Page 205 - Curvature and Homology

5.9. EXAMPLES OF KAEHLER MANIFOLDS 187
is, its coefficients ail.. j,. . . j, are harmonic functions which are invariant
sl
by r. Consequently, these functions are the images by v* of harmonic
functions on T,. But a harmonic function on a compact manifold is a
constant function, and so clr E r\?(Cn).
4. On a bounded open set M contained in C, there exists a well-
defined 2-form invariant by the group of complex automorphisms of M.
This is a consequence of the theory of Bergman. One can construct
canonically from this form a 2-form R having the Kaehler property,
namely, dR = 0 [72].
5. Complex projective n-space P,: By identifying pairs of antipodal
points of the sphere
n

contained in C,,, we obtain P,. For every index j, let Uj be the open
subspace of P, defined by ti # 0 where to, tl;-., tn denote the homo-
geneous coordinates of the points of P,. The map

is a holomorphic isomorphism of Uj onto C,. It is easily checked that
these maps for j = 0, 1, ..., n define a complex structure on P,.
Consider the functions yj = 2 a?: defined in each open set Uj
i=O
of the covering. On U,n Uk we have

z,! = Z:/Z; (k not summed)
and
n n
vk = 2 zyk = 2 (&:)zkL = vjz&j (j, k not summed)
i=O 6-0
where zi is a holomorphic function in Uk, and hence in Uj n Uk. The
yj define a real closed form R of bidegree (1,l) on P,. Indeed, in Uj n Uk
d'd"(l0g tpj - log 9,) = 0.
Hence, R is given by
A2 = 47 d'd" log 9,
in each open set Ui. In particular in Uo

SZ = 1/---i d'd" log g+,.