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Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture

Lie Fu, Zhiyuan Li, Teppei Takamatsu, Haitao Zou

TL;DR

This work advances the geometric analogue of the Shafarevich problem for primitive symplectic varieties by proving finiteness of generic fibers for pointed families over a fixed curve and, under semi-ampleness, finiteness of projective families as well. The authors develop a global moduli framework, deploy a uniform Kuga–Satake construction to reduce to polarized cases, and leverage relative cone conjectures and Matsusaka–Mumford techniques to control birational and deformation types. They establish finiteness results for generic fibers, NS lattices, and polarized models, and provide counterexamples to demonstrate the optimality of their hypotheses in the non-projective setting. The results significantly extend finiteness phenomena from hyper-Kähler manifolds to singular primitive symplectic varieties and lay groundwork for a robust moduli-theoretic treatment of these geometries.

Abstract

We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve $(B, 0)$ and any fixed primitive symplectic variety $X$, among all locally trivial families of $\mathbb{Q}$-factorial and terminal primitive symplectic varieties over $B$ whose fiber over $0$ is isomorphic to $X$, we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-Kähler manifolds of known deformation types, we show that there are only finitely many such projective families up to isomorphism. These results are optimal since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-Kähler varieties over some pointed curve $(B, 0)$ such that they are all isomorphic over the punctured curve $B\backslash \{0\}$ and have isomorphic fibers over the base point $0$.

Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture

TL;DR

This work advances the geometric analogue of the Shafarevich problem for primitive symplectic varieties by proving finiteness of generic fibers for pointed families over a fixed curve and, under semi-ampleness, finiteness of projective families as well. The authors develop a global moduli framework, deploy a uniform Kuga–Satake construction to reduce to polarized cases, and leverage relative cone conjectures and Matsusaka–Mumford techniques to control birational and deformation types. They establish finiteness results for generic fibers, NS lattices, and polarized models, and provide counterexamples to demonstrate the optimality of their hypotheses in the non-projective setting. The results significantly extend finiteness phenomena from hyper-Kähler manifolds to singular primitive symplectic varieties and lay groundwork for a robust moduli-theoretic treatment of these geometries.

Abstract

We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve and any fixed primitive symplectic variety , among all locally trivial families of -factorial and terminal primitive symplectic varieties over whose fiber over is isomorphic to , we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-Kähler manifolds of known deformation types, we show that there are only finitely many such projective families up to isomorphism. These results are optimal since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-Kähler varieties over some pointed curve such that they are all isomorphic over the punctured curve and have isomorphic fibers over the base point .

Paper Structure

This paper contains 28 sections, 35 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Hyperbolicity of moduli spaces and polarized Shafarevich conjecture
  3. Beyond moduli spaces: unpolarized Shafarevich conjecture
  4. Geometric unpolarized Shafarevich conjecture
  5. Main results
  6. Strategy of proof and challenges
  7. Primitive symplectic varieties
  8. Primitive symplectic varieties
  9. Locally trivial families of primitive symplectic varieties
  10. $\mathbb{Q}$-factoriality in locally trivial families
  11. Period map and Local Torelli Theorem
  12. Cone conjectures for primitive symplectic varieties
  13. Néron--Severi lattices and cones
  14. Kawamata--Morrison cone conjectures over a field
  15. Consequences of cone conjectures: finiteness results
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(B,0)$ be a pointed smooth connected curve defined over an algebraically closed field $k$ of characteristic 0. Let $X$ be a $\mathbb{Q}$-factorial terminal primitive symplectic variety over $k$.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • ...and 74 more