Page 191 - Curvature and Homology

5.5. KAEHLER MANIFOLDS

Lemma 5.4.1. In C,
and

In the first place, it is easily checked that

and

Pre-multiplying the first of these equations by i(8,) and post-multiplying
the second by i(9,) one obtains after subtracting and summing with
respect to k
Ad - d/l = g(8' - 8'3

since i(9,) commutes with a,. The desired formulae are obtained by
separating the components of different types.

5.5. Kaehler manifolds
Let M be a complex manifold with an hermitian metric g. Then,
in general, there does not exist at each point P of M a local complex
coordinate system which is geodesic, that is a local coordinate system
(29 with the property that g is equal to 22, dsi @ dZi modulo terms
of higher order. (Two tensors coincide up to the order k at P E M
if their coefficients, as well as their partial derivatives up to the order k,
coincide at P. A complex geodesic coordinate system at P should have
the property that g coincide with 2& dzi @ up to the order 1 at P.)
We seek a condition to ensure that such local coordinates exist.
Let&, ..a, 8,) be a base for the forms of bidegree (1,O) on M with
the property that g may be expressed in the form

(cf. 5.3.30).
Our problem is to find n 1-forms w, of bidegree (1,O) such that
(i) wi(P) = 8,(P), i = 1, -, n;
(ii) g = 2 2, wt @ (3% modulo terms of higher order; and
(iii) dw,(P) = 0, i = 1, --, n.