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Hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds

Bastien Philippe

TL;DR

The paper investigates hyperbolicity phenomena for moduli spaces of marked hyperkähler manifolds by focusing on deformations along directions with positive Hodge-bundle curvature. It develops a lifting framework that realizes prescribed period maps from arbitrary curves into the period domain and then into the moduli space, extending prior results to higher dimensions and noncompact base curves. By analyzing 2-dimensional K3-type period domains and employing Nevanlinna theory, it derives a vanishing result for the positive Kobayashi pseudo-distance on the moduli space when the marking lattice has rank at least 4 and provides jet-analytic restrictions on entire curves in period domains. The work connects period-domain geometry, MBM/class-wall structures, and complex-analytic techniques to constrain families of hyperkähler manifolds and their moduli, with implications for directed hyperbolicity and period-map dynamics.

Abstract

We study the hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds along directions corresponding to families having positivity properties for their Hodge bundle. In particular, we show that the Kobayashi pseudo-distance computed using disks tangent to these directions vanishes. As an intermediate step, we establish the existence of families of marked hyperk{ä}hler manifolds over arbitrary curves having a prescribed period map to the corresponding period domain. This generalizes a recent theorem of Greb and Schwald [GS24] to the case of hyperk{ä}hler manifolds of arbitrary dimensions and nonnecessarily compact curves. Finally, using Nevanlinna theory, we establish restrictions on families of hyperk{ä}hler manifolds over C having positive Hodge bundle.

Hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds

TL;DR

The paper investigates hyperbolicity phenomena for moduli spaces of marked hyperkähler manifolds by focusing on deformations along directions with positive Hodge-bundle curvature. It develops a lifting framework that realizes prescribed period maps from arbitrary curves into the period domain and then into the moduli space, extending prior results to higher dimensions and noncompact base curves. By analyzing 2-dimensional K3-type period domains and employing Nevanlinna theory, it derives a vanishing result for the positive Kobayashi pseudo-distance on the moduli space when the marking lattice has rank at least 4 and provides jet-analytic restrictions on entire curves in period domains. The work connects period-domain geometry, MBM/class-wall structures, and complex-analytic techniques to constrain families of hyperkähler manifolds and their moduli, with implications for directed hyperbolicity and period-map dynamics.

Abstract

We study the hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds along directions corresponding to families having positivity properties for their Hodge bundle. In particular, we show that the Kobayashi pseudo-distance computed using disks tangent to these directions vanishes. As an intermediate step, we establish the existence of families of marked hyperk{ä}hler manifolds over arbitrary curves having a prescribed period map to the corresponding period domain. This generalizes a recent theorem of Greb and Schwald [GS24] to the case of hyperk{ä}hler manifolds of arbitrary dimensions and nonnecessarily compact curves. Finally, using Nevanlinna theory, we establish restrictions on families of hyperk{ä}hler manifolds over C having positive Hodge bundle.

Paper Structure

This paper contains 19 sections, 39 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Families of hyperkähler manifolds
  4. Moduli space of marked hyperkähler manifolds
  5. Pseudo-hermitian metrics
  6. Period domains of $\mathrm{K3}$-type
  7. Variation of Hodge structures of $\mathrm{K3}$-type
  8. The Griffiths-Schmid metric
  9. Domains of dimension 2
  10. Families with prescribed period map
  11. Global Torelli theorem and MBM classes
  12. Proof of Theorem \ref{['thm lift']}
  13. Kobayashi pseudo-distance
  14. Positive Kobayashi pseudo-distance
  15. Vanishing of the positive Kobayashi pseudo-distance
  16. ...and 4 more sections

Key Result

Theorem 1.2

Assume that $\mathrm{rk}(\Lambda) \geqslant 4$. Then the positive Kobayashi pseudo-distance vanishes on every connected component of $\mathcal{M}_\Lambda$.

Theorems & Definitions (97)

  • Theorem 1.2: $\subset$ Theorem \ref{['thm distance koba espace de mod']}
  • Theorem 1.3: = Theorem \ref{['thm lift']}
  • Theorem 1.5: Corollary \ref{['coro Nevanlinna 0jet']} + Proposition \ref{['prop nevanlinna 1jet']}
  • Theorem 1.6: = Theorem \ref{['thm SMT']}
  • Definition 2.1
  • Proposition 2.2: HuybrechtsBourbaki Proposition 4.3
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 3.1
  • ...and 87 more