Dynamics of Projectivized Toric Vector Bundles

TL;DR

This paper investigates the structure and dynamics of endomorphisms of projectivized toric vector bundles, classifying their behaviors, establishing the Kawaguchi--Silverman conjecture in certain cases, and proving non-existence results for specific bundles.

Contribution

It provides a structural classification of endomorphisms of projectivized bundles, proves the Kawaguchi--Silverman conjecture for non-split equivariant bundles, and shows tangent and cotangent bundle projectivizations admit no certain surjective endomorphisms.

Findings

Classified all surjective endomorphisms of Hirzebruch surfaces.

Established the Kawaguchi--Silverman conjecture for non-split equivariant bundles.

Proved no non-automorphic surjective endomorphisms exist for tangent and cotangent bundle projectivizations.

Abstract

We study surjective endomorphisms of projective bundles over toric varieties, achieving three main results. First, we provide a structural theorem describing endomorphisms of projectivized split bundles over arbitrary base varieties, which we use to classify all surjective endomorphisms of Hirzebruch surfaces and construct novel families of examples. Second, for non-split equivariant bundles over toric varieties, we prove that the dynamical degree of an endomorphism of the projectivization is controlled by the base morphism; as a consequence, we establish the Kawaguchi--Silverman conjecture for such bundles. Third, using an explicit transition function method, we prove that projectivizations of tangent and cotangent bundles of smooth toric varieties admit no non-automorphic surjective endomorphisms commuting with toric morphisms on the base.