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.