Abelian varieties not isogenous to any Jacobian

It is well known that in dimension $g\ge 4$ there exist complex abelian varieties not isogenous to any Jacobian. A question of Katz and Oort asked whether one can find such examples over the field of algebraic numbers. This was answered affirmatively by Oort-Chai under the André-Oort conjecture, and by Tsimerman unconditionally. They gave examples within Complex Multiplication. In joint work with Masser, by means of a completely different method, we proved that in a sense the general abelian variety over $\overline{\mathbb{Q}}$ is indeed not isogenous to any Jacobian. I shall illustrate the basic principles of the proofs.