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On symplectic geometry of tangent bundles of Hermitian symmetric spaces

Johanna Bimmermann

Abstract

We explicitly construct a symplectomorphism that relates magnetic twists to the invariant hyperkähler structure of the tangent bundle of a Hermitian symmetric space. This symplectomorphism reveals foliations by (pseudo-) holomorphic planes, predicted by vanishing of symplectic homology. Furthermore, in the spirit of Weinstein's tubular neighborhood theorem, we extend the (Lagrangian) diagonal embedding of a compact Hermitian symmetric space to an open dense embedding of a specified neighborhood of the zero section. Using this embedding, we compute the Gromov width and Hofer-Zehnder capacity of these neighborhoods of the zero section.

On symplectic geometry of tangent bundles of Hermitian symmetric spaces

Abstract

We explicitly construct a symplectomorphism that relates magnetic twists to the invariant hyperkähler structure of the tangent bundle of a Hermitian symmetric space. This symplectomorphism reveals foliations by (pseudo-) holomorphic planes, predicted by vanishing of symplectic homology. Furthermore, in the spirit of Weinstein's tubular neighborhood theorem, we extend the (Lagrangian) diagonal embedding of a compact Hermitian symmetric space to an open dense embedding of a specified neighborhood of the zero section. Using this embedding, we compute the Gromov width and Hofer-Zehnder capacity of these neighborhoods of the zero section.

Paper Structure

This paper contains 16 sections, 68 theorems, 256 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of tangent bundles
  4. Magnetic systems
  5. Coadjoint orbits
  6. Riemannian symmetric spaces
  7. Hermitian symmetric spaces
  8. Symplectic identifications
  9. Proof of Theorem \ref{['twisted to hyperkähler']}
  10. Proof of Theorem \ref{['hyperkähler to constant']}
  11. Proof of Theorem \ref{['diagonal']}
  12. Computing some capacities
  13. Hamiltonian circle actions and capacities
  14. Hermitian symmetric spaces of compact type
  15. Proof of Theorem \ref{['capacompact']}
  16. ...and 1 more sections

Key Result

Theorem A

If $M\cong G/K$ is a Hermitian symmetric space of compact type there is a $G$-equivariant symplectomorphism identifying If $M$ is a Hermitian symmetric space of non-compact type, then

Figures (2)

  • Figure 1: In order to satisfy $\mathrm{ad}_{Z_i}^2=-\mathrm{id}$, the norm of $Z_i$ needs to be one. We see that $(Z_i,x_i)$ is equal to the height function, which generates a circle action of period $2\pi$. In particular $\nu$ generates a circle action of period one.
  • Figure 2: The figure schematically shows how the poly-spheres sits in an affine copy of $\mathfrak{su}(2)^r$.

Theorems & Definitions (122)

  • Remark 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Corollary 1.3
  • proof
  • Theorem E
  • Definition 2.1: Horizontal & vertical lift
  • Proposition 2.2: D62, Lemma 2
  • ...and 112 more