Minimal rational curves on equivariant compactifications of symmetric spaces

TL;DR

This paper studies minimal rational curves on equivariant compactifications of symmetric spaces, showing they are orbit-closures of 1-parameter subgroups under certain geometric conditions, and extends known results to a broader class of compactifications.

Contribution

It establishes a criterion linking minimal rational curves to 1-parameter subgroup orbits via Gauss-nondegeneracy of VMRT, and generalizes previous results to arbitrary equivariant compactifications.

Findings

Minimal rational curves are orbit-closures of 1-parameter subgroups under Gauss-nondegeneracy.

Gauss-nondegeneracy of VMRT holds for compactifications of simple algebraic groups.

VMRT is the closure of an adjoint orbit, extending known results.

Abstract

Let $G/H$ be a symmetric space of a complex linear algebraic group $G$ and let $X$ be a nonsingular equivariant compactification of $G/H$. We investigate the question: when are minimal rational curves on $X$ orbit-closures of 1-parameter subgroups of $G$? We show that this is the case if the variety of minimal rational tangents (VMRT) at a base point in $G/H \subset X$ is Gauss-nondegenerate. Our method combines algebraic geometry of minimal rational curves with differential geometry of symmetric spaces: orbits of 1-parameter subgroups arise as holomorphic geodesics of an invariant torsion-free affine connection on $G/H$. We prove furthermore that the Gauss-nondegeneracy of VMRT holds for nonsingular equivariant compactifications of simple algebraic groups regarded as symmetric spaces. In this case, we also show that the VMRT is the closure of an adjoint orbit, which generalizes a result of Brion and Fu's on wonderful compactifications to arbitrary equivariant compactifications.