The existence of an abelian variety over $\overline{\mathbb{Q}}$ isogenous to no Jacobian

Abstract

We prove the existence of an abelian variety $A$ of dimension $g$ over $\overline{\mathbb{Q}}$ that is not isogenous to any Jacobian, subject to the necessary condition $g\!>\!3$. Recently, C. Chai and F. Oort gave such a proof assuming the André-Oort conjecture. We modify their proof by constructing a special sequence of CM points for which we can avoid any unproven hypotheses. We make use of various techniques from the recent work of Klingler-Yafaev et al.