TITLE: Hodge Index theorem for Complex Manifolds
QUESTION [7 upvotes]: The Riemann-Roch theorem is a result about Riemann Surfaces that was extended to the Hirzebruch–Riemann–Roch theorem, a result about compact complex manifolds. The Hodge Index theorem is a result about Riemann surfaces (I'm just worried about the complex case) that is proved using Riemann-Roch. Has the Hirzebruch–Riemann–Roch theorem been used to extend the Hodge Index theorem to a result about compact complex manifolds.

REPLY [3 votes]: For a compact Kähler manifold $X$ of complex dimension $2m$, the signature is given by
$$\sigma(X) = \sum_{p=0}^{2m}\sum_{q=0}^{2m}(-1)^ph^{p,q}(X).$$
As algori points out, this formula does not hold for all even-dimensional compact complex manifolds. However, for compact Kähler manifolds, $h^{p,q}(X) = h^{q,p}(X)$ so the signature is also given by
$$\sigma(X) = \sum_{p=0}^{2m}\sum_{q=0}^{2m}(-1)^qh^{p,q}(X).$$
Surprisingly, this formula does hold for all even-dimensional compact complex manifolds (check algori's example). This can be deduced using the properties of Hirzebruch's $\chi_y$ genus, as is shown here.