A superpotential for Grassmannian Schubert varieties

TL;DR

This paper introduces a superpotential for Grassmannian Schubert varieties, linking it to toric degenerations and Newton-Okounkov bodies, extending mirror symmetry concepts to singular varieties.

Contribution

It generalizes the Marsh-Rietsch superpotential to Schubert varieties, constructs Newton-Okounkov bodies, and relates them via tropicalization, providing new toric degenerations and desingularizations.

Findings

Superpotential $W^{ ext{\lambda}}$ governs toric degenerations.

Newton-Okounkov bodies coincide with superpotential polytopes.

Existence of small toric desingularizations and Fano partial desingularizations.

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