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Marked points of families of hyperbolic automorphisms of smooth complex projective varieties

Yugang Zhang

TL;DR

This work develops a two-pronged framework to quantify stability for marked points in flat families of smooth projective varieties with fiberwise hyperbolic automorphisms of positive entropy. It constructs both geometric geometric canonical heights (via relative divisors D^± and their λ-Scaling) and analytic Green currents (with continuous potentials) and proves their coincidence when the base is a curve, yielding a unified stability theory. The paper establishes a weak Northcott property and a function-field Kawaguchi–Silverman result for non-isotrivial families, and it develops a relative divisor toolkit (including the augmented base locus) alongside Chow- and Hilbert-scheme parameter spaces to study the sparsity and density of stable marked points. These results link dynamical complexity, arithmetic growth, and geometric positivity, providing a mechanism to transfer dynamical stability information to function-field height considerations and to draw KS-type conclusions in this setting.

Abstract

Let $π: X\to Λ$ be a flat family of smooth complex projective varieties parameterized by a smooth quasi-projective variety $Λ$, and let $f: X\to X$ be a family of automorphisms with positive topological entropy. Suppose $σ: Λ\to X$ is a marked point, i.e., it is a rational section of $π$. We propose two methods to measure the stability, normality, or periodicity of the family given by $t \mapsto f_t^n(σ(t))$. First, from an algebraic perspective, we construct geometric canonical height functions that have desirable properties. Second, from an analytic viewpoint, we construct a positive closed $(1,1)$-current with continuous local potential. When $Λ$ is a curve, we demonstrate that these two constructions actually coincide, providing a unified approach to understanding the dynamical behavior of the family. As an application of the algebraic method, we prove a special case of the Kawaguchi-Silverman conjecture over complex function fields.

Marked points of families of hyperbolic automorphisms of smooth complex projective varieties

TL;DR

This work develops a two-pronged framework to quantify stability for marked points in flat families of smooth projective varieties with fiberwise hyperbolic automorphisms of positive entropy. It constructs both geometric geometric canonical heights (via relative divisors D^± and their λ-Scaling) and analytic Green currents (with continuous potentials) and proves their coincidence when the base is a curve, yielding a unified stability theory. The paper establishes a weak Northcott property and a function-field Kawaguchi–Silverman result for non-isotrivial families, and it develops a relative divisor toolkit (including the augmented base locus) alongside Chow- and Hilbert-scheme parameter spaces to study the sparsity and density of stable marked points. These results link dynamical complexity, arithmetic growth, and geometric positivity, providing a mechanism to transfer dynamical stability information to function-field height considerations and to draw KS-type conclusions in this setting.

Abstract

Let be a flat family of smooth complex projective varieties parameterized by a smooth quasi-projective variety , and let be a family of automorphisms with positive topological entropy. Suppose is a marked point, i.e., it is a rational section of . We propose two methods to measure the stability, normality, or periodicity of the family given by . First, from an algebraic perspective, we construct geometric canonical height functions that have desirable properties. Second, from an analytic viewpoint, we construct a positive closed -current with continuous local potential. When is a curve, we demonstrate that these two constructions actually coincide, providing a unified approach to understanding the dynamical behavior of the family. As an application of the algebraic method, we prove a special case of the Kawaguchi-Silverman conjecture over complex function fields.
Paper Structure (18 sections, 25 theorems, 79 equations)

This paper contains 18 sections, 25 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Some definitions and notations
  4. Height functions
  5. Kawaguchi--Silverman Conjecture
  6. Green currents
  7. Examples
  8. Relative positive divisors
  9. Definitions and basic properties
  10. The augmented base locus
  11. Geometric canonical height functions
  12. Sparsity of stable marked points
  13. Parameter spaces
  14. Chow variety
  15. Hilbert scheme
  16. ...and 3 more sections

Key Result

Theorem 1.1

Under assumption (H1), the canonical height functions $\hat{h}_f^\pm$ and $\hat{h}_f$ are well-defined, non-negative, and uniquely determined by the following two properties.

Theorems & Definitions (45)

  • Theorem 1.1: cf. Propositions \ref{['constructionHeight']} and \ref{['stablebound']}
  • Theorem 1.2
  • Theorem 1.3: cf. Corollary \ref{['corK3']}
  • Theorem 1.4: cf. Corollary \ref{['cor:KSconj']}
  • Theorem 1.5: cf. Proposition \ref{['constructionGreenCurrents']} and Proposition \ref{['equalityMassHeight']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 35 more