Constructing Abelian Varieties over $\overline{\mbthbb{Q}}$ Not Isogenous to a Jacobian

We discuss the following question of Nick Katz and Frans Oort: Given an

Algebraically closed field K , is there an Abelian variety over K of

dimension g which is not isogenous to a Jacobian? For K the complex

numbers

its easy to see that the answer is yes for g>3 using measure theory, but

over a countable field like $\overline{\mbthbb{Q}}$ new methods are required. Building on

work

of Chai-Oort, we show that, as expected, such Abelian varieties exist for

$K=\overline{\mbthbb{Q}}$ and g>3 . We will explain the proof as well as its connection to

the

Andre Oort conjecture.