Bott–Samelson resolution

In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is independently due to and.

Definition

Let G be a connected reductive complex algebraic group, B a Borel subgroup and T a maximal torus contained in B.
Let Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:
so that. Let be the subgroup generated by B and a representative of. Let be the quotient:
with respect to the action of by
It is a smooth projective variety. Writing for the Schubert variety for w, the multiplication map
is a resolution of singularities called the Bott–Samelson resolution. has the property: and In other words, has rational singularities.
There are also some other constructions; see, for example,.