Characterization of $Q$-complex tori via endomorphisms -- an addendum to "Int-amplified endomorphisms of compact K\"ahler spaces''

TL;DR

This paper establishes a criterion for identifying $Q$-complex tori among compact K"ahler spaces with pseudo-effective canonical divisors, based on the existence of int-amplified endomorphisms, with applications to threefolds.

Contribution

It provides a dynamical criterion linking endomorphisms to the structure of K"ahler spaces, specifically characterizing $Q$-complex tori and rationally connected threefolds.

Findings

Spaces with int-amplified endomorphisms are $Q$-complex tori.

Simply connected K"ahler threefolds with such endomorphisms are rationally connected.

The criterion applies to spaces with pseudo-effective canonical divisors.

Abstract

In this short note, we consider a normal compact K\"ahler klt space $X$ whose canonical divisor $K_X$ is pseudo-effective, and give a dynamical criterion for $X$ to be a $Q$-complex torus. We show that, if such $X$ admits an int-amplified endomorphism, then $X$ is a $Q$-complex torus. As an application, we prove that, if a simply connected compact K\"ahler (smooth) threefold admits an int-amplified endomorphism, then it is (projective and) rationally connected.