Page 172 - Curvature and Homology

154 V. COMPLEX MANIFOLDS

It is easy to see that an element of GL(2n, R) belongs to the real repre-
sentation of GL(n, C), if and only if, it commutes with J,.
A metric may be defined on V by prescribing a positive definite
symmetric tensor g on V (cf. § 1.9). In terms of a given basis of V we
denote the components of g by gA,. Suppose V is given the complex
structure J. Then, an hermitian structure is given to V by insisting
that J be an isometry, that is, for any v E V

An equivalent way of expressing this in terms of the prescribed base is

The tensors g and J are then said to commute. The space V endowed
with the hermitian structure defined by J and the hermitian metric g
is called an hermitian vector space. It is immediate from (5.2.7) and
J2 = - I that for any vector a, the vectors v and Jv are orthogonal.
Let FAB = FAC gBC and consider the 2-form 52 on V defined in terms
of a given basis of V by
l2 =&FABwA A wB (5.2.9)

where the uA(A = 1, ..., 2n) are elements of the dual base. We define
an operator which is again denoted by J on the space of real tensors t
of type (092) by
( JZ)AB = ~AC FB'. (5.2.10)
Denoting by J once again the induced map on 2-forms and taking
account of (5.2.8) we may write JSZ = g.
The metric of any Euclidean vector space with a complex structure
can be modified in such a way that the space is given an hermitian
structure. To see this, let V be an Euclidean vector space with a complex
structure defined by the linear transformation J. Define the tensor k
in terms of J and the metric h of V as follows:

Since the metric of V is positive definite, so is the quadratic form k
defined by h, and therefore, the metric defined by

is also positive definite. A computation yields