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

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}).

35 pages. arXiv admin note: substantial text overlap with arXiv:1610.00812, arXiv:1908.05595