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)