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In the Wikipedia article on the Hodge Standard Conjecture it is written:

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 0 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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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. –  JS Milne Jun 12 '10 at 1:13
    
Is it known for Grassmannians? –  Dyke Acland Jun 12 '10 at 18:30
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