SPECIAL DIVISOR S O N ALGEBRAI C CURVE S 5 Finally, we shall recall a little bit o f Hodg e theory. I t i s clear tha t ther e i s an inclu- sion H0^1) C //^R(C) in fact, H°(£2l) give s a ^-dimensional subspace of the 2^-dimensiona l vector spac e H^R(C), on e that ha s the followin g propert y (whos e proof i s also clear): (1.9) / / we consider the cup-product pairing (1-10) Q: H^R{C) ® H*K(C) C , then H0^1) is an isotropic subspace and the nondegenerate pairing (1.10) gives the duality (1.11) H^n'r^H^iO/H 0 ^1). With the tool s (1.4) , (1.8), and (1.11 ) we are read y t o discus s the proble m o f com - puting the dimensio n 1(D) o f L(D). An y function/ G L(D) is determined u p t o a n additiv e constant b y it s differentia l df. W e denote by D' th e diviso r o f df, an d observ e that dfE H°(£ll(D')) ha s no residue s an d is zero in H^R(C). Conversely , suppose w e consider al l possible candidates fo r th e Lauren t development s o f differential s o f functions / G L(D). B y (1.4) there i s a global meromorphi c differentia l o f th e 2n d kin d 0 having a given such Lau - rent development , an d 0 is unique u p to additio n o f a holomorphic differential . Findin g a function / with df=(p mo d H0^1) i s therefore equivalen t t o solvin g the proble m [0+co o ] = 0 in H^ R (C) for som e co 0 G H°(£ll). B y (1.11), this in turn i s equivalent t o (1.12) Q(0 , co) = 0 fo r al l co G H°(£l l ). It remain s t o comput e th e cu p product (1.12) . T o d o this we suppose tha t i where the pt ar e distinct , and we choose a local coordinate z t aroun d pr I f th e Lauren t de - velopment aroun d p ( o f ou r desire d functio n / is N fi = Z a vJZ " v = l then we note tha t th e conditio n (1.13) J ] Res p .(w/f) = 0 fo r al l co eH°(£l l ) i is intrinsic an d is satisfied i n case / exists . I n fac t i t i s not to o difficul t t o establis h equalit y of the left han d side s of (1.12 ) and (1.13) , so that thes e latter equation s giv e exactly th e conditions tha t th e prescribe d "Lauren t tails " ft come fro m a global meromorphic function . In particular, if w e define th e inde x of specialit y i(D) to b e the dimensio n o f th e sub- space ^(^(-D^CH0^1)
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