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Classification of Equivariant Legendrian Embeddings of Rational Homogeneous Spaces into Nilpotent Orbits

Minseong Kwon

Abstract

For a complex semi-simple Lie algebra, every nilpotent orbit in its projectivization comes with a complex contact structure. For each nilpotent orbit, we classify projective Legendrian subvarieties that are homogeneous under the actions of their stabilizers in the adjoint group. In particular, we present a classification of equivariant Legendrian embeddings of rational homogeneous spaces into adjoint varieties.

Classification of Equivariant Legendrian Embeddings of Rational Homogeneous Spaces into Nilpotent Orbits

Abstract

For a complex semi-simple Lie algebra, every nilpotent orbit in its projectivization comes with a complex contact structure. For each nilpotent orbit, we classify projective Legendrian subvarieties that are homogeneous under the actions of their stabilizers in the adjoint group. In particular, we present a classification of equivariant Legendrian embeddings of rational homogeneous spaces into adjoint varieties.

Paper Structure

This paper contains 11 sections, 28 theorems, 35 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results and outline
  4. Conventions
  5. Contact geometry of nilpotent orbits
  6. Legendre moduli space
  7. Classification of isotropy irreducible pairs
  8. Homogeneous Legendrian subvarieties arising from isotropy representations
  9. Homogeneous Legendrian subvarieties of adjoint varieties as linear sections
  10. Proof of Theorems \ref{['main thm: adjoint']}--\ref{['main thm: semi simple']} and Corollaries
  11. Tables

Key Result

Theorem 1.1

Let $\mathfrak{s}$ be a simple Lie algebra, $S_{\text{ad}}$ its adjoint group, and $Z \subset \mathbb{P}(\mathfrak{s})$ the adjoint variety. Assume that $O$ is a projective Legendrian subvariety of $Z$. If $O$ is homogeneous under the $\text{Stab}_{S_{\text{ad}}}(O)$-action, then $O$ is the highest

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4: Beauville1998FanoContact
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 53 more