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

Abstract

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 Q-factorial normal, every minimal model program is equivariant relative to the monoid 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 SEnd(X) such that G acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group Aut(X) has finitely many connected components; and every amplified endomorphism is int-amplified.