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The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties

Ariyan Javanpeykar, Ruiran Sun, Kang Zuo

Abstract

Motivated by Lang-Vojta's conjectures on hyperbolic varieties, we prove a new version of the Shafarevich conjecture in which we establish the finiteness of pointed families of polarized varieties. We then give an arithmetic application to the finiteness of integral points on moduli spaces of polarized varieties.

The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties

Abstract

Motivated by Lang-Vojta's conjectures on hyperbolic varieties, we prove a new version of the Shafarevich conjecture in which we establish the finiteness of pointed families of polarized varieties. We then give an arithmetic application to the finiteness of integral points on moduli spaces of polarized varieties.

Paper Structure

This paper contains 16 sections, 11 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Motivation: Lang-Vojta's conjecture on hyperbolicity
  4. Arithmetic application
  5. An open problem
  6. A remark on stackyness
  7. The moduli space of pointed maps
  8. Boundedness
  9. Infinitesimal deformations
  10. Deforming pointed families of polarized varieties
  11. Starting point: a non-rigid pointed morphism
  12. The construction of the map $\xi$
  13. Viehweg-Zuo's graded Higgs bundle
  14. The upper bound on $\# B$
  15. Proof of the Weak-Pointed Shafarevich conjecture
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of characteristic zero, let $h\in \mathbb{Q}[t]$, let $U$ be a quasi-projective variety over $k$, and let $U\to \mathcal{M}_{h}\otimes_{\mathbb{Q}} k$ be a quasi-finite morphism. Then $U$ has the following finiteness property: If $C$ is a smooth quasi-project then the set of non-constant morphisms $f:C\to U$ with $f(c_1) = u_1, \ldots, f(c_N) = u_N$ is fini

Theorems & Definitions (30)

  • Theorem 1.1: Weak-Pointed Shafarevich Conjecture
  • Conjecture 1.2: Lang-Vojta
  • Conjecture 1.3: Persistence Conjecture
  • Theorem 1.4: Persistence Conjecture holds for $U$
  • Conjecture 1.5: Pointed Shafarevich Conjecture
  • Conjecture 1.6: Uniformizability of $\mathcal{M}$
  • Proposition 2.1: Scheme structure
  • Theorem 2.2: Boundedness
  • Proposition 2.3: Infinitesimal deformations
  • proof
  • ...and 20 more