The existence of an abelian variety over Q isogenous to no Jacobian

We prove the existence of an abelian variety A of dimension g over 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 Andr-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.