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 .
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