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Polarizations and Grothendieck’s standard conjectures. (English) Zbl 1055.14008

Summary: We prove that Grothendieck’s Hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes, we prove the Hodge standard conjecture unconditionally.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K15 Arithmetic ground fields for abelian varieties
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