# Contents

## Idea

The Hodge numbers ${h}^{p,q}\left(X\right)$ of a compact complex manifold $X$ are the dimensions of certain cohomology spaces of $X$.

## Defition

The $\left(p,q\right)$-Hodge number of a compact complex manifold $X$ is

${h}^{p,q}\left(X\right):={\mathrm{dim}}_{ℂ}{H}^{q}\left(X;{\Omega }^{p}\right),$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 ${H}^{q}\left(X;{\Omega }^{p}\right)$ is the corresponding abelian sheaf cohomology.

## Properties

If ${\mathrm{dim}}_{ℂ}\left(X\right)=n$, then ${h}^{p,q}\left(X\right)={h}^{n-p,n-q}\left(X\right)$; in particular, ${h}^{n,n}\left(X\right)=1$.

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

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

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

• ${b}_{k}\left(X\right)={\sum }_{p+q=k}{h}^{p,q}\left(X\right)$, where ${b}_{k}$ is the $k$-th Betti number of $X$.

Revised on June 17, 2010 05:57:47 by Urs Schreiber (87.212.203.135)