nLab
Hodge number

Contents

Idea

The Hodge numbers h p,q(X)h^{p,q}(X) of a compact complex manifold XX are the complex dimensions of the Dolbeault cohomology groups of XX; the complex geometry-analog of the Betti numbers.

Definition

Via the Dolbeault theorem, the (p,q)(p,q)-Hodge number of a compact complex manifold XX is

h p,q(X):=dim H q(X;Ω p), h^{p,q}(X) := dim_\mathbb{C}H^q(X;\Omega^p),

where Ω p\Omega^p is the sheaf of holomorphic p-forms on XX and H q(X;Ω p)H^q(X; \Omega^p) is the corresponding abelian sheaf cohomology.

Properties

If dim (X)=ndim_\mathbb{C}(X)=n, then h p,q(X)=h np,nq(X)h^{p,q}(X)=h^{n-p,n-q}(X); in particular, h n,n(X)=1h^{n,n}(X)=1.

When XX is a Kähler manifold, then Hodge numbers have a number of additional nontrivial properties:

  • they are symmetric, i.e., h p,q(X)=h q,p(X)h^{p,q}(X)=h^{q,p}(X);

  • h p,p1h^{p,p}\geq 1 for any p=1,,np=1,\dots,n;

  • b k(X)= p+q=kh p,q(X)b_k(X)=\sum_{p+q=k}h^{p,q}(X), where b kb_k is the kk-th Betti number of XX.

Revised on June 3, 2014 06:23:58 by Urs Schreiber (89.204.155.45)