Fields of definition of abelian subvarieties

In this paper we study the field of definition abelian subvarieties $B\subset A_{\overline{K}}$ for an variety $A$ over a $K$ characteristic $0$. We show that, provided that no isotypic component $A_{\overline{K}}$ is simple, there are infinitely many with $K_A$, endomorphisms $A_{\overline{K}}$. This result combined earlier work R\'emond gives explicit maximum minimal degree extension which subvariety defined varying fixed dimension and