Abelian varieties isogenous to no Jacobian

A question of N. Katz and F. Oort asked whether (for any g >4), there exist abelian varieties of dimension g defined over the algebraic numbers and isogenous to no Jacobian. J. Tsimerman gave the first unconditional proof, however providing abelian varieties with CM, so special in another sense. In the paper, by a completely different method. we prove that, in a precise sense, the "general" abelian variety of dimension g>4 defined over the algebraic numbers is not isogenous to any Jacobian. The methods lead to several other results; they rely among other things use the Pila-Wilkie counting and isogeny estimates of Masser-Wuestholz.