Conormal varieties on the cominuscule Grassmannian-II

Autor: Rahul Singh
Rok vydání: 2020
Předmět:
Zdroj: Mathematische Zeitschrift. 298:551-576
ISSN: 1432-1823
0025-5874
DOI: 10.1007/s00209-020-02620-7
Popis: Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let $\mu:T^*X\rightarrow\mathcal N$ be the Springer map from the cotangent bundle of $X$ to the nilpotent cone $\mathcal N$. In this paper, we construct a resolution of singularities for the conormal variety $T^*_XX_w$ of $X_w$ in $X$. Further, for $X$ the usual or symplectic Grassmannian, we compute a system of equations defining $T^*_XX_w$ as a subvariety of the cotangent bundle $T^*X$ set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties $\mu(T^*_XX_w)$. Inspired by the system of defining equations, we conjecture a type-independent equality, namely $T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu(T^*_XX_w))$. The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C.
Comment: Version 1 included similar results in type D. However, there were serious errors in the proof. All claims to results in type D have been removed in this version. We are grateful to Anna Melnikov for pointing us to these errors
Databáze: OpenAIRE
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