Applications of Homogeneous Fiber Bundles to the Schubert Varieties

This article explores the relationship between Schubert varieties and equivariant embeddings, using the framework of homogeneous fiber bundles over flag varieties. We show that the homogenous fiber bundles obtained from Bott-Samelson-Demazure-Hansen varieties are always toroidal. Furthermore, we identify the wonderful varieties among them. We give a short proof of a conjecture of Gao, Hodges, and Yong for deciding when a Schubert variety is spherical with respect to an action of a Levi subgroup. By using BP-decompositions, we obtain a characterization of the smooth spherical Schubert varieties. Among the other applications of our results are: 1) a characterization of the spherical Bott-Samelson-Demazure-Hansen varieties, 2) an alternative proof of the fact that, in type A, every singular Schubert variety of torus complexity 1 is a spherical Schubert variety, and 3) a proof of the fact that, for simply laced algebraic groups of adjoint type, every spherical $G$-Schubert variety is locally rigid, that is to say, the first cohomology of its tangent sheaf vanishes.