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A note on the Nielsen realization problem for hyper-Kähler manifolds

Simone Billi

Abstract

We give an answer to the Nielsen realization problem for hyper-Kähler manifolds in terms of the same invariant used for K3 surfaces. We determine that, for some of the known deformation types, the representation of the mapping class group on the second cohomology admits a section on its image.

A note on the Nielsen realization problem for hyper-Kähler manifolds

Abstract

We give an answer to the Nielsen realization problem for hyper-Kähler manifolds in terms of the same invariant used for K3 surfaces. We determine that, for some of the known deformation types, the representation of the mapping class group on the second cohomology admits a section on its image.
Paper Structure (2 sections, 5 theorems, 7 equations)

This paper contains 2 sections, 5 theorems, 7 equations.

Table of Contents

  1. Sections of the representation map
  2. Nielsen realization

Key Result

Theorem 1

Let $X$ be an hyper-Kähler manifold such that $\mathop{\mathrm{Aut}}\nolimits(X)\rightarrow \mathop{\mathrm{\mathrm{O}}}\nolimits^+(\mathop{\mathrm{\mathbf{\Lambda}}}\nolimits_X)$ is injective, then the restriction of $\rho\colon\mathop{\mathrm{Mod}}\nolimits(X)\rightarrow \mathop{\mathrm{\mathrm{O}

Theorems & Definitions (12)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Remark 1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • proof : Proof of \ref{['thm1']}
  • proof : Proof of \ref{['cor:max_monodromy']}
  • ...and 2 more