Bott–Samelson resolution explained

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

w \in W = N _{G(T)/T.} Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:\underline{w} = (s _{i _{1}} , s _{i _{2}} , \ldots, s _{i _{\ell}} )

so that

w = s _{i _{1}} s _{i _{2}} … s _{i _{\ell}} . (ℓ is the length of w.) Let P _{i _{j}} \subset G be the subgroup generated by B and a representative of s _{i _{j}} . Let Z _{\underline{w} } be the quotient:Z _{\underline{w} } = P_ \times \cdots \times P_/B^\ell

with respect to the action of

B ^{\ell} by(b _{1,} \ldots, b _{\ell)} ⋅ (p _{1,} \ldots, p _{\ell)} = (p _{1} b ^{-1} _{1} , b _{1} p _{2} b ^{-1} _{2} , \ldots, b _{\ell-1} p _{\ell} b ^{-1} _{\ell} ).

It is a smooth projective variety. Writing

X _{w} = \overline{BwB} / B = (P _{i _{1}} … P _{i _{\ell}} )/B for the Schubert variety for w, the multiplication map\pi: Z _{\underline{w} } \to X_w

is a resolution of singularities called the Bott–Samelson resolution.

\pi has the property: \pi _{*} l{O} _{Z _{\underline{w}} } = \mathcal_ and R ^{i} \pi _{*} l{O} _{Z _{\underline{w}} } = 0, \, i \ge 1. In other words, X _{w} has rational singularities.

There are also some other constructions; see, for example, .

References

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