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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.

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    $\begingroup$ The negativity lemma (lemma 3.39 in Koll\'ar-Mori) is very useful in the minimal model program $\endgroup$
    – Hacon
    Oct 19, 2017 at 19:31
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    $\begingroup$ 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. $\endgroup$
    – byu
    Oct 19, 2017 at 22:30
  • $\begingroup$ 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 $\endgroup$
    – roy smith
    Oct 20, 2017 at 3:50
  • $\begingroup$ @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? $\endgroup$
    – rmdmc89
    May 1, 2021 at 21:14

1 Answer 1

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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 $H^2(S, \, \mathbb{Q})$.

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    $\begingroup$ Isn't this exactly the statement? You mean that the statement itself is the reason why it is important? :p $\endgroup$
    – diverietti
    Oct 19, 2017 at 20:08
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    $\begingroup$ The original statement is about the signature of the intersection form in the Neron-Severi group. The statement I gave is the form in which the result is generally used in the context of surfaces. In fact, in Barth-Peters-Van de Ven book it is explicitly stated as an important corollary. $\endgroup$
    – Francesco Polizzi
    Oct 19, 2017 at 20:14

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