Page 213 - Curvature and Homology

EXERCISES 195

let fi be a real-valued function of class 00 with no zeros in Ui. If, for each
pair of integers (i, j) there exists a holomorphic function hij on Ui n Uj such that

then, there exists a real closed form Q of bidegree (1,l) on M such that
Q = 2/--li d'd" log f,
on each open set Ui.
2. Let {u,} be an open covering of M by coordinate neighborhoods with complex
coordinates (xi) and Y a real 2n-form of maximal rank 2n on M. Then, the
restriction Yi of Y to each Ui is given by

where fi is either a real or purely imaginary function with no zeros in Ui;
moreover, on Ui n Uj

where hi, is a holomorphic function on Ui n U,. Show that Y determines a
real, closed Zform of bidegree (1,l) and maximal rank on M.
Bergman has shown that on every bounded open subset S of C, there exists
a well-defined real form of degree 2n invariant under the complex automor-
phisms of S and independent of the imbedding. With respect to this form we
may construct a 2-form Q on S whose associated metric is Kaehlerian.

E. The fundamental commutativity formulae. Topology of Kaehler
manifolds [SO, 721

1. Let M be an hermitian manifold with metric g. Assume that T1eO is a free
F-module; this is certainly the case if M is holomorphically isomorphic
with an open subset of Cn. Let {x,, ---, Xn} be a basis of TIw0; then, {XI, .a*, x,,}
is a basis of TOe1. By employing the Schmidt orthonormalization process the
Xi, i = 1, .-., n may be chosen so that

(cf. equations (5.2.13) and (5.3.1)). Consider this basis of TC = T1*O @ TOJ
and denote by {&,&I, = 1, .-a, n the dual basis. Then,
i

and