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.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 arxiv.org/abs/1604.01998, by toric methods