Some results on equivariant contact geometry for partial flag varieties

TL;DR

This paper investigates equivariant contact structures on partial flag varieties for type ADE groups, proving a special case of the LeBrun-Salamon conjecture using properties of G-equivariant vector bundles, and describes a canonical contact structure on isotropic Grassmannians.

Contribution

It provides an independent proof of a special case of the LeBrun-Salamon conjecture for partial flag varieties of type ADE, utilizing properties of G-equivariant vector bundles.

Findings

Proves a special case of the LeBrun-Salamon conjecture for type ADE partial flag varieties.

Provides a canonical description of the G-invariant contact structure on isotropic Grassmannians.

Establishes a new approach independent of Boothby's classification for contact structures on these varieties.

Abstract

We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby's classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of $G$-equivariant vector bundles on $G/P$. A byproduct of our argument is a canonical, global description of the unique $SO_{2n}(\mathbb C)$-invariant contact structure on the isotropic Grassmannian of $2$-planes in $\mathbb C^{2n}$.