On symplectic geometry of tangent bundles of Hermitian symmetric spaces

TL;DR

This paper constructs a symplectomorphism linking magnetic twists to hyperk"ahler structures on tangent bundles of Hermitian symmetric spaces, revealing geometric foliations and computing symplectic capacities.

Contribution

It explicitly constructs a symplectomorphism connecting magnetic twists with hyperk"ahler structures and extends the diagonal embedding to compute symplectic capacities.

Findings

Revealed foliations by holomorphic planes in tangent bundles.

Computed Gromov width and Hofer-Zehnder capacity of neighborhoods.

Extended Lagrangian diagonal embedding near the zero section.

Abstract

We explicitly construct a symplectomorphism that relates magnetic twists to the invariant hyperk\"ahler 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.