Workshop on Combinatorics of Enumerative Geometry

Abstract: While mirror symmetry for flag varieties and Grassmannians has been extensively studied, Schubert varieties in the Grassmannian are singular, and hence standard mirror symmetry statements are not well-defined. Nevertheless, in joint work with Konstanze Rietsch we introduce a ``superpotential'' for each Grassmannian Schubert variety, generalizing the Marsh-Rietsch superpotential for Grassmannians, and we show that it governs many toric degenerations of the Schubert variety. We also prove a ``polytopal mirror theorem'' for Schubert varieties: namely, for any cluster seed for the Schubert variety, we construct a corresponding Newton-Okounkov convex body, and show that it coincides with the superpotential polytope, that is, it is cut out by the inequalities obtained by tropicalizing an associated Laurent expansion of the superpotential. Finally, we observe that the lattice points of our Newton-Okounkov bodies have an interpretation in terms of quantum cohomology.