Cotangent Bundles of Partial Flag Varieties and Conormal Varieties of their Schubert Divisors

TL;DR

This paper constructs natural compactifications of cotangent bundles and conormal varieties of Schubert divisors within affine Schubert varieties, establishing their geometric properties such as normality and Cohen-Macaulayness.

Contribution

It introduces a new geometric framework linking cotangent bundles and conormal varieties to affine Schubert varieties, providing their compactifications and analyzing their properties.

Findings

Cotangent bundle of G/P compactified as a closed subvariety of an affine Schubert variety.

Conormal variety of a Schubert divisor compactified as an affine Schubert variety.

Proved that these varieties are normal, Cohen-Macaulay, and Frobenius split.

Abstract

Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle $T^*G/P$. Restricting this identification to the conormal variety $N^*X(w)$ of a Schubert divisor $X(w)$ in $G/P$, we show that there is a compactification of $N^*X(w)$ as an affine Schubert variety. It follows that $N^*X(w)$ is normal, Cohen-Macaulay, and Frobenius split.