Normal projective varieties admitting polarized or int-amplified endomorphisms

Let $X$ be a normal projective variety admitting a polarized or int-amplified endomorphism $f$. We list up characteristic properties of such an endomorphism and classify such a variety from the aspects of its singularity, anti-canonical divisor and Kodaira dimension. Then we run the equivariant minimal model program with respect to not just the single $f$ but also the monoid $SEnd(X)$ of all surjective endomorphisms of $X$, up to finite-index. Several applications are given. We also give both algebraic and geometric characterizations of toric varieties via polarized endomorphisms.

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