格拉斯曼舒伯特变体的超势能

Konstanze Rietsch, Lauren Williams
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引用次数: 0

摘要

虽然对旗变和格拉斯曼的镜像对称性已有广泛研究,但格拉斯曼中的舒伯特变是奇异的,因此标准的镜像对称性声明并不明确。然而,在这篇文章中,我们为每个格拉斯曼中的舒伯特变$X_{\lambda}$引入了一个 "超势能"$W^{\lambda}$,概括了格拉斯曼的马什-里奇超势能,并证明了$W^{\lambda}$支配着$X_{\lambda}$的多子退化。我们还推广了先前工作中针对格拉斯曼的 "多顶镜像定理":即对于 $X_{\lambda}$ 的任何簇种子 $G$,我们构造了一个相应的牛顿-奥孔科夫凸体 $/Delta_G^{/lambda}$,并证明它与超势能多面体 $\Gamma_G^{\lambda}$ 重合,也就是说,它是通过对 $W^{\lambda}$ 的相关劳伦展开进行热带化而得到的不等式切割出来的。这样,我们就得到了舒伯特变元 $X_{\lambda}$ 到牛顿-奥孔科夫体(奇异)变元 $Y(\mathcal{N}_{\lambda})$的环状退化。最后,对于一个特定的簇种子 $G=G^\lambda_{\mathrm{rec}}$,我们展示了环综 $Y(\mathcal{N}_\{lambda})$ 有一个小的环去奇化,并且我们描述了一个中间部分去奇化 $Y(\mathcal{F}_\lambda)$,它是戈伦斯坦法诺的。我们的许多结果都可以推广到格拉斯曼中更多的一般 varieties 上。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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'' $W^{\lambda}$ 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.
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