2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61146We 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.12 pages published versionAlgebraic GeometryNumber Theory14K99; 11G99text