Bott–Samelson resolution - Alchetron, the free social encyclopedia

In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by Bott & Samelson (1958) in the context of compact Lie groups. The algebraic formulation is due to Hansen (1973) and Demazure (1974).

Definition

Let G be a connected reductive complex algebraic group, B a Borel subgroup and T a maximal torus contained in B.

Let w ∈ W = N G ( T ) / T . Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:

w _ = ( s i 1 , s i 2 , … , s i l )

so that w = s i 1 s i 2 ⋯ s i l . (l is the length of w.) Let P i j ⊂ G be the subgroup generated by B and a representative of s i j . Let Z w _ be the quotient:

Z w _ = P i 1 × ⋯ × P i l / B l

with respect to the action of B l by

( b 1 , … , b l ) ⋅ ( p 1 , … , p l ) = ( p 1 b 1 − 1 , b 1 p 2 b 2 − 1 , … , b l − 1 p l b l − 1 ) .

It is a smooth projective variety. Writing X w = B w B ¯ / B = ( P i 1 ⋯ P i l ) / B for the Schubert variety for w, the multiplication map

π : Z w _ → X w

is a resolution of singularities called the Bott–Samelson resolution. π has the property: π ∗ O Z w _ = O X w and R i π ∗ O Z w _ = 0 , i ≥ 1. In other words, X w has rational singularities.

There are also some other constructions; see, for example, Vakil (2006).

References

Bott–Samelson resolution Wikipedia

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