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Minimal rational curves on complete symmetric varieties

Michel Brion, Shin-young Kim, Nicolas Perrin

TL;DR

The paper advances the theory of minimal rational curves on complete symmetric varieties by linking the VMRT of a covering family to the orbit geometry in adjoint symmetric spaces. It shows that, for indecomposable non-exceptional cases, there is a unique MR-family whose VMRT is homogeneous, and it provides a detailed correspondence with the restricted root system, including a notable negative answer to Hwang’s question in several Hermitian non-exceptional types. The approach unifies the analysis across group, Hermitian, and simple types, leveraging highest weight curves and the restricted-root framework to describe both MR-classes and their VMRT embeddings, and culminates in a complete classification table for these objects. Practically, this reduces VMRT problems on complete symmetric varieties to representation-theoretic data of the associated symmetric spaces and clarifies how toroidal adjoint compactifications govern MR-geometry. The results also yield Fano/non-Fano distinctions and explicit VMRT decompositions in key cases, informing rigidity and deformation questions in higher Picard-number settings.

Abstract

We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents (VMRT). In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves, and hence a unique VMRT. We relate these results to the restricted root system of the associated symmetric space. In particular we answer by the negative a question of Hwang: for certain Fano wonderful symmetric varieties, the VMRT has two connected components.

Minimal rational curves on complete symmetric varieties

TL;DR

The paper advances the theory of minimal rational curves on complete symmetric varieties by linking the VMRT of a covering family to the orbit geometry in adjoint symmetric spaces. It shows that, for indecomposable non-exceptional cases, there is a unique MR-family whose VMRT is homogeneous, and it provides a detailed correspondence with the restricted root system, including a notable negative answer to Hwang’s question in several Hermitian non-exceptional types. The approach unifies the analysis across group, Hermitian, and simple types, leveraging highest weight curves and the restricted-root framework to describe both MR-classes and their VMRT embeddings, and culminates in a complete classification table for these objects. Practically, this reduces VMRT problems on complete symmetric varieties to representation-theoretic data of the associated symmetric spaces and clarifies how toroidal adjoint compactifications govern MR-geometry. The results also yield Fano/non-Fano distinctions and explicit VMRT decompositions in key cases, informing rigidity and deformation questions in higher Picard-number settings.

Abstract

We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents (VMRT). In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves, and hence a unique VMRT. We relate these results to the restricted root system of the associated symmetric space. In particular we answer by the negative a question of Hwang: for certain Fano wonderful symmetric varieties, the VMRT has two connected components.
Paper Structure (31 sections, 75 theorems, 55 equations)

This paper contains 31 sections, 75 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Rational curves and symmetric spaces
  3. Families of rational curves
  4. Almost homogeneous varieties
  5. Symmetric spaces
  6. Maximal tori of fixed type.
  7. Maximal tori of split type.
  8. The normalizer of a symmetric subgroup
  9. The adjoint symmetric space
  10. Highest weight curves
  11. Complete symmetric varieties
  12. Wonderful and complete symmetric varieties
  13. Relation between families of minimal rational curves
  14. Minimal rational curves for group and Hermitian types
  15. Some cases of simple type
  16. ...and 16 more sections

Key Result

Theorem 1.1

${\mathcal{K}}_x$ is smooth and $\tau_x: {\mathcal{K}}_x \to {\mathcal{C}}_x$ is an isomorphism.

Theorems & Definitions (155)

  • Theorem 1.1: Theorem \ref{['thm:VMRT=orbit']}
  • Theorem 1.2
  • Theorem 1.3: Proposition \ref{['prop:red']} and Remark \ref{['rem-2-ou-1']}
  • Proposition 1.4: Proposition \ref{['prop-psi']}
  • Theorem 1.5: Corollary \ref{['coro:vir-cov']}
  • Proposition 1.6
  • Theorem 1.7: Theorem \ref{['theo-adj']}
  • Corollary 1.8
  • Lemma 2.1
  • proof
  • ...and 145 more