Hodge numbers - Imported Knowlpedia Content

Definition

Let $X$ be a compact Kähler manifold of complex dimension $n$ . Its Hodge number of bidegree $(p,q)$ is

h^{p,q}(X)=\dim_{\mathbb C}H_{\bar\partial}^{p,q}(X), \qquad 0\leq p,q\leq n.

Equivalently, $h^{p,q}(X)$ is the dimension of the $(p,q)$ -summand in the Hodge decomposition of $H^{p+q}_{\mathrm{dR}}(X;\mathbb C)$ . The finite array $\{h^{p,q}\}$ , conventionally displayed with $p+q$ constant along diagonals, is called the Hodge diamond. A Hodge number is a numerical invariant, whereas the corresponding Hodge component retains its vector-space structure. Only indices in the displayed range occur.

Symmetries and Betti numbers

Complex conjugation and Hodge duality give

h^{p,q}=h^{q,p}, \qquad h^{p,q}=h^{n-p,n-q}.

The diagonal sums recover the Betti numbers:

b_k(X)=\sum_{p+q=k}h^{p,q}(X).

These identities are consequences of compact Kähler Hodge theory Huybrechts, Chapter 3, §3.2.

Examples

For $\mathbb{CP}^n$ , one has $h^{p,p}=1$ for $0\leq p\leq n$ and all off-diagonal Hodge numbers vanish. A compact Riemann surface of genus $g$ has $h^{0,0}=h^{1,1}=1$ and $h^{1,0}=h^{0,1}=g$ . These examples show respectively that the diamond can be diagonal or can record complex-analytic information not visible in a single cohomological degree.

Conventions and scope

For an arbitrary compact complex manifold, authors still define $h^{p,q}=\dim H_{\bar\partial}^{p,q}$ , but the diagonal-sum formula and the second symmetry need not hold. In algebraic geometry, “Hodge numbers” may also refer to dimensions of graded pieces of a mixed Hodge structure; those require a weight index in addition to $(p,q)$ .

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: Chapter 3, §3.2, Hodge decomposition and Hodge numbers.
Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Publisher record. Relevant: Chapter 6 for Kähler Hodge decomposition and Chapter 7 for Hodge structures.