Polarizations and Grothendieck's Standard Conjectures

J. S. Milne
2002 Annals of Mathematics  
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. Introduction. In examining Weil's proofs (Weil 1948) of the Riemann hypothesis for curves and abelian varieties over finite fields, Grothendieck was led to state two "standard" conjectures (Grothendieck
more » ... res (Grothendieck 1969), which imply the Riemann hypothesis for all smooth projective varieties over a finite field, essentially by Weil's original argument. Despite Deligne's proof of the Riemann hypothesis, the standard conjectures retain their interest for the theory of motives. The first, the Lefschetz standard conjecture (Grothendieck 1969, §3), states that, for a smooth projective variety V over an algebraically closed field, the operators Λ rendering commutative the diagrams (0 ≤ r ≤ 2n, n = dim V )
doi:10.2307/3062126 fatcat:4i7ilazy3nfzves2au62awx3gy