Page 214 - Curvature and Homology

196 V. COMPLEX MANIFOLDS

Establish the formulae

and

Hint: Employ C.4.
2. Using the above formulae for d',d",S1, and 8" as well as formula (5.4.2)
derive the fundamental lemma 5.6.1.
3. Establish the formulae
S'L - LS' = ad".
and
S"L - La" = - d'.
These relations are the duals of those in lemma 5.6.1.
4. For a complex manifold M, A*C(M) is a direct sum of the subspaces A a*',
that is any u E A*C(M) may be uniquely expressed as a sum of pure forms +,,
of bidegrees (q,r), respectively. Consider the map

sending u into ua,,. If M is Kaehlerian denote by A the algebra of operators
generated by *, d, L, and P,,,. Show that A belongs to the center of A. If M is
compact prove that the operators H and G associated with the underlying
Riemannian structure also belong to the center. In particular, A, H and G
commute with d', d", S', S", and A.
5. Prove that the harmonic part H[a] of a pure form a of bidegree (q,r) on a
compact hermitian manifold is itself of bidegree (q,r) (cf. II.B.3).
6. Let DQ*'(M) denote the quotient space of the space of d-closed forms of
bidegree (q,r) on the compact Kaehler manifold M by the space of exact forms
of bidegree (q,r). Prove that D'(M) is the direct sum of the spaces Dqer(M)
with q + r = p. (Note that this decomposition is independent of the Kaehler
metric.)
The map a -+ & induces an isomorphism of DQvV(M) onto Dr*"M). Hence,
b,,, = b,,, where b,,, = dim DQ-,(M).
In terms of the complex structure on DHM) (the pth cohomology space
constructed from the subspace of real forms) induced by that of M, it may be
shown once again that b, is even for p odd.
Hint: Extend the complex structure J of M to p-forms on M and prove that
p = (- lyI where j denotes the induced map on A'; then, prove that
j and A commute.