Page 240 - Curvature and Homology

222 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

de Rham cohomology (d-cohomology) of a differentiable manifold.
The reason for considering cohomology with the differential operator d"
is clear. Indeed, it yields information regarding holomorphic forms.
We remark that in this section to every statement regarding the
operator d" there is a corresponding statement for the operator d'.
Thus, there is a corresponding cohomology theory defined by d'.

Lemma 6.10.1. For every form a of bidegree (q, Y) and any fl

To see this, it is only necessary to apply the operator d to a A fl
and compare the bidegrees in the resulting expansion.
Let Ag*r denote the linear space of forms of bidegree (q, r) on M.
Consider the sequence of maps

where for the moment we write dl;,, = d" I Aqvr.. Now, put
kernel d", ,
~,r M
H' ( ) = image d\r-l '
then,

Proposition 6.10.1.
H;*'(M) = kernel d",, .

For, if a E image d",,,-, it must come from a form of bidegree
(q, r - 1). Let a be a form of bidegree (p, 0). Then, its image by
d''q,r-l must be 0.

Corollary. Hf*O(M;) is the linear space of holomorphic p-forms.
Now, by lemma 6.10.1 if a and fl are holomorphic forms, so is a A p.
Define

then, by the remark just made, H,(M). has a ring structure.
It is now shown that the d"-cohomology ring of a compact complex
parallelisable manifold A4 depends only upon the local structure of its
universal covering space.