Page 222 - Curvature and Homology

Corollary 1. The betti numbers of Pn are

Since P, is connected, it is only necessary to show that Pn is compact.
The following proof is instructive: In Cn+, with the canonical metric g
define the sphere

Consider the equivalence relation
eb - eo
defined by

where g, is a real-valued function. Pn is thus the quotient space of S2"+l
by this equivalence relation. In fact, P, may be identified with the
quotient space U(?t + l)/U(n) x U(1). To see this, consider the unitary
frame (eA, e,,), A = 0, 1, -.a, n obtained by adjoining to eo, n vectors
e, in such a way that the frames obtained from (eA, eA,) by a trans-
formation of U(n + 1) are unitary. Since the frames obtained from
(ec, e,,), i = 1, ..a, n by means of the group U(n) are unitary, Pn has
the given representation. That Pn is compact now follows immediately
from the fact that the unitary group is compact [27].
Incidentally, this gives another proof that P,, is a Kaehler manifold.
For, by the compactness of U(n + 1) we may construct an invariant
hermitian metric by 'averaging' over U(n + I). The fundamental form
52 is thus invariant. Hence, since U(n + 1)/ U(n) x U(1) is a symmetric
space, that is, the curvature tensor associated with this metric has
vanishing covariht derivative, 52 is closed (cf. V1.E for the definition
of a symmetric space). We have invoked the theorem that an invariant
form in a symmetric space is closed. (That Pn is a symmetric space
follows directly from the fact that with the Fubini metric it is a manifold
of constant holomorphic curvature). The reader is referred to V1.E
for further details.

Corollary 2. There are no holomorphic p-forms, 0 < p 5 n on Pn.
In degree 0 the holomorphic forms are constant functions.
Indeed, by (5.6.4) the pth betti number