Abelian Varieties not Isogenous to Jacobians

Katz and Oort raised the following question: Given an algebraically closed field k, and a positive integer g>3, does there exist an abelian variety over k not isogenous to a Jacobian over k? There has been much progress on this question, with several proofs now existing over $\overline{\mathbb{Q}}$. We discuss recent work with Ananth Shankar, answering this question in the affirmative over $\overline{\mathbb{F}_q(T)}$. Our method introduces new types of local obstructions, and can be used to give another proof over $\overline{\mathbb{Q}}$.