Characterizing subadjoint varieties among Legendrian varieties

TL;DR

This paper characterizes subadjoint Legendrian varieties within the broader class of Legendrian varieties, using isotropy representations and properties of their projective third fundamental forms.

Contribution

It provides a new characterization of subadjoint varieties among Legendrian varieties based on isotropy representations and geometric features.

Findings

Characterization of subadjoint varieties among Legendrian varieties.

Relation between third fundamental forms and lines on Legendrian varieties.

Identification of automorphism groups of nonsingular Legendrian varieties.

Abstract

For a symplectic vector space $V$, a projective subvariety $Z \subset {\bf P} V$ is a Legendrian variety if its affine cone $\widehat{Z} \subset V$ is Lagrangian. In addition to the classical examples of subadjoint varieties associated to simple Lie algebras, many examples of nonsingular Legendrian varieties have been discovered which have positive-dimensional automorphism groups. We give a characterization of subadjoint varieties among such Legendrian varieties in terms of the isotropy representation. Our proof uses some special features of the projective third fundamental forms of Legendrian varieties and their relation to the lines on the Legendrian varieties.