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 ( w ˜ ) be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression w ˜ of w with respect to the data ( G , B , T ) . 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.