Page 206 - Curvature and Homology

188 V. COMPLEX MANIFOLDS

Clearly, 52 is a closed 2-form, and since

The associated metric tensor g (sometimes called the Fubini metric) is
given by

or, more explicitly by

We remark that the fundamental form 52 of any Kaehler manifold may
be written in the form (5.9.1). For, by 5 5.3, since the metric tensor g is
(locally) expressible as
ay
gij* =

for some real-valued function f,

6. Let M be a Kaehler manifold and M' a complex manifold holo-
morphically imbedded (that is, without singularities) in M. The metric g
on M induces an hermitian metric on M'. The associated 2-form 52'
on M' coincides with the form induced by 52 and is therefore closed.
In this way, the induced complex structure on M' is Kaehlerian
(cf. 5 5.8).
7. Let G(n, k) denote the Grassman manifold of k-dimensional projective
subspaces of P, [24. It can be shown that it is a non-singular irreducible
rational variety in a P, for sufficiently large N. Moreover, its odd-
dimensional betti numbers vanish whereas b,, is the number of partitions
of p = a, + a, + .*. + a, (a,: integers) such that 0 I a, a, $ .-.
Sa,Sn-k.