Toric degenerations of spherical varieties

TL;DR

This paper demonstrates that spherical varieties can be degenerated into toric varieties, providing conditions for Gorenstein Fano limits and connecting to Mirror Symmetry and the Minimal Model Program.

Contribution

It establishes a general degeneration process for spherical varieties into toric varieties and explores their properties as boundary points in moduli spaces.

Findings

Any affine or polarized projective spherical variety admits a flat degeneration to a toric variety.

Conditions are given for the limit toric variety to be Gorenstein Fano.

Examples illustrating the degeneration process and its applications are provided.

Abstract

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.