Minimal rational curves on equivariant compactifications of symmetric spaces

Let $G\mathrm{/}H$ be a symmetric space of a complex linear algebraic group $G$ and let $X$ be a nonsingular equivariant compactification of $G\mathrm{/}H$. 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 $G\mathrm{/}H\subset X$ 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 $G\mathrm{/}H$. 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.