A superpotential for Grassmannian Schubert varieties

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 this article we introduce a ``superpotential'' W^ for each Grassmannian Schubert variety X_, generalizing the Marsh-Rietsch superpotential for Grassmannians, and we show that W^ governs many toric degenerations of X_. We also generalize the ``polytopal mirror theorem'' for Grassmannians from our previous work: namely, for any cluster seed G for X_, we construct a corresponding Newton-Okounkov convex body G^, and show that it coincides with the superpotential polytope G^, that is, it is cut out by the inequalities obtained by tropicalizing an associated Laurent expansion of W^. This gives us a toric degeneration of the Schubert variety X_ to the (singular) toric variety Y (N_) of the Newton-Okounkov body. Finally, for a particular cluster seed G=G^ₑ₄₂ we show that the toric variety Y (N_) has a small toric desingularisation, and we describe an intermediate partial desingularisation Y (F_) that is Gorenstein Fano. Many of our results extend to more general varieties in the Grassmannian.