ENDOMORPHISMS OF PROJECTIVE VARIETIES

Abstract

This thesis is devoted to studies of various aspects of endomorphisms of projective varieties. It consists of five parts.

1. Working over fields of characteristic 0, we show that the family of (not-necessarily linear) algebraic groups is uniformly Jordan and hence the automorphism group of a projective variety is Jordan.

2. We describe the building blocks of a polarized endomorphism f of a projective variety X over a field of characteristic 0. Precisely, we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

3. By the recent work of MMP in positive characteristic, we generalize part of the results of polarized endomorphisms to the case of positive characteristic.

4. We generalize the results of polarized endomorphisms to int-amplified endomorphisms.

5. Via polarized endomorphisms, we characterize toric varieties in both algebraic and geometric ways.