Int-amplified endomorphisms of compact Kähler spaces

Abstract

Let $X$ be a normal compact Kähler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^{\ast }\xi -\xi =\eta$ for some Kähler 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 Kähler 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.

Changes in the structure; Asian Journal of Mathematics (to appear)

References in corpus (3)

• Questions on self maps of algebraic varieties
• Totally invariant divisors of int-amplified endomorphisms of normal projective varieties
• Minimal model theory for log surfaces in Fujiki's class $\mathcal{C}$

Cited by in corpus (3)

• Compact Kähler threefolds with the action of an abelian group of maximal rank
• Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
• Existence of the equivariant minimal model program for compact Kähler threefolds with the action of an abelian group of maximal rank