Page 251 - Curvature and Homology

Since the functions fa8 are holomorphic in Ua n U8, it follows that
a2 log a, - a2 log a@
--- asi a$
azi a~i in U, n Ug.
Thus, the 2-form
a2 log a,
yij* dz' A d9 = = d9
dzi
A
azi a9
is defined over the whole manifold M (cf. V.D).
4 form 4 (form of bidegree (p, 9)) with coefficients in B is a system
{$,I differential forms (forms of bidegree (p, q)) defined in {U,} such
of
that
Following 5 5.4 we define complex analogs d', d", 8' and 6" of the
operators d and 6 for a form 4 = {+,I with coefficients in B:

(a : not summed)-the star operator * being defined as usual by the
Kaehler metric of M. In terms of these operators it can be shown that
A = 2(d'S1 + S'd')

is the correct operator for the analogous Hodge theory - being called
harmonic if it is a solution of A$ = 0.
If M is compact it is known that Hp(M, A@(B)) g HqpP(B)-the vector
space of all harmonic forms of bidegree (q, p) with coefficients in B [47]. It
follows that dim Hp(M, A q(B)) is finite for all p and q.
Since fa@ fay fya = 1 in Ua n Up n U,,

is a constant in U, n Ug n U,, where caBy E 2. The system {cap,) C Z
defines a 2-cocycle on the nerve N( 8) of the covering % (cf. Appendix A
and [72]). It therefore determines a cohomology class cN E H2(N( a), 2);
indeed, by taking the direct limit
H2(M,Z) = lim H2(N(%),Z)
4
we obtain an element c = c(B) E H2(M, Z) called the characteristic class
of the principal bundle associated with B.