Page 212 - Curvature and Homology

194 V. COMPLEX MANIFOLDS

Hint: In a Riemannian parallelisable manifold, the map

where {Xi} and {P) are dual bases is an anti-derivation. Show that d and d agree
on AO(M) and Al(M), and hence on A(M).
If any of these conditions is satisfied, the manifold is Kaehlerian and 52 is the
fundamental form defining the Kaehlerian structure. Note that
g(X, JY) + g( JXY) = 0-
Incidentally, from the formula

we may derive the formula

Hence, (&) (X, Y) = Xa(Y) - Ya(X) - a([X, Y]) (cf. formula (3.5.2)).
5. If M is Kaehlerian, show that DxA qer(M) C A Qmr(M) for every pair of inte-
gers (w) and any X E T*
6. Let M be a complex manifold, J the linear endomorphism of T defining the
complex structure of M and 52 a real form of bidegree (1,l) on M. Then,

for any X,Y E T. Show that the 'metric' g defined by

is symmetric, hermitian and real; hence if 8 is closed and g is positive definite,
the metric is Kaehlerian.

D. The 2-form D
1. The form
8 = 1/Td'd"f
where f is a real-valued function of class oo on the complex manifold M is real,
closed and of bidegree (1,l). Let {(I,) be an open covering of M. For each i