A note on toric degeneration of a Bott–Samelson–Demazure–Hansen variety

In this paper, we study the geometry of toric degeneration of a Bott–Samelson–Demazure–Hansen (BSDH) variety, which was algebraically constructed by Pasquier (J Algebra 323(10):2834–2847, 2010). We give some applications to BSDH varieties. Precisely, we classify Fano, weak Fano and log Fano BSDH varieties and their toric limits in Kac–Moody setting. We prove some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties. We also recover the results in (Parameswaran and Karuppuchamy, Toric degeneration of Bott–Samelson–Demazure–Hansen varieties. arXiv:1604.01998, 2016) and extend them to the Kac-Moody setting, by toric methods.