Log abelian varieties over a log point

TL;DR

This paper investigates the structure and properties of log abelian varieties over a log point, establishing key theorems about their endomorphisms, duality, and reducibility in the context of log geometry.

Contribution

It introduces new results on the dual short exact sequence, Poincaré reducibility, and semi-simplicity of endomorphism algebras for log abelian varieties over a log point.

Findings

Proved the dual short exact sequence for isogenies.

Established Poincaré complete reducibility theorem for log abelian varieties.

Showed the semi-simplicity of endomorphism algebras.

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.