Hodge number ($h^{p,q}$)

The Hodge number of a compact complex manifold is $h^{p,q}={\dim }_{\mathbb{C}}{H}_{\bar{\partial }}^{p,q}(X)$.

For a compact Kähler manifold, these dimensions are the graded dimensions in its Hodge structure.

On a compact Kähler manifold of complex dimension $n$, the conjugate-linear Hodge star gives a bijection of harmonic forms in complementary bidegrees. The Dolbeault Hodge decomposition identifies these harmonic spaces with Dolbeault cohomology and yields the displayed equality.

On a compact Kähler manifold, complex conjugation commutes with the real Hodge Laplacian and exchanges harmonic forms of bidegrees $(p,q)$ and $(q,p)$. The identification of Dolbeault cohomology with harmonic forms therefore gives the symmetry of Hodge numbers.