Log abelian varieties over a log point

Abstract

We study (weak) log abelian varieties with constant degeneration in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In particular, we prove the dual short exact sequence for isogenies, Poincar\'e complete reducibility theorem for log abelian varieties, and the semi-simplicity of the endomorphism algebras of log abelian varieties.