Minimal rational curves on equivariant compactifications of symmetric spaces

Jun-Muk Hwang; Qifeng Li

Paper

Minimal rational curves on equivariant compactifications of symmetric spaces

Abstract

Let

be a symmetric space of a complex linear algebraic group

and let

be a nonsingular equivariant compactification of

. We investigate the question: when are minimal rational curves on

orbit-closures of 1-parameter subgroups of

? We show that this is the case if the variety of minimal rational tangents (VMRT) at a base point in

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

. 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.