The Hodge numbers of a compact complex manifold are the complex dimensions of the Dolbeault cohomology groups of ; the complex geometry-analog of the Betti numbers.
Via the Dolbeault theorem, the -Hodge number of a compact complex manifold is
where is the sheaf of holomorphic p-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 3, 2014 06:23:58
by Urs Schreiber