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Birational automorphism groups in families of hyper-Kähler manifolds

Francesco Antonio Denisi, Claudio Onorati, Francesca Rizzo, Sasha Viktorova

TL;DR

The paper analyzes how birational automorphism groups of hyper-Kähler manifolds behave in families, proving that finiteness on very general fibers persists across the very general locus while special fibers can and do acquire infinite birational automorphism groups on a dense countable subset for all known deformation types. Central to the approach are monodromy constraints, the movable cone interior characterization, and lattice-theoretic constructions that detect when birational automorphism groups become infinite. The results show that the Mori dream space property, conjecturally tied to finiteness of $ ext{Bir}$ for hyper-Kähler manifolds, is not an open condition in families, and this non-openness extends to nonprojective families as well. The work combines deformation theory, period maps, and wall-and-chamber analyses to establish both upper-semicontinuity for very general fibers and density of infinite birational symmetry among special fibers.

Abstract

We study the behavior of birational automorphism groups in families of projective hyper-Kähler manifolds.

Birational automorphism groups in families of hyper-Kähler manifolds

TL;DR

The paper analyzes how birational automorphism groups of hyper-Kähler manifolds behave in families, proving that finiteness on very general fibers persists across the very general locus while special fibers can and do acquire infinite birational automorphism groups on a dense countable subset for all known deformation types. Central to the approach are monodromy constraints, the movable cone interior characterization, and lattice-theoretic constructions that detect when birational automorphism groups become infinite. The results show that the Mori dream space property, conjecturally tied to finiteness of for hyper-Kähler manifolds, is not an open condition in families, and this non-openness extends to nonprojective families as well. The work combines deformation theory, period maps, and wall-and-chamber analyses to establish both upper-semicontinuity for very general fibers and density of infinite birational symmetry among special fibers.

Abstract

We study the behavior of birational automorphism groups in families of projective hyper-Kähler manifolds.
Paper Structure (9 sections, 10 theorems, 61 equations)

This paper contains 9 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattice theory and hyper-Kähler manifolds
  4. Monodromy operators and cones in the Néron--Severi space
  5. Birational automorphism group of the very general fiber
  6. Proof of Theorem \ref{['thm:main1']}:
  7. Birational automorphism group of the special fiber
  8. Proof of Theorem \ref{['thm:main2']}:
  9. Proof of Corollary \ref{['cor:corollary']}:

Key Result

Theorem 1.1

Let $\varphi\colon\mathscr{X}\to \Delta$ be a nontrivial polarized family of hyper-Kähler manifolds, over a small disk $\Delta$. There exists a (possibly empty) finite subset $\mathcal{F}\subseteq \mathcal{S}$, a group $G^0$ and a constant $N \coloneqq N(\varphi)$ such that and In particular, the map is "upper-semicontinuous" with respect to the co-finite topology on $\Delta\smallsetminus \math

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Example 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 15 more