Page 246 - Curvature and Homology

A representative c, of the (n - r + 1)"t Chern class of an hermitian
manifold is given in terms of the curvature forms 83 by means of the
formula 1211

The theorem invoked above may be stated as follows:
The Euler characteristic of a compact hermitian manifold M is given
by the Gauss-Bonnet formula

As in 9 6.11, in each coordinate neighborhood U there exists N
holomorphic functions f' such that

by means of which M is mapped locally, (1-1) into CN. Moreover, the
metric g of M defined by the matrix of coefficients

is induced by the flat Kaehler metric

of C, where

is the rth abelian integral of the first kind on M.
To compute the curvature tensor of the metric g we proceed as
follows: In the first place, from (5.3.19) the only non-vanishing com-
ponents are given by

From (6. lU), since the functions a(:), r = 1, ..., N; i = 1, -.-, n are
holomorphic,