Building blocks of polarized endomorphisms of normal projective varieties

By Sheng Meng, De-Qi Zhang. 2016

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Let $X$ be a normal projective variety over an algebraically closed field $k$ of characteristic $0$. We consider an int-amplified surjective endomorphism $f$ of $X$, that is $f^\ast L - L =H$ for some ample Cartier divisors $L$ and $H$. First, we show that the anti-canonical divisor $-K_X$ is weakly numerically equivalent to an effective Weil $\mathbb{Q}$-divisor; and the Albanese morphism of $X$ is an equi-dimensional algebraic fibre space. Suppose $X$ is $\mathbb{Q}$-Gorenstein. We show that $X$ has at worst lc singularities. Suppose either $X$ is klt and $K_X$ is pseudo-effective or $X$ is non-uniruled. We show that $X$ is a $Q$-abelian variety (i.e.~quasi-\'etale quotient of an abelian variety). Finally, we show that one can run the minimal model program (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for a mildly singular $X$ and reach a $Q$-abelian variety $Y$. As a consequence, the building blocks of int-amplified endomorphisms are those of $Q$-abelian varieties and those of Fano varieties of Picard number one. Along the way, we show that a power of $f$ acts diagonally on the Neron-Severi group of $X$ (modulo torsion) provided that $X$ is smooth and rationally connected.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b23a96acc03aa6d100df5b3a29829bc7d0cb8b8a