Building blocks of polarized endomorphisms of normal projective varieties

Abstract

An endomorphism $f$ of a projective variety X is polarized (resp. quasi-polarized) if $f^*H$ is linearly equivalent to $qH$ 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. Along the way, we show that $f$ always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that the pullback of a power of $f$ acts as a scalar multiplication on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected. Partial answers about X being of Calabi-Yau type, or Fano type are also given with an extra primitivity assumption on $f$ which seems necessary by an example.