4

P. A. GRIFFITH S

and the orde r o f pol e N an d residu e a__

x

ar e both independen t o f th e local holomorphic co -

ordinate z . B y definition th e pola r diviso r is

(A)~ = 2/P, .

i

where 0 has pole o f orde r Nt a t p

i

G C.

We shall us e the standar d algebro-geometri c notatio n H°(^l

1(D))

fo r th e vecto r spac e

of meromorphi c differential s whos e pola r diviso r i s no wors e than a given diviso r D. A basic

existence theore m give s the dimensio n h

0(£ll(D))

o f

H°m1(D)),

fo r an y D, a s

(1.3) A°(n 1.l-d+g=)(/))

More precisely, the followin g holds :

(1.4) Given any divisor D = ^N^, there exists a meromorphic differential 0 having

preassigned Laurent developments (1.2) around the points p

(

, provided only that the residue

theorem

(1.5) ZRes

p

,(*) = 0

is satisfied. Any such 0 is unique up to addition of a holomorphic differential co G H

0^1),

and the dimension h°(£l

l)

of this space is equal to the genus g.

Next, we recall that a meromorphic differentia l 0 is said t o b e of th e secon d kin d i f i t

has no residues . I n this cas e the integra l

(1.6) jd

7

over an y close d curv e 7 supported awa y fro m th e singularitie s o f 0 depend s onl y o n th e

homology clas s [7 ] ^H^C, Z ) o f 7 i n the whol e Rieman n surface , an d consequentl y 0 de-

fines a linear functio n [0 ] o n H^C, Z) . I t i s clea r tha t [0 ] = 0—i.e. , the integra l (1.6) is

zero fo r al l closed curve s 7—if and onl y i f th e differentia l 0 is exact; this mean s tha t

0 = 4 f

for som e meromorphi c functio n / o n C, and moreove r an y suc h / i s unique u p t o a con-

stant. I n othe r words , if we define th e algebrai c d e Rham cohomolog y b y

H^R(C) = {differential s o f th e 2n d kind}/{exac t forms} ,

then ther e i s an injectio n

(1.7)

# D R ( C ) - *

/ /

1

( C Z ) * ® C .

A second basi c resul t i s the

(1.8) (Algebraic de Rham theorem for curves): The mapping (1.7) is an isomorphism.

Actually, (1.8) is somewhat o f a misnomer i n tha t thi s result fo r curve s is elementary .