Building blocks of polarized endomorphisms of normal projective varieties

摘要

An endomorphism f of a projective variety X is polarized (resp. quasi-polarized) if f*H similar to qH (linear equivalence) for some ample (resp. nef and big) Cartier divisor H and integer q 1. First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi-etale quotient of an abelian variety). Finally, we show that 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. As a consequence, the building blocks of polarized endomorphisms are those of Q-abelian varieties and those of Fano varieties of Picard number one.