Legendrian cone structures and contact prolongations

TL;DR

This paper investigates Legendrian cone structures on contact manifolds, characterizing subadjoint varieties via contact prolongations, and demonstrates a holomorphic horizontal splitting of the canonical distribution on the associated contact G-structure.

Contribution

It introduces a new characterization of subadjoint varieties among Legendrian submanifolds using contact prolongations and proves a splitting property of the canonical distribution.

Findings

Characterization of subadjoint varieties among Legendrian submanifolds.

Existence of a holomorphic horizontal splitting on the contact G-structure.

Insight into the structure of Legendrian cone structures on contact manifolds.

Abstract

We study a cone structure ${\mathcal C} \subset {\mathbb P} D$ on a holomorphic contact manifold $(M, D \subset T_M)$ such that each fiber ${\mathcal C}_x \subset {\mathbb P} D_x$ is isomorphic to a Legendrian submanifold of fixed isomorphism type. By characterizing subadjoint varieties among Legendrian submanifolds in terms of contact prolongations, we prove that the canonical distribution on the associated contact G-structure admits a holomorphic horizontal splitting.