On Fano and weak Fano Bott-Samelson-Demazure-Hansen varieties

Abstract: Let $G$ be a simple algebraic group over the field of complex numbers. Fix a maximal torus $T$ and a Borel subgroup $B$ of $G$ containing $T$. Let $w$ be an element of the Weyl group $W$ of $G$, and let $Z(\tilde w)$ be the Bott-Samelson-Demazure-Hansen (BSDH) variety corresponding to a reduced expression $\tilde w$ of $w$ with respect to the data $(G, B, T)$. In this article we give complete characterization of the expressions $\tilde w$ such that the corresponding BSDH variety $Z(\tilde w)$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.