Abstract

Toric degenerations play an important role in mirror symmetry, allowing constructions for toric varieties to be extended to other varieties. For example, the Gelfand—Cetlin toric degeneration of the Grassmannian gives a Laurent polynomial mirror to the Grassmannian. The Calabi—Yau hypersurface  in the Grassmannian is expected to mirror a compactification of the fibers of the Laurent polynomial. These fibers can easily be compactified to a  hypersurface in a singular toric variety using toric mirror symmetry, but how can the toric degeneration be unwound on mirror side? How can we leave the toric context? Using the combinatorics of the Gelfand—Cetlin polytope, I’ll propose an answer to this question, which surprisingly turns out to be a natural generalization to Grassmannians of the oldest mirror symmetry constructions: the quintic mirror. This is joint work with Tom Coates and Charles Doran.