Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties

TL;DR

This paper proves the Kawaguchi-Silverman conjecture and related conjectures for endomorphisms on various algebraic varieties, including Fano type, toric, and linear algebraic groups, and suggests a new approach via equivariant MMP.

Contribution

It establishes the conjecture for multiple classes of varieties and group endomorphisms, advancing understanding of dynamical properties in algebraic geometry.

Findings

Proved KSC for varieties of Fano type and toric varieties.

Confirmed KSC for group endomorphisms of linear algebraic groups.

Proposed a new approach using equivariant MMP.

Abstract

We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group endomorphisms of linear algebraic groups. We also propose a possible approach to the conjecture using equivariant MMP.