Int-amplified endomorphisms of compact K\"ahler spaces

TL;DR

This paper extends the concept of int-amplified endomorphisms to compact K"ahler spaces, showing that under certain conditions, such spaces are $Q$-tori or can be simplified via the minimal model program.

Contribution

It generalizes the notion of int-amplified endomorphisms to K"ahler spaces and classifies the structure of spaces admitting such endomorphisms with pseudo-effective canonical divisors.

Findings

For smooth, surface, or threefold K"ahler spaces with mild singularities, admitting an int-amplified endomorphism implies the space is a $Q$-torus.

The minimal model program can be run $f$-equivariantly on certain threefolds with terminal singularities.

Such spaces either become $Q$-tori or Fano varieties of Picard number one after the MMP.

Abstract

Let $X$ be a normal compact K\"ahler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*\xi-\xi=\eta$ for some K\"ahler classes $\xi$ and $\eta$. First, we show that this definition generalizes the notion in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact K\"ahler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a positive power, we can run the minimal model program (MMP) $f$-equivariantly for such $Y$ and reach either a $Q$-torus or a Fano (projective) variety of Picard number one.