Abelian varieties isogenous to no Jacobian

Abstract

We prove among other things the existence of Hodge generic abelian varieties defined over the algebraic numbers and not isogenous to any Jacobian. Actually, we also show that in various interpretations these abelian varieties make up the majority, and we give certain uniform bounds on the possible degree of the fields of definition. In particular, this yields a new answer (in strong form) to a question of Katz and Oort, compared to previous work of Chai and Oort (2012, conditional on the André-Oort Conjecture) and by Tsimerman (2012 unconditionally); their constructions provided abelian varieties with complex multiplication (so not ``generic"). Our methods are completely different, and they also answer a related question posed by Chai and Oort in their paper.