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Kummer Rigidity for Hyperkähler Automorphisms

Seung uk Jang

Abstract

We show that a holomorphic automorphism on a projective hyperkähler manifold that has positive topological entropy and has volume measure as the measure of maximal entropy, is necessarily a Kummer example, partially extending the analogous results in (Cantat-Dupont 2020)(Filip-Tosatti 2018) for complex surfaces. A trick with Jensen's inequality is used to show that stable and unstable distributions exhibit uniform rate of contraction and expansion, and with them our hyperkähler manifold is shown to be flat. A result in (Greb-Kebekus-Peternell 2016) then implies that our hyperkähler manifold is birational to a torus quotient, giving the Kummer example structure.

Kummer Rigidity for Hyperkähler Automorphisms

Abstract

We show that a holomorphic automorphism on a projective hyperkähler manifold that has positive topological entropy and has volume measure as the measure of maximal entropy, is necessarily a Kummer example, partially extending the analogous results in (Cantat-Dupont 2020)(Filip-Tosatti 2018) for complex surfaces. A trick with Jensen's inequality is used to show that stable and unstable distributions exhibit uniform rate of contraction and expansion, and with them our hyperkähler manifold is shown to be flat. A result in (Greb-Kebekus-Peternell 2016) then implies that our hyperkähler manifold is birational to a torus quotient, giving the Kummer example structure.

Paper Structure

This paper contains 37 sections, 26 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Hyperkähler Kummer Examples
  3. Statement of the Result
  4. Examples of Hyperkähler Kummer Examples
  5. Generalized Kummer Variety
  6. Hilbert Scheme of a Kummer Surface
  7. Outline of the Proof
  8. Example: dilating metric on a Hilbert Scheme of a Kummer Surface
  9. Constructing the metric $\omega_0$
  10. Flatness of $\omega_0$ and Verifying a Kummer Example
  11. The Hypothesis $\mu=\mathrm{vol}$
  12. Contents of the Paper
  13. Preliminaries on Hyperkähler Manifolds and their Dynamical Structures
  14. Hyperkähler Structures
  15. Invariance
  16. ...and 22 more sections

Key Result

Theorem 1

Let $X$ be a projective hyperkähler manifold. Let ${f}\colon X\to X$ be a holomorphic automorphism that has positive topological entropy. Suppose the volume measure is the ${f}$-invariant measure of maximal entropy. Then the underlying hyperkähler manifold $X$ is a normalization of a torus quotient,

Theorems & Definitions (49)

  • Definition : Kummer Example
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem : oguiso2009
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 39 more