2024/09/01 by Konstanze Rietsch, Lauren Williams, Rietsch, Konstanze +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2409.00734
openalex publication_date 2024/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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'' Wλ for each Grassmannian Schubert variety Xλ, generalizing the Marsh-Rietsch superpotential for Grassmannians, and we show that Wλ governs many toric degenerations of Xλ. We also generalize the ``polytopal mirror theorem'' for Grassmannians from our previous work: namely, for any cluster seed G for Xλ, we construct a corresponding Newton-Okounkov convex body ΔGλ, and show that it coincides with the superpotential polytope ΓGλ, that is, it is cut out by the inequalities obtained by tropicalizing an associated Laurent expansion of Wλ. This gives us a toric degeneration of the Schubert variety Xλ to the (singular) toric variety Y(Nλ) of the Newton-Okounkov body. Finally, for a particular cluster seed G=Gλrec we show that the toric variety Y(Nλ) has a small toric desingularisation, and we describe an intermediate partial desingularisation Y(Fλ) that is Gorenstein Fano. Many of our results extend to more general varieties in the Grassmannian.