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Symplectic maps and hyperKähler moment map geometry

Yann Rollin

Abstract

We obtain a correspondence between the group of symplectic diffeomorphisms of a 4-dimensional real torus and the vanishing locus of a certain hyperKähler moment map. This observation gives rise to a new flow, called the modified moment map flow. The construction can be adapted to the polyhedral setting, for which we prove a Duistermaat type theorem. This paper lays out the ground work for some effective polyhedral symplectic geometry and for a potential Morse-Bott theory, with applications to the topology of the space of symplectic maps of the 4-torus.

Symplectic maps and hyperKähler moment map geometry

Abstract

We obtain a correspondence between the group of symplectic diffeomorphisms of a 4-dimensional real torus and the vanishing locus of a certain hyperKähler moment map. This observation gives rise to a new flow, called the modified moment map flow. The construction can be adapted to the polyhedral setting, for which we prove a Duistermaat type theorem. This paper lays out the ground work for some effective polyhedral symplectic geometry and for a potential Morse-Bott theory, with applications to the topology of the space of symplectic maps of the 4-torus.

Paper Structure

This paper contains 54 sections, 70 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Notations
  4. Statement of results in the smooth setting
  5. Statement of results in the polyhedral setting
  6. Acknowledgments
  7. Kähler moment map
  8. Smooth symplectic maps
  9. Quotient Hermitian torus
  10. Differentials and differentiation
  11. Cohomology
  12. Bundle structures
  13. An involution
  14. Gauge group action
  15. Infinitesimal action
  16. ...and 39 more sections

Key Result

Theorem 1

We consider a hyperKähler quotient torus $M=V/\Gamma$ of real dimension $4$, endowed with its canonical symplectic form $\omega_M$ and conjugate hyperKähler structure, given by the almost complex structures $I, J$ and $K$, with associated Kähler forms $\hat{\omega}_\bullet$ for $\bullet =I,J,K$. T where ${\mathcal{G}}$ is a Euclidean inner product compatible with the almost complex structures ${

Theorems & Definitions (142)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.3.1
  • Remark 1.3.2
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 132 more