[Paper Review] Building blocks of polarized endomorphisms of normal projective varieties

Sheng Meng, De‐Qi Zhang|arXiv (Cornell University)|Jun 4, 2016

Algebraic Geometry and Number Theory43 references3 citations

TL;DR

This paper establishes that quasi-polarized endomorphisms of normal projective varieties are always polarized, and shows that such endomorphisms descend via equivariant rational maps. The main contribution is proving that the building blocks of polarized endomorphisms are Fano varieties of Picard number one and Q-abelian varieties (quasi-étale quotients of abelian varieties), achieved via f-equivariant minimal model program (MMP) and MRC fibrations.

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-étale 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.

Motivation & Objective

• To prove that every quasi-polarized endomorphism of a normal projective variety is actually polarized.
• To show that the polarized property descends along any equivariant dominant rational map.
• To establish that a maximal rationally connected (MRC) fibration can be made f-equivariant, leading to a polarized endomorphism on the base, which is a Q-abelian variety.
• To demonstrate that the minimal model program (MMP) can be run f-equivariantly for mildly singular varieties, terminating in either a Q-abelian variety or a Fano variety of Picard number one.
• To classify the fundamental building blocks of polarized endomorphisms as Fano varieties of Picard number one and Q-abelian varieties.

Proposed method

• Using cone analysis to prove that quasi-polarized endomorphisms are polarized, based on numerical equivalence and big divisors.
• Applying Nakayama's construction to make the MRC fibration f-equivariant, ensuring the base inherits a polarized endomorphism.
• Proving that the base of the MRC fibration is a Q-abelian variety via the structure of the endomorphism and numerical properties.
• Running the minimal model program (MMP) in an f-equivariant manner for varieties with mild singularities, using the cone theorem and relative ampleness.
• Showing that the pullback of f acts as a scalar multiplication on the Néron–Severi group modulo torsion when X is smooth and rationally connected.
• Using the ramification divisor and divisorial contraction analysis to prove that −K_X is big when the endomorphism is primitive and not of Q-abelian type.

Experimental results

Research questions

• RQ1 Are all quasi-polarized endomorphisms necessarily polarized?
• RQ2 Does the polarized property descend through equivariant dominant rational maps?
• RQ3 Can the maximal rationally connected (MRC) fibration of a variety with a polarized endomorphism be made f-equivariant, and what is the nature of the base?
• RQ4 Can the minimal model program be performed in an f-equivariant way for mildly singular projective varieties?
• RQ5 What are the fundamental building blocks of polarized endomorphisms in terms of birational geometry and classification?

Key findings

• Quasi-polarized endomorphisms are always polarized, as shown via cone analysis and numerical equivalence arguments.
• The polarized property descends along any equivariant dominant rational map, with degree relation (deg f)^dim(Y) = (deg g)^dim(X).
• The base of the f-equivariant MRC fibration is a Q-abelian variety, i.e., a quasi-étale quotient of an abelian variety.
• For mildly singular projective varieties, the f-equivariant MMP terminates in either a Q-abelian variety or a Fano variety of Picard number one.
• The building blocks of polarized endomorphisms are precisely Fano varieties of Picard number one and Q-abelian varieties.
• When X is smooth and rationally connected, the pullback of a power of f acts as scalar multiplication on the Néron–Severi group modulo torsion.