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 (/))) = g + d - l . 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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