Conormal Varieties on the Cominuscule Grassmannian - II

TL;DR

This paper constructs a resolution of singularities for conormal varieties on cominuscule Grassmannians, computes defining equations for these varieties, and proposes a conjecture relating conormal and orbital varieties across types.

Contribution

It provides explicit resolutions and equations for conormal varieties on cominuscule Grassmannians and introduces a conjecture linking these to orbital varieties in a type-independent manner.

Findings

Constructed a resolution of singularities for conormal varieties.

Computed defining equations for conormal varieties in specific Grassmannians.

Proposed a conjecture relating conormal and orbital varieties across types.

Abstract

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.