Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism

Let $X$ be a projective variety admitting a polarized (or more generally, int‑amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $\mathrm{SEnd}(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth, we show: there is a finite-index submonoid $G$ of $\operatorname{SEnd}(X)$ such that $G$ acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group $\operatorname{Aut}(X)$ has finitely many connected components; and every amplified endomorphism is int‑amplified.

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The first named-author is supported by a Research Assistantship of NUS. The second named-author is supported by an Academic Research Fund of NUS.

Received 18 August 2018

Accepted 20 December 2018

Published 8 June 2020