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Gap theorems and achirality for automorphisms of K3 surfaces and Enriques surfaces

Kohei Kikuta, Yuta Takada, Taiki Takatsu

Abstract

We prove gap theorems for entropy norms on automorphism groups of K3 surfaces, Enriques surfaces, and irreducible holomorphic symplectic manifolds. We also study the achirality of automorphisms of K3 surfaces and Enriques surfaces in terms of genus-one fibrations.

Gap theorems and achirality for automorphisms of K3 surfaces and Enriques surfaces

Abstract

We prove gap theorems for entropy norms on automorphism groups of K3 surfaces, Enriques surfaces, and irreducible holomorphic symplectic manifolds. We also study the achirality of automorphisms of K3 surfaces and Enriques surfaces in terms of genus-one fibrations.

Paper Structure

This paper contains 26 sections, 41 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Gap theorems and achirality
  3. Main results
  4. Gap theorem for entropy norm
  5. Achirality of parabolic automorphisms
  6. Organization of the paper
  7. Preliminaries
  8. Hyperbolic spaces
  9. Norms and quasimorphisms
  10. Acylindrical actions and WPD elements
  11. Hyperbolic spaces associated with surfaces
  12. Irreducible holomorphic symplectic manifolds
  13. Gap theorems
  14. Achiral elements
  15. Uniform estimate for quasimorphisms
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $X$ be a K3 surface or an Enriques surface over an algebraically closed field of characteristic different from $2$, or a projective irreducible holomorphic symplectic manifold. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (83)

  • Theorem 1.1: \ref{['Gap_thm: ent_Aut']}
  • Theorem 1.2: \ref{['prop:exsistenceof-1', 'th:para_K3']}
  • Theorem 1.3: \ref{['th:para-auto-of-Enriques']}
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4: Bowditch2008Osin
  • Theorem 2.5: Bowditch2008, DGO17
  • ...and 73 more