Conormal Varieties on the Cominuscule Grassmannian

TL;DR

The paper constructs a compactification of conormal varieties on cominuscule Grassmannians using Schubert varieties and explores conditions for these compactifications to be Schubert subvarieties, with applications to determinantal varieties.

Contribution

It introduces a new compactification method for conormal varieties on cominuscule Grassmannians via Schubert varieties and characterizes when these are Schubert subvarieties.

Findings

Compactification of conormal varieties as Schubert subvarieties.

Characterization of smoothness conditions for subvarieties.

Application to conormal fibers in determinantal varieties.

Abstract

Let $G$ be a simply connected, almost simple group over an algebraically closed field $\mathbf k$, and $P$ a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification $\phi:T^*G/P\rightarrow X(u)$, where $X(u)$ is a Schubert variety corresponding to the loop group $LG$. Let $N^*X(w)\subset T^*G/P$ be the conormal variety of some Schubert variety $X(w)$ in $G/P$; hence we obtain that the closure of $\phi(N^*X(w))$ in $X(u)$ is a $B$-stable compactification of $N^*X(w)$. We further show that this compactification is a Schubert subvariety of $X(u)$ if and only if $X(w_0w)\subset G/P$ is smooth, where $w_0$ is the longest element in the Weyl group of $G$. This result is applied to compute the conormal fibre at the zero matrix in any determinantal variety.