The Hodge numbers of a compact complex manifold are the dimensions of certain cohomology spaces of .
The -Hodge number of a compact complex manifold is
where is the sheaf of holomorphic -forms on and is the corresponding abelian sheaf cohomology.
If , then ; in particular, .
When is a Kähler manifold, then Hodge numbers have a number of additional nontrivial properties:
they are symmetric, i.e., ;
for any ;
, where is the -th Betti number of .
Revised on June 17, 2010 05:57:47
by Urs Schreiber