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

In this article we give complete characterization of the expressions w˜ such that the corresponding BSDH variety Z(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.