Minimal symplectic atlases of Hermitian symmetric spaces

TL;DR

This paper determines the minimal number of Darboux charts required to cover Hermitian symmetric spaces of compact type, linking it to their embedding degrees in complex projective space, using recent symplectic geometry methods.

Contribution

It introduces a method to compute minimal symplectic atlases for Hermitian symmetric spaces based on their embedding degrees, expanding understanding of their symplectic covering properties.

Findings

Computed minimal number of Darboux charts for various Hermitian symmetric spaces

Established a relationship between embedding degree and symplectic covering

Applied symplectic geometry tools to derive new bounds

Abstract

In this paper we compute the minimal number of Darboux chart needed to cover a Hermitian symmetric space of compact type in terms of the degree of their embeddings in $\mathbb{C} P^N$. The proof is based on the recent work of Y. B. Rudyak and F. Schlenk [18] and on the symplectic geometry tool developed by the first author in collaboration with A. Loi and F. Zuddas [12]. As application we compute this number for a large class of Hermitian symmetric spaces of compact type.