-vanishing on X - {PI,"" Pd, and, by definition of the valuation vP;, we have for any i = 1, ... ,k:

This shows that

k

divs

=L

i=1

niPi

=D

and finally we get

D = div[(£',s)]. By the injectivity of div, the pair (£', s) we have just defined is well defined, up to isomorphism, by this last relation. It is often denoted by (O(D), lo(D»)' Observe that, again by the injectivity of div, for any holomorphic line bundle £, on X and for any non-zero meromorphic section s of £, over X, we have an isomorphism of holomorphic line bundles

(B.3.3)

£, ~ O(divs).

The reader will check easily that, for any D E Div(X) and any open subset U C X, the trivialization of O(D) on X -IDI defines an isomorphism

(B.3.4)

HO(U; O(D)) ~ {f E M(U) I VP E U, vp(f)

2: -mp(D)}.

In particular

(B.3.5)

HO(X; O(D)) ~ {f E M(X)

I div(f) 2:

-D},

and the space of meromorphic functions considered at the beginning of this

(X;

0 (2::=1 miPi - 2:~=1 n jQj ) ). Section is nothing else than HO Observe finally that the tensor product provides a commutative group law on E / rv, defined by 19

Observe that (i < j and Ui n Uj =f. 0) =? (i = 0 and j:::: 1). Thus we only need to consider the k transition functions attached to the intersectIOns Uo nUl, ... , Uo n Uk. Moreover, since i < j < k =? Ui n Uj n Uk = 0, no cocycle condition needs to be checked.

J. -B. Bost

[(£, s)]

+ [(M, t)]

123

= [(£ 0 M, st)],

and that div is then an isomorphism of groups. In particular, for any two divisors Dl and D2 on X, we have isomorphisms

O(D}) 0 O(D 2 ) ~ O(DI

(B.3.6)

+ D2 )

O(Dd* ~ O(-Dd·

(B.3.7)

B.4. The degree of a line bundle In this Section, we discuss how the classification of Coo line bundles over a closed oriented surface may be realized by means of a single numerical invariant, the degree. Consider a Coo line bundle £ over an oriented Coo surface X, P a point of X and s a continuous section of £ over a neighbourhood of P which does not vanish on U - {P}. Let us choose a trivialization of £ on a neigbourhood V of P and let be a small closed contour in U n V which goes once around P, anticlockwise under the orientation of X. Using this trivialization, the restriction of s to may be identified with a continuous map -+ C* , and the winding number of this map is easily seen not to depend on the choices of the trivialization and of It is called the index of s at P and is denoted Indps. It is non-zero only if s(P) = O. See figure 26 for examples where X = C and £ is Tx, the holomorphic tangent bundle.

r

r

r

r.

Proposition -Definition B.4.1. Let X be a closed oriented surface (ef. Appendix A) and let £ be a Coo line bundle over X. There exist (infinitely many) sections s E Coo(X; £) which vanish only on a finite subset of X and the integer

L

Indps

PEX

s(p)=o

associated to any s~lch section is independent of s. It is called the degree of £ and is denoted degx£.

By definition, the trivial line bundle has degree 0 and two isomorphic Coo line bundles have the same degree. If £1 and £2 are two Coo line bundles and if SI and S2 are sections of £} and £2 with finite sets Fl and F2 of zeros, then S1 0 32 is a section of £1 0 £2 which vanishes only on Fl U F2 and for any P, we have This implies that

(B.4.1) This identity applied to £1 = £ and £2 = £* shows that

124

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

degxC* = -degxC.

(B.4.2)

The following proposition explains the significance of the degree of Coo line bundles over a closed surface. Proposition B.4.2. Lei X be an oriented closed surface and C 1 and C 2 be two Coo line bundles over X. The following three conditions are equivalent: i) C 1 and C2 are isomorphic as topological line bundles; (i. e., there exists a continuous isomorphism from C 1 to C2 .) ii) C 1 and C 2 are isomorphic as Coo line bundles; (i. e., there exists a Coo isomorphism from C1 to C2 .) iii) degx C 1 = degx C 2 .

Index 1

Index 1

Index -1

Index 2

Index 3

Fig. 26. The index of some vector fields.

According to (B.4.1), the Z-valued map degx, defined on the set of isomorphism classes of Coo line bundles over X, is in fact an isomorphism of Abelian groups, when the set of isomorphism classes of Coo line bundles over X is equipped with the structure of Abelian group defined by the tensor product 20 . In the sequel, we will consider the degrees only of holomorphic line bundles over a compact connected Riemann surface. 20

The only non-trivial point in the verification of the group axioms is the existence of an inverse to the class of a line bundle [: it is given by the class of [*, since [* lSI [ is isomorphic to the trivial bundle.

J. -B. Bost

125

1£ X is such a Riemann surface, we may for instance consider the integers degxwx and degxTx. According to (BA.2), we have (BA.3)

and according to the Poincare-Hopf formula, degxTx may be expressed in terms of the genus g of X: (BAA)

degxTx = 2 - 2g.

Finally, the degree of the line bundle associated to some divisor on X is easily computed:

L:t=l niPi be a divisor on a compact connected Riemann surface. Then the degree of the line bundle OeD) is L:t=l ni.

Proposition B.4.3. Let D =

Indeed consider the section lO(D) of OeD) whose divisor is D and choose a Coo Hermitian metric II lion OeD). An elementary computation shows that s .-

10 (D)

---::--'--"--~

.- 1+ 111 0 (D) 112

is a Coo section of OeD) which vanishes exactly on P E IDI is the multiplicity of P in D.

IDI

and whose index at

d

We will call degree of the divisor D the degree

L:

n; of line bundle OeD).

;=1

Exercise. Use (B.4.3), (BAA) and Proposition BA.3 to compute the genus of the Fermat curve (cf. §I.4.2). Together with the isomorphism (B.3.3), Proposition BA.3 has the following immediate consequence:

Corollary BA.4. A holomorphic line bundle on a compact connected Riemann surface X whose degree is negative has no non-zero holomorphic section over

X. B.5. The operators

8 and 8£

Let U be an open subset of
af ( Oz x

.) _ ~ (af(x + iy) 2 ax

+ zy -

thus,

8f

-af:= of Oz· dz; is an element ofCoo(U;wu).

E Coo(U; q, one defines

.af(x + iY)) ay

+Z

and

(B.5.1)

f

126

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

The operator (j so defined clearly satisfies the Leibniz formula (B.5.2) Moreover

8f vanishes iff f satisfies the Cauchy-Riemann equation

of ox

.af _ 0

+ loy

-

,

i. e., iff f is holomorphic. These remarks show that (B.5.3)

8( h . f)

= h . 8f

if h is holomorphic.

Let now c.p : U -) V be a holomorphic map between two open subsets of
8(f 0 c.p) = c.p*(8f).

(B.5.4)

This implies that the differential operator 8 may be defined on the Coo functions over any Riemann surface, in such a way that (B.5.1) holds locally for any holomorphic coordinate z. A direct intrinsic definition of 8f is simply:

Clearly the relations (B.5.2) and (B.5.3) are still true on any Riemann surface, as well as the characterization of holomorphic functions as the Coo functions f such that 8 f = O. Consider now a holomorphic line bundle C on a Riemann surface X. Let U be an open set in X such that, over U, C possesses a holomorphic trivialization t (in other words, t is a non-vanishing holomorphic section of Cover U). Then any s E Coo(U, C) may be written s = ft where f E Coo(U; q and the relation (B.5.3) easily implies that the element 8f· t of coo(U; C®w x) does not depend on the choice of t. This shows that there exists a unique first order differential operator acting on the section of C with values in the sections of C ® wx which sends s to 8f . t for any such U, s, t and f. It is called the 8-operator with coefficients in C and denoted by 8 c.. By construction of 8e, for any s E Coo(U; C), we have

8 es

= 0

{::?

s is holomorphic on U.

In particular HO (X; C) is the kernel of

8c. : Coo(X; C) -) Coo(X; C ® wx). The cokernel of this map, i. e., the quotient vector space

J.-B.Bost

127

is called the first Dolbeault cohomology group of £ and is denoted by HI (X; £) (HO(X; £) is also called the zero-th cohomology group of C). Contrary to HO(X; C), the significance of the group HI(X; £) is not very intuitive a priori. The various theorems stated in the next Sections should make clear its great usefulness. Let us only say that it will appear as a vector space 'measuring the obstructions to build holomorphic sections of £ over X'. Suppose now that £ and wx are equipped with Coo Hermitian metrics II 1I.e and II Ilwx· These metrics determine a Coo Hermitian metric on £ 0 wx, such that

II R0a 11.c0wx=1I R 1I.e ·11 a IIwx for any R E £p and any a E wx,p, and a positive 2-form p, on X given by (B.2.2). Using p" II . 1I.e and IIlIwx, we can define on Coo(X; £) and on Coo(X; £0 wx) some L2 Hermitian norms II 11£2: for any s E Coo(X; £) and any t E Coo(X; £ 0 wx),

Ix II 111,2:= Ix II

II s 111,2:=

sex)

112

p,(x)

II t

t(x)

112

p,(x).

These L2 structures on COO (X; £) andCoo(X; £0wx) allow to define the adjoint &~ of &.e by the identity

(&.es, t) = (s, &~t) where ( , ) denote the scalar products associated with the Hermitian norms

II 11£2. This being set, it is possible to give a heuristic interpretation of Hl(X; £) which may be appealing to some mathematical physicists: formally, HI(X; £) may be identified with the kernel of &~ - the 'zero-modes of the adjoint of &.e' in the language of physicists. Indeed, in the finite dimerisional case, if T : E -+ F is a linear map between two finite dimensional vector spaces E and F endowed with Hermitian scalar products, then the kernel of the adjoint T* of T, defined by (Tx, y) = (x, T*y) for any (x, y) E E x F, is isomorphic with coker T := F/T(E), via the map ker T* -+ coker T Y

f-+

[y].

B.6. The finiteness theorem The following theorem is a most basic fact concerning holomorphic sections of holomorphic line bundles on a compact Riemann surface.

128

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

Theorem B.6.1 (Finiteness Theorem). Let X be a compact Riemann s'U,rface and let £ be a holomorphic line bundle on X. The Dolbeault cohomology groups HO(X;£) and Hl(X;£) are finite dimensional vector spaces.

To show the significance of the finite dimensionality of Hl(X;£), let us prove the following Proposition B.6.2. Let us keep the notations of Theorem B.6.1. For any P EX, there exists a meromorphic section s of £ which is holomorphic on X - {P} and which has a pole at P of order at least 1 and at most 1 + dimH 1 (X; C).

Consider a local holomorphic coordinate z at P, a non-vanishing holomorphic section t of £ on an open neighbourhood U of P and a function p E COO(X; q such that p == 1 near P and such that p vanishes outside a compact subset of U. For any positive integer i, the section 8p· z-it (resp. pz-it) of 1: (8) wx (resp. of £) over U - {P} extended by zero defines an element of COO(X; £ (8) wx) (resp. of COO(X - {P}, C)). Moreover, on X - {P}, we have

(B.6.1) Linear algebra shows that, if d ::::: dimH 1(X; C), there exists (AI"'" Ad+1) E Cd+ 1 - {O} such that the class of d+l Q

= LAJ'jp. z-i t i=l

in Hl(X;£) vanishes. Then there exists (3 E COO(X;£) such that

(B.6.2)

Q

= 8c (3,

and, according to (B.6.1) and (B.6.2), d+1

s=

L AiPZ-i t -

(3

i=1

is an element of COO(X - {P}, £) such that 8c s = O. Thus s is holomorphic on X - {P} and its 'polar part' at P is d+l

LAi z - it i=l

which is non-zero and of order at most d + 1 by construction. Observe that, according to (B.3.3), Proposition B.6.2 implies the following Corollary B.6.3. Any holomorphic line bundle £ on a compact connected Riemann surface X is isomorphic, as a holomorphic line bundle, to the line bundle OeD) associated to some divisor D on X.

J. -B. Bost

129

Observe also that, applied to the trivial line bundle 0, Proposition B.6.2 shows that for any P E X there exists a non-constant meromorphic function f on X which is holomorphic on X - {P} and has a pole of order at most 1 + dimH 1 (X; 0) at P. In particular, this prove that meromorphic functions separate the points of X (compare with Theorem I.4.3). B.7. HI(X;C) and polar parts of merom orphic sections

The construction used to prove Proposition B.6.2 may be extended to provide an alternative description of the cohomology group H J (X; C) in terms of meromorphic sections of C and of their 'polar parts' (hence a purely 'holomorphic' description). To do this we need to introduce a few notations: • M(X; C) will denote the vector space of meromorphic sections of Cover X', • for any x E X, P(x;C) is defined as the quotient of the set M(x;C) of pairs (U, s), where U is an open neighbourhood of x in X and sis meromorphic section of C on U, by the equivalence relation v defined by (U, s)

rv

(V, t) {:} t - s is holomorphic at x.

The space P(x; C) is the space of 'polar parts at x' of meromorphic sections of C. Indeed, if t is a non-vanishing holomorphic section of C on a neighbourhood of x and if z is a local coordinate at x such that z( x) = 0, then one defines an isomorphism between the vector space of polynomials in z-l without constant . ~d' ~d' term and P(x; C) by sendmg L..-i=l aiz-' to the class of L..-i=l aiz-'i. • P(X; C) will denote the subspace of ITXEx P(x; C) defined by the condition (Px)xEX E P(X;C) {:} Px -=I 0 only for a finite set of x's.

• Pc will denote the linear map from M(X; C) to P(X; C) which associates to a meromorphic section s of C over X the element (px )xEX of P(X; C), where Px is the class of sin P(x; C). (In other words Pe(s) is the family of 'polar parts' of s.) Consider now (Px)xEX E P(X;C) and let {PI"" ,Pn } = {x E X

I Px

-=I OJ.

Choose for each i = 1, ... , n a meromorphic section 1; on some neighbourhood Ui of Pi whose class in P(Pi; C) is PPi and a function Pi E C=(X; q such that Pi == 1 near Pi and such that Pi vanishes outside some compact subset of Ui. Then, the section fi' lJPi of C ®wx over Ui - {P;} extended by zero defines an element of C=(X; C ®wx). One easily checks that the class of 2:7=1 fi . lJPi in Hl(X;C) only depends on the class of (Px)xEX in cokerPe := P(X; C)! Pc(M(X; C)). (This space is called the space of repartition classes of X). In that way, one defines a linear map

130

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

Ie : cokerPe

-+

[(PX)XEX]

f->

Hl(X; £,)

[t

J; .8Pi ].

Theorem B.7.!. The kernel of Pe is HO(X; £'). The map Ie is an isomorphism from the eokernel of Pe onto Hl(X;£,). The first assertion is clear. The second one means that Hl (X; £,) 'measures the obstruction' to finding meromorphic sections of £, with prescribed polar parts. See §C.4 for a proof. Observe that, when £, = WX, COO(X;£,) is a subspace of the space of Coo complex I-forms over X, COO(X; £, ® wx) may be identified with the space of Coo complex 2-forms over X (ef. (B.2.I)), and up to a sign, 8e is nothing else than the exterior differential d acting from I-forms to 2-forms (indeed, in local coordinates we have (B.7.I)

d(fdz) = df 1\ dz = 8f(0,1) 1\ dz

because of 1\ dz = 0, since dz 1\ dz = 0). In particular, the image of 8e is formed by exact two forms and their integral over X vanishes. This shows that the following map is well defined:

The insertion of the factor (21l'i)-1 as well as the notation Res are explained by the following result, which we leave as an exercise.

Lemma B.7.2. For any (ax)xEx E P(X;wx), we have Res 0 Pwx ([(ax)xEx]) =

L

Resxa x.

xEX (In this equality [( ax)xEx] denotes the class of (ax )xEX in cokerPe and Resxa x the residue at x of any meromorphic differential representing ax).

B.8. Serre duality An important complement to the Finiteness Theorem B.6.I is the Serre duality Theorem. To formulate it, we need a few preliminaries.

J. -B. Bost

131

The Leibniz rule (B.5.2) generalizes immediately as follows: for any two hololllorphic line bundles £I and £2 on X, and for any h E COO(X;£d and any 12 E COO(X; £2), then (B.8.1)

8C 0C (h ·h)=h·8 h+8 Jl·h (ECOO(X;£10£20wx)). 1

In particular, if

c

c2

2

h

E

HO(X,£d, we have

8C10C2(h . h) = h ·8c h. 2

This shows that the bilinear map

HO(X; £d X COO(X; £20 wx) (h, (2)

--* f--t

COO(X; £10 £20 wx) fl . a2

yields a quotient map

HO(X; £d (B.8.2)

X

H1(x; £2) (h, [a2])

--* f--t

H1(X; £10 £2) := [h . a2J.

h . [a2J

We can now state:

Theorem B.8.1 (Serre duality). Let X be a compact Riemann surface and let £ be a holomorphic line bundle on X. The bilinear map

obtained by composing the product (B.8.2)

HO(X;£* 0 wx) x H1(X;£)

--*

H 1(X;C* 0£0wx) ~ H1(X;wx)

with the map Res is a perfect pairinll .

Observe that, by the definitions of Res and of the product (B.8.2), for any

s E HO(X; £) and a E COO(X; £* 0 Wx 0 wx), we have 1 (s, [aJ) = -2. [ sa, 7rl

Jx

where sa is a section of £0£* 0wx 0wx, identified with (A2T* X)c. Theorem B.8.1 asserts that this pairing defines an isomorphism

and implies that (B.8.3) 21

Recall that, if E and F are two finite dimensional vector spaces, a bilinear map (0,.) : Ex F' -7 C is called a perfect pairing when the map (E -7 F*,x f---7 (x,,)), or equivalently the map (F -7 E*, Y f---7 ( ' , y)), is bijective.

132

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

Corollary B.8.2. Let X be a compact connected Riemann surface of genus g. 1) There exists a canonical isomorphism

In particular

(B.8.4)

dimHl(X; 0) = g .

Moreover the map S;P (X)

-+

Hl(X; 0)

deduced from the inclusion Ql (X) '-+ COO(X; Wx) is an isomorphism. In other words COO(X;wx) = QI (X) EB COO(X; C).

a

2) The map is an isomorphism.

The first assertion in 1) (resp. the assertion 2)) follows from Theorem B.8.1 applied to C = 0 (resp. C = wx). The equality (B.8.4) is then a consequence of Corollary (B.2.6). The last assertion in 1) is implied by the equality of dimensions dim Ql(X) = dim Ql(X)* = dimHl(X;O) and by the injectivity of the map Ql(X) -+ HI(X; 0), which follows from the identity 1 (:8, [;3]) = -2. [73 1\;3 < 0 7rZ

Jx

for any ;3 E QI (X). Exercise: Deduce from Corollary B.2.4, Proposition B.6.2, and (B.8.4) that any compact connected Riemann surface of genus 0 is biholomorphic to jp'1
B.9. Riemann-Roch theorem Recall that, for any two vector spaces E and F, a linear map T : E -+ F is said to have an index if the kernel kerT and the cokernel coker T := FjT(E) of T are finite dimensional vector spaces. The index of T is then defined as the integer ind T = dim ker T - dim coker T (compare [Bel] §1.4). According to the very definition of the cohomology groups HO(X; C) and HI (X; C), the Finiteness Theorem B.6.1 precisely asserts that the operator

J. -D.

8c : COO(X; C)

--t

Bo~t

1;);!

COO(X; C ® wx)

has an index. Its index, namely dim ker8 c - dim coker8c = dimHO(X;C) - dimH 1 (X; C), is given by the famous Riemann-Roch theorem:

Theorem B.9.1. Let X be a compact connected Riemann surface of genus 9 and let C be a holomorphic line bundle on X. Then:

(B.9.I) Corollary B.9.2 (Riemann's inequality). Let D = E~=l niPi be a divisor on X. Then k

(B.9.2)

dim{f E M(X) I divf ~ -D} ~ 1- 9 + Lni. i=l

This follows immediately from (B.9.I) applied to C = OeD) and from (B.3.5) and Proposition (B.4.3). Observe that Corollary B.9.2 is an existence statement concerning meromorphic functions with prescribed zeroes and poles, which answers the question discussed at the beginning of §B.3. Indeed (B.9.2) implies that if E~=l ni > g, there exists a non-constant meromorphic function f on X such that divf ~ -D. Classically, Riemann-Roch formula (B.9.1) is combined with the equalit.y (B.8.3) and is written as

(B.9.3) Observe that, applied to C = w x, this equality becomes 9- 1= 1- 9

+ degx C

since dimHO(X; wx) = 9 and dimHO(X; 0) = 1 (cf. Corollary B.2.6 and §B.1.7). This shows that the degree of Wx is 2g - 2 as mentioned above (see (BA.3) and (BAA)). Consequently, if degxC > 2g - 2, then degxC* ®wx < 0 and HO(X; C* ® wx) = 0 (by (BA.I), (BA.2) and Corollary BAA) hence Hl(X;C) = 0 (by (B.8.3)) and Riemann-Roch formula reads (B.9A)

Of course, this equality has an interpretation in terms of divisors, analogous to (B.9.2). At this point of our discussion, the proof of Riemann-Roch Theorem is not difficult. Since any holomorphic line bundle over X is isomorphic to the line

134

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

bundle O(D) associated to some divisor D, a simple induction shows that to prove (B.9.I), we only need to prove that: i) (B.9.I) is true for I: = 0; ii) for any holomorphic line bundle over X, (E.9.I) is true for I: = N iff (B.9.I) is true for I: == N ® O( -P). However the validity of (B.9.I) rewritten as (B.9.3) is clear when I: = 0, since dim HO(X; 0) = 1 and dimHO(X; wx) = g. As for ii), since

it amounts to proving that

dimHO(X;N ® O(-P))- dimH 1 (X;N ® O( -P)) + 1 = dimHO(X;N) - dimH 1 (X;N) that is, according to Theorem B.7.I, (B.9.5)

indP.c00(-P)

+1=

indP.c.

The proof of (B.9.5) will be based on the following lemma of linear algebra, which we leave as an exercise 22 • Lemma B.9.3. Let E, F and G be vector spaces and let u : E - t F and v : F - t G be linear maps with index. Then v 0 u : E - t G is a linear map with index and

ind v

0

u = ind u

+ ind v.

Observe that the meromorphic (resp. holomorphic) sections of N 0 O( - P) over an open subset U C X may be identified with the meromorphic sections of N over U (resp. with the holomorphic sections of N over U which vanish at P if P E U). These identifications provide linear bijections ~

pO: M(X;N ® O(-P))

M(X;N)

and

M(x;N ® O(-P)) ~ M(x;N) and linear maps

P(x;N ® O( -P))

-t

P(x;N)

and

pI : P(X;N ® O( -P))

-t

P(X;N).

The following properties of pO and pI are easily checked: • the following diagram 22

We will use Lemma B.9.3 only when u is an isomorphism or when v is onto and dim ker v = 1. In these two cases, the proof of Lemma B.9.3 is very simple.

J. -B. Bost

135

PJI!0 0 (-P)

M(X;N@ O( -P))

)

P(X;N @ O( -P))

1 po

(B.9.6)

1pi PJI!

M(X;N)

----t

P(X;N)

is commutative (i. e., pI 0 P.N00( -P) = P.N 0 pO) . • pI is onto and the kernel of pI is generated by the class of (Px )xEX, where Px = 0 if x =I P and PP has a simple pole at P. In particular ind pI = 1. Thus the four maps which occur in (B.9.6) are maps with index and Lemma B.9.3 shows that indPN00(-P) + 1 = ind P.N00(-P) +ind pI

= ind

pI 0 P.N00( -P)

= ind P.N 0 pO = ind PN + ind pO = ind PN,

as was to be proved. Finally, let us give a consequence of the results of this Appendix which will playa key role in the construction of the Jacobian embedding of compact Riemann surfaces. Proposition B.9.4. Let X be a compact connected Riemann surface of genus g 2:: 1. Then for any P E X, there exists wE QI(X) such that w(P) =I O.

Assume the contrary. Then the injection

is a bijection and since degx Wx 0 O( -P) = degx Wx - 1, Riemann-Roch formula (B.9.I) and Corollary B.8.2, 2) show that dim Hl(X;wX

@

O( -P)) = dim HI(X;wx) + 1 = 2.

Therefore, by Serre duality dim HO(X; O(P)) = 2 and there exists a nonconstant meromorphic function on X, whose only pole is P, and is a simple pole. By Corollary B.2.4, this implies that X is isomorphic to ]p'IC, hence has genus O.

136

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

Appendix C. Analysis on compact Riemann surfaces This appendix is devoted to the proofs of the Finiteness Theorem B.6.1, of the Serre duality Theorem B.S.l, of Theorem B.7.1 (isomorphism between HI and repartition classes) and of Theorem B.2.5 (Hodge decomposition).

C.l. Regularizing operators Let C I and C 2 be two Coo line bundles on a compact manifold X. Let k E cOO(X x X; C2 [3] (Cr 0 1/\ Ix)) and f E cOO(X; C1 ). For any x E X, Y 1-+ k(x, y)f(y) is a Coo section of C 1 ,x 01/\ Ix, and the integral

(C.1.1)

Kf(x):=

f

JyEx

k(x,y)f(y)

is a well defined element of C I ,x. Moreover, the section K f of C 1 over X so defined is Coo. (This is clear by the elementary differentiability properties of integrals depending on a parameter when f is supported by a chart on which C2 is trivial. In general, partitions of unity reduce to this case). The section k is easily seen to be uniquely determined by the operator

K: cOO(X;Ct}

~

c OO (X;C 2 )

and is called its kernel. The linear maps associated in this way to kernels in 1/\ Ix)) are called the regularizing operators from

COO (X x X; C2 [3] (Cr 0 cOO(X;CJ) to cOO(X;C 2 ).

A special class of regularizing operators are the regularizing operators of finite rank, which are defined by kernels k given by finite sums N

(C.1.2)

k(x, y) =

L
where
1/Ji

E cOO(X;£:i 01/\ Ix), and have the form

The Finiteness Theorem B.6.1 and the Serre duality Theorem B.S.l are simple consequences of the following two propositions:

Proposition C.l.I. Lei C be a Coo line bundle on a compact manifold X and let

K : cOO(X; C)

~

cOO(X; C)

be a regularizing operator. Then: i) Id +K is an operator with index from cOO(X; C) to itself

J. -B. Bast

ii) For any linear map A: COO(X;£) --; C such that Ao(Id+K) exists I! E COO(X; £* ® 1/\ Ix) such that, for any f E COO(X; C),

A(f) =

fx

= 0,

137

thf-rf-

I!(x)f(x)dx.

Proposition C.1.2. For any holomorphic line bundle £ over a compact Riemann surface X, there exists a linear map

P: COO(X;£ 0wx) --; COO(X;£) and rf-gularizing operators

and such that

and

Proposition C.l.1 is a variant of Fredholm's theory. The operator P, whose existence is asserted in Proposition C.l.2, is traditionally called, after Hilbert, a parametrix of Be. Proof of Theorems B.6.1 and B.S.1 (taking Propositions C.l.I and C.l.2 for granted): • ker Be is clearly contained in ker Po Be = ker(Id +KJ). This space is finite dimensional according to Proposition C.l.I i), applied to [(1. Hence HO(X;£) is finite dimensional. • Be (COO(X; c»~ clearly contains

This space has finite codimension in COO (X; £ 0 wx) according to Proposition C.l.1 i) applied to K = [(2. Hence the same is true for Be (COO (X; C»~, i.e., HI (X; £) is finite dimensional. This proves Theorem B.6.l. • Consider an element A in HI(X;£)* i.e., a linear map

138

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

which vanishes on the image of 8e. Then A vanishes on the image of 8e 0 p = Id+J(2. Therefore, according to Proposition C.l.l ii) applied to J( = K 2 , there exists e E Coo(X; (C ® wx)* ® 1/\ Ix) such that, for any f E Coo(X; C ® wx)

A(f) =

(C.l.3)

1.

C(x)f(x).

",EX

The section C may be seen as a Coo section of C* ® w x. Indeed,

and

Wx ® 1/\ Ix ': : '. wx,

since (ef. §B.l.9 and (B.2.I»

1/\ Ix

'::::'./\~

': : '. Wx

®wx.

The relation (C.1.3) uniquely determines C, and the map ,\ f-t e so defined is an isomorphism from Hl(X; C)* onto the subspace of Coo(X; C ® wx) formed by the sections C such that

This subspace is nothing else than HO (X; C* ®w x), as follows from the following Lemma C.1.3. For any s E Coo(X; C) and anye E Coo(X; C* ® wx), we have

LC.8es=- L8e*0 w X C. S . Indeed

L e· 8e s + L 8e*0wx C· s = L 8(e· s) =

L

d(C· s)

= 0

(ef.

(B.8.I»

(ef· (B.7.1» (Stokes formula).

This proves Theorem B.8.I, since the bijection

HO(X; C ® wx) -+ Hl(X; C)* Cf-t A coincides (up to a factor 21l"i) with the linear map HO(X; C®wx) -+ Hl(X; C)* defined by the bilinear pairing (-, .), which therefore is non-degenerate.

J. -B. Bost

139

C.2. Fredholm's theory This Section is devoted to the proof of Proposition C.l.l. We will indicate the main lines of the proof and leave (easy) details to the reader. We begin with a few observations: • If Kl and K2 are two regularizing operators from COO(Xi C) to itself, with kernel kl and k2' then Kl 0 K2 is a regularizing operator, whose kernel is (C.2.I) In particular, if Kl or K2 is a regularizing operator of finite rank, then 1(101(2 is also such an operator. • Choose Coo Hermitian metrics II 11.e and II 1III\Ix on the line bundles C and 1/\ Ix. By duality and tensor product, these metrics define a Hermitian metric II . lion C [8J (C* 01/\ Ix). Using II II, we can define a norm III Ilion the space CO(X x Xi C [8J (C* 01/\ Ix)) of continuous kernels as III k III =

II k(x,y) II .

sup (x,y)EXXX

Equipped with this norm, CO(X x Xi C [8J (C* 01/\ Ix)) is a Banach space. Moreover, one easily checks that the composition of kernels (C.2.I) is still well defined for continuous kernels and that the following estimate holds, for any two continuous kernels kl and k2 : (C.2.2)

IIIkJ * kzlll

:::;

Mlllktllllllkzlll,

where M is the integral over X of the density p defined locally on X by Idxl /\ ... /\ dXnl -:-:--c:-'--"---:--""";":- II dXJ /\ ... /\ dXn 1III\Ix .

p -

The proof of Proposition C.l.I is based on three preliminary lemmas. Lemma C.2.I. The kernels defining the regularizing operators of finite rank (i.e., the kernels of the form (C.l.2)) are dense in COO(X x X, C [8J (C* 0 1/\ Ix)) equipped with the norm III III·

Using partitions of unity, charts and local trivializations of C, this follows from the fact that, if DJ and Dz are two (closed) balls in ]Rn, any Coo function on Dl X Dz may be uniformly approximated by functions of the form N

(x, y)

f-+

L Ji(X)gi(Y) i=1

where Ji E COO(DJ) and gi E COO(D z ). This fact may be deduced, for instance, from the existence of uniform polynomial approximations.

140

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

Lemma C.2.2. Let kernel k satisfies

J{:

COO(X;£) ....... COO(X;£) be a reg1darizing operator whose 1

(0.2.3)

III k III < M'

Then Id +J{ is invertible and there exists a regularizing operator

L: COO(X;£) ....... COO(X;£) such that (Id+J{)-l = Id+L.

Let us define

* ... * k

kn = k

(n factors).

Then it follows from (0.2.2) that

Mlllknill ::; (Mlllkilit and from (0.2.3) that 00

L

IIIknlll < +00.

n=l Therefore

00

n=l

is a well defined element of CO(X x X; £ !IJ (£* ® 1/\ Ix )). Moreover, the continuity of the composition product * with respect to the norm I I I I (cf. (0.2.2)) implies that 00

L (-1 t

k + C* k = k +

(0.2.4)

kn

*k

n=l 00

n=l

= -C and similarly that

k + k * C = -C.

(0.2.5)

Inserting (0.2.4) in (0.2.5), we get: C = -k

+ k * k + k * C* k

On the other hand, the formula

shows that

J.~

* .e * k is

Coo. Therefore.

e is COO.

.

J. -B. Bost

141

If L denotes the operator of kernel €, the relations (C.2.4) and (C.2.5) are equivalent to

K+L+LK=O

K+L+KL=O

~d

that is to (Id+L)(Id+K) = (Id+K)(Id+L) = Id. This proves Lemma C.2.2. Lemma C.2.3. Proposition C.l.l is true when K is a regularizing operator of finite rank.

Indeed, if 'Pi E COO(X; C) and ¢i E COO (X; C* ®

1/\ Ix), i

= 1, ... , N, are

N

such that the kernel of K is (x,y)

1--+

2: 'Pi(X)¢i(Y),

then

i=1

ker(Id+K) = {f E COO (X; C) If = -Kf} is contained in the vector space spanned by the 'Pi'S, and (Id+K)(Coo(X;C)) contains ker K and a fortiori the subspace of COO (X; C) defined by the vanishing of the linear forms

f

1--+

fx

¢i . f

, i

= 1, ... ,N.

This clearly shows that Id + K has an index. Moreover, if a linear form A - t
Coo (X; C)

A(.f) = - A 0 K(.f) N

~ A('Pi)

= -

fx

fx

¢i . f

€. f

where N

€=

-

L A('Pi)¢i. i=1

Finally, we can prove Proposition C.l.l. Let K : Coo(X; C) - t Coo(X; C) be a regularizing operator. According to Lemma C.2.1, we can write K = KI + K2 where K2 is regularizing of finite rank and where IIIKIIII < Then Lemma C.2.2 shows the existence of a regularizing operator L such that

it.

Now we get:

142

Chapter 2. Compact Riemann Surfaces, Jacobians and Abelian Varieties

Id+J{ = (Id+J{l +J{2)(Id+L)(Id+J{1) = (Id +J{2 + J{2L )(Id +J{l).

(C.2.6)

According to Lemma C.2.3 applied to /(2 + /(2L, which is regularizing of finite rank, ker(Id +/() = (Id +/(d- 1 (ker(Id +/(2 + J{2 L)) is therefore finite dimensional and

has finite co dimension in COO(X; C). Finally, if ,\ is a linear form such that ,\ 0 (Id +/() = 0, then (C.2.6) shows that (Id+/(2 +

,\ 0

/(2L) =

0

and, again by Lemma C.2.3, ,\ is of the required form. C.3. A parametrix for

8e

This Section is devoted to the proof of Proposition C.1.2. Our proof is based on the fact that an 'inverse' of 8 acting on Coo functions with compact supports on
FOT

any