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.



Hodge Index theorem for Complex Manifolds
The Hodge index theorem IS a result on compact Kahler manifolds of complex dimension 2n.


It states that the signature of the intersection form on $H^{2n}(X, \mathbb{R})$ equals $\sum (-1)^a h^{a, b}(X)$, where $h^{a, b}$ are the Hodge numbers.


See Voisin, Hodge theory and complex algebraic geometry I, theorem 6.33


By the way, the result you probably have in mind is on complex surfaces, which are NOT Riemann surfaces (these are complex curves).


