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Dynamics of Projectivized Toric Vector Bundles

Javier González-Anaya, Brett Nasserden, Sasha Zotine

TL;DR

The paper addresses the dynamics of surjective endomorphisms on projectivized toric bundles by introducing a transition-function framework that translates endomorphisms into explicit polynomial data on local trivializations. It distinguishes split versus non-split equivariant bundles on toric bases, yielding a structural classification and Hirzebruch-surface examples in the split case, while establishing fibre-base dynamical-degree relations and KS conjecture validity for non-split toric bundles. A central no-go theorem shows that P(T_X) and P(Ω_X) admit no non-automorphic base-toric endomorphisms on smooth toric X not isomorphic to a product of P^1’s, with a detailed analysis via tangent/cotangent transition data and case-splitting along invariant curves. The transition-function method provides a powerful, explicit toolkit for encoding endomorphisms as degree-d polynomial data and checking surjectivity and compatibility across charts. Overall, the work bridges toric geometry, vector-bundle theory, and holomorphic/arithmetic dynamics, delivering explicit classifications, dynamical-control results, and obstruction principles with potential for arithmetic applications.

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.

Dynamics of Projectivized Toric Vector Bundles

TL;DR

The paper addresses the dynamics of surjective endomorphisms on projectivized toric bundles by introducing a transition-function framework that translates endomorphisms into explicit polynomial data on local trivializations. It distinguishes split versus non-split equivariant bundles on toric bases, yielding a structural classification and Hirzebruch-surface examples in the split case, while establishing fibre-base dynamical-degree relations and KS conjecture validity for non-split toric bundles. A central no-go theorem shows that P(T_X) and P(Ω_X) admit no non-automorphic base-toric endomorphisms on smooth toric X not isomorphic to a product of P^1’s, with a detailed analysis via tangent/cotangent transition data and case-splitting along invariant curves. The transition-function method provides a powerful, explicit toolkit for encoding endomorphisms as degree-d polynomial data and checking surjectivity and compatibility across charts. Overall, the work bridges toric geometry, vector-bundle theory, and holomorphic/arithmetic dynamics, delivering explicit classifications, dynamical-control results, and obstruction principles with potential for arithmetic applications.

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.

Paper Structure

This paper contains 17 sections, 34 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Dynamics
  4. Toric geometry
  5. Properties of Based Maps
  6. Chern classes of bundles admitting based endomorphisms
  7. The transition function method
  8. The split case
  9. Specialization to smooth projective toric varieties
  10. Explicit classifications of morphisms on Hirzeburch surfaces
  11. The tangent and cotangent bundles of a toric variety
  12. Cases \ref{['i:overall-case1']} and \ref{['i:overall-case2']}: nef tangent bundle
  13. The transition function method
  14. The transition functions of tangent and cotangent bundles of a toric variety
  15. Case \ref{['i:overall-case3']}: using the transition functions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $X$ be projective variety and $\varphi\colon X\rightarrow X$ be a surjective endomorphism. Consider line bundles ${\mathcal{L}}_i$ on $X$ for each $0\leq i\leq r$, and define ${\mathcal{E}}=\bigoplus_{i=0}^r{\mathcal{L}}_i$. Let $\pi\colon \mathbf{P}({\mathcal{E}})\rightarrow X$ denote the natur The relation between both of the previous perspectives is that $\psi$ restricted to a fibre $\pi^{-

Theorems & Definitions (75)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Definition 2.2: lesieutresatriano2021ksc
  • Definition 2.3
  • Proposition 2.4: NasserdenZotine2023
  • ...and 65 more