nLab Hodge number

Idea

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

Definition

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

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

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

Properties

If $dim_\mathbb{C}(X)=n$, then $h^{p,q}(X)=h^{n-p,n-q}(X)$; in particular, $h^{n,n}(X)=1$.

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

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

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

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

Related concepts

• Betti number