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A smooth Birman-Hilden theory for hyperkähler manifolds

Sidhanth Raman

TL;DR

The paper develops a smooth, higher-dimensional Birman–Hilden theory in the hyperkähler setting, proving that for a smooth regular $D$-cover $p:X\to Y$ with $X$ irreducible hyperkähler, the symmetric mapping class group modulo $D$ is isomorphic to the downstairs mapping class group: $\mathrm{SMod}(X)/D\cong \mathrm{Mod}(Y)$. It builds this theory via a Teichmüller-space framework for Enriques manifolds, establishing a local and global Torelli correspondence by constructing Teichmüller spaces, period domains, and a key Main Lemma that enables the descent of isomorphisms through families. The work further derives Nielsen realization results for Enriques surfaces, showing criteria under which finite subgroups act by isometries or automorphisms, and proving non-realizability phenomena for certain Dehn twists, all tied to liftings to the hyperkähler cover. A central contribution is the global Torelli theorem for Enriques manifolds, together with an injective lifting map on Teichmüller spaces, which together yield the BH-type isomorphism and connect Torelli questions to linear-algebraic invariants of the BBF lattice. The paper also states a conjecture that the Enriques smooth Torelli group is trivial, linking it to the corresponding K3 Torelli problem and highlighting deep structural connections between moduli, Teichmüller theory, and 4-manifold topology.

Abstract

This article grew out of an effort to understand the smooth mapping class groups of certain 4-manifolds in a geometric manner. We prove a smooth analog of the Birman-Hilden theorem for manifolds that admit a hyperkähler structure. This allows us to probe the smooth mapping class groups associated to certain manifolds with nontrivial fundamental groups. Along the way, and of independent interest, we prove a global Torelli theorem for generalized Enriques manifolds. The techniques used are analogous to Teichmüller-theoretic methods in the classical theory of mapping class groups. We apply this hyperkähler Birman-Hilden theorem to obtain results regarding smooth, metric, and complex Nielsen realization on Enriques surfaces.

A smooth Birman-Hilden theory for hyperkähler manifolds

TL;DR

The paper develops a smooth, higher-dimensional Birman–Hilden theory in the hyperkähler setting, proving that for a smooth regular -cover with irreducible hyperkähler, the symmetric mapping class group modulo is isomorphic to the downstairs mapping class group: . It builds this theory via a Teichmüller-space framework for Enriques manifolds, establishing a local and global Torelli correspondence by constructing Teichmüller spaces, period domains, and a key Main Lemma that enables the descent of isomorphisms through families. The work further derives Nielsen realization results for Enriques surfaces, showing criteria under which finite subgroups act by isometries or automorphisms, and proving non-realizability phenomena for certain Dehn twists, all tied to liftings to the hyperkähler cover. A central contribution is the global Torelli theorem for Enriques manifolds, together with an injective lifting map on Teichmüller spaces, which together yield the BH-type isomorphism and connect Torelli questions to linear-algebraic invariants of the BBF lattice. The paper also states a conjecture that the Enriques smooth Torelli group is trivial, linking it to the corresponding K3 Torelli problem and highlighting deep structural connections between moduli, Teichmüller theory, and 4-manifold topology.

Abstract

This article grew out of an effort to understand the smooth mapping class groups of certain 4-manifolds in a geometric manner. We prove a smooth analog of the Birman-Hilden theorem for manifolds that admit a hyperkähler structure. This allows us to probe the smooth mapping class groups associated to certain manifolds with nontrivial fundamental groups. Along the way, and of independent interest, we prove a global Torelli theorem for generalized Enriques manifolds. The techniques used are analogous to Teichmüller-theoretic methods in the classical theory of mapping class groups. We apply this hyperkähler Birman-Hilden theorem to obtain results regarding smooth, metric, and complex Nielsen realization on Enriques surfaces.
Paper Structure (21 sections, 32 theorems, 61 equations)

This paper contains 21 sections, 32 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Reviewing Birman--Hilden Theory
  3. Why Hyperkähler?
  4. Nielsen Realization Results
  5. Related Works
  6. Outline
  7. Acknowledgements
  8. Smooth Lifting Lemmas
  9. Definitions and Lifting Isotopies
  10. Reformulating the Birman--Hilden property
  11. A Review of The Twistor Construction
  12. Enriques Manifolds and The Kuranishi Family
  13. Cohomology and Families
  14. Examples of Enriques Manifolds
  15. Period Domains and Local Torelli
  16. ...and 6 more sections

Key Result

Theorem A

Let $p: X \to Y$ be a smooth regular $D$-covering of a manifold $Y$ admitting a complex structure, where $X$ is a manifold admitting an irreducible hyperkähler structure. Then the cover $p$ has the smooth Birman--Hilden property. That is, $\mathop{\mathrm{SMod}}\nolimits(X)/D \cong \mathop{\mathrm{M

Theorems & Definitions (73)

  • Definition 1.2
  • Theorem A: Hyperkähler Birman--Hilden Theorem
  • Definition 1.4
  • Corollary 1.5
  • Theorem B: Global Torelli Theorem for Enriques Manifolds
  • Theorem 1.6: Birman--Hilden Theorem
  • Remark 1.7
  • Theorem C: Enriques Metric/Complex Nielsen Realization
  • Theorem D: Non-realizability of Dehn twists
  • Theorem E: Ricci-flat implies complex
  • ...and 63 more