# Important consequences of the Hodge Index Theorem

The Hodge Index Theorem for compact Kaehler manifolds seems to be a big deal in complex geometry. See here for the surface version of the result.

https://en.wikipedia.org/wiki/Hodge_index_theorem

I personally can't see why this result is important, and I would like to know what are some important consequences of this result.

• The negativity lemma (lemma 3.39 in Koll\'ar-Mori) is very useful in the minimal model program Oct 19, 2017 at 19:31
• Another consequence: If $A,B$ are nef divisors on a surface and $A\cdot B=0$, then $A$ and $B$ are numerically proportional. This trick is often used in the study of linear systems on surfaces.
– byu
Oct 19, 2017 at 22:30
• if you believe the Riemann hypothesis for curves is important, that seems to follow from Hodge (and Riemann -Roch), via Mattuck-Tate, Castelnuovo-Severi, Grothendieck. arxiv.org/pdf/1509.00797.pdf Oct 20, 2017 at 3:50
• @byu Using Hodge index theorem in the situation you described, I can see that $A^2>0$ implies $B$ numerically trivial and, by symmetry, $B^2>0$ implies $A$ numerically trivial. How do you conclude $A,B$ are numerically proportional? May 1, 2021 at 21:14

Let $S$ be a smooth projective surface and $H$ a $\mathbb{Q}$-divisor with $H^2>0$ (for instance, an ample divisor).
If $D$ is another $\mathbb{Q}$-divisor such that $HD=0$, then $D^2 \leq 0$ and equality holds if and only if the class of $D$ is zero in $H^2(S, \, \mathbb{Q})$.