Marked points of families of hyperbolic automorphisms of smooth complex projective varieties
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913638878543872 |
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| author | Zhang, Yugang |
| author_facet | Zhang, Yugang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12342 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Marked points of families of hyperbolic automorphisms of smooth complex projective varieties Zhang, Yugang Dynamical Systems Algebraic Geometry Complex Variables 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. |
| title | Marked points of families of hyperbolic automorphisms of smooth complex projective varieties |
| topic | Dynamical Systems Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2409.12342 |