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In the Wikipedia article on the Hodge Standard Conjecture it is written (note [Oct. 2015]: it has since been fixed):

In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge theory. In positive characteristic the Hodge standard conjecture is known only for surfaces and abelian varieties.

I have three questions:

(1) Is the characteristic zero version the Hodge Index Theorem?

(2) If so, what is a good reference for an algebraic geometry proof? I know it can be proved for a surface using the Riemann-Roch Theorem. Does this continue to be true for higher dimensions?

(3) Is the conjecture really only known in positive characteristic for surfaces and abelian varieties? Surely it should be possible to at least compute a result for projective $n$-space.

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    $\begingroup$ Actually the Hodge standard conjecture is not even known in positive characteristic for abelian varieties --- it is only known that it is implied by the Hodge conjecture for complex CM abelian varieties (see my 2002 Annals paper). Of course, the conjecture is trivial for projective n-space. A good reference for these things is Kleiman's article in the proceedings of the Motives conference, Seattle 1991, published by AMS 1994. $\endgroup$ – JS Milne Jun 12 '10 at 1:13
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    $\begingroup$ The case of abelian fourfolds has recently be settled by Giuseppe Ancona, see arxiv.org/abs/1806.03216 . $\endgroup$ – Olivier Benoist Jun 16 '18 at 12:44
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Probably too late, but...

(1) Not quite. The Hodge index theorem only works for $\mathbb{C}$. To extend it to all characteristic $0$ ground fields, you need the Lefschetz principle and the comparison theorem.

(2) As J. S. Milne mentioned, the wikipedia article was mistaken (it has since been fixed). It isn't known how to extend Segre-Grothendieck to higher dimension varieties.

(3) It seems that only surfaces are known so far. The Lefschetz conjecture for projective spaces and Grassmannians follows from intersection theory, but I'm not aware of such results for the Hodge standard conjecture.

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(1) No, the Hodge index theorem, valid in arbitrary characteristic, is the Hodge standard conjecture for divisors on surfaces. Taking 2-dimensional linear sections, one can deduce the Hodge standard conjecture for divisors on arbitrary smooth projective varieties.

The Hodge standard conjecture over $\mathbf{C}$ is not a consequence of the Hodge index theorem but rather of the fact that the space of algebraic cycles is contained in the space of Hodge classes $H^{p,p}(X)\cap H^{2p}(X, \mathbf{Q})$, and by Hodge theory (Riemann bilinear relations), the Lefschetz pairing is definite on the primitive parts of those a priori bigger spaces.

(2) I do not think there is an algebraic proof, since the above arguments requires some "positivity" (a definite pairing on a $\mathbf{Q}$-vector space remains definite after restricting to any subspace).

Speculation: Maybe the Weil cohomology theory for varieties over $\overline{\mathbf{F}}_p$ with values in the Kottwitz category ${\rm Kt}_{\mathbf{R}}$ whose existence was recently conjectured by Scholze could make such an argument possible in positive characteristic.

(3) Tetsushi Ito in "Weight-monodromy conjecture for $p$-adically uniformized varieties" (Invent. Math. 2005) crucially proved the Hodge standard conjecture for the varieties obtained from $\mathbf{P}^n$ over $\mathbf{F}_q$ by successively blowing up all $\mathbf{F}_q$-points, the strict transforms of the lines between those points etc. The cohomology of these varieties is generated by algebraic cycles, but they do not lift to characteristic zero.

(Obviously, if a variety in characteristic $p$ whose cohomology is generated by algebraic cycles lifts to characteristic zero "together with all cycles", then the Hodge standard conjecture holds.)

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  • $\begingroup$ Do you have a source for this speculation by Scholze? It's very interesting & I hadn't heard of it before. $\endgroup$ – dhy May 3 at 12:44
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    $\begingroup$ @dhy 9.5 here arxiv.org/pdf/1712.03708.pdf $\endgroup$ – Will Sawin May 3 at 13:03
  • $\begingroup$ As to (1), that is the Hodge index theorem. This is the version you find in Voisin, and although I don't have access to Hodge's book, I suspect it is stated there in this form as well. $\endgroup$ – R. van Dobben de Bruyn May 3 at 18:42
  • $\begingroup$ Do you know if Ito's result is related to Adiprasito–Huh–Katz by using matroids that are realisable over $\mathbf F_q$ but not $\mathbf C$? Or is it a little different? $\endgroup$ – R. van Dobben de Bruyn May 3 at 19:12

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