# Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbb{C})$

## TL;DR

This paper investigates the rigidity of Bott-Samelson-Demazure-Hansen varieties associated with the group PSO(2n+1, C), focusing on cohomology vanishing of tangent bundles for specific reduced expressions of the longest Weyl group element.

## Contribution

It characterizes all reduced expressions of the longest Weyl group element for which the tangent bundle's higher cohomology vanishes, revealing rigidity conditions.

## Key findings

- All reduced expressions with vanishing higher cohomology are described.
- The study provides conditions for the rigidity of these varieties.
- Cohomology modules of tangent bundles are explicitly computed.

## Abstract

Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced expressions of $w_{0}$ in terms of a Coxeter element such that all the higher cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i})$ vanish (see Theorem \ref{theorem 8.1}).