Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the Nielsen realization problem for Enriques manifolds

Simone Billi

Abstract

We give a numerical criterion for the Nielsen realization problem for Enriques manifolds, based on the recent developments on the Birman-Hilden theory for hyper-Kähler manifolds and on Nielsen realization for hyper-Kähler manifolds. We apply the criterion to known examples of Enriques manifolds to get explicit groups that can be realized or not realized, and comment on questions related to the Nielsen realization problem.

A note on the Nielsen realization problem for Enriques manifolds

Abstract

We give a numerical criterion for the Nielsen realization problem for Enriques manifolds, based on the recent developments on the Birman-Hilden theory for hyper-Kähler manifolds and on Nielsen realization for hyper-Kähler manifolds. We apply the criterion to known examples of Enriques manifolds to get explicit groups that can be realized or not realized, and comment on questions related to the Nielsen realization problem.

Paper Structure

This paper contains 6 sections, 2 theorems, 18 equations.

Table of Contents

  1. Proof of the numerical criterion
  2. The known examples
  3. Enriques manifolds from K3 surfaces
  4. Enriques manifolds from abelian surfaces
  5. Some examples of realizations and non-realizations
  6. A non-trivial Birman-Hilden central extension

Key Result

Theorem 1

Let $Y$ be an Enriques manifold with universal cover $p\colon X\to Y$ and $D=\pi_1(Y)$. Let $G\leq \mathop{\mathrm{Mod}}\nolimits(Y)$ be a finite subgroup and $\widetilde{G}=\tilde{p}^{-1}(G)\leq \mathop{\mathrm{SMod}}\nolimits_D(X)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (6)

  • Theorem 1
  • Proposition 2
  • proof : Proof of \ref{['thm1']}
  • Example 3.1: Some realizations
  • Example 3.2: Some non-realizations
  • proof : Proof of \ref{['thm:central_extension']}