Misplaced Pages

Bott–Samelson resolution

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Bott–Samelson varieties)

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 independently 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 . {\displaystyle 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:

w _ = ( s i 1 , s i 2 , … , s i ℓ ) {\displaystyle {\underline {w}}=(s_{i_{1}},s_{i_{2}},\ldots ,s_{i_{\ell }})}

so that w = s i 1 s i 2 ⋯ s i ℓ {\displaystyle w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell }}} . (ℓ is the length of w.) Let P i j ⊂ G {\displaystyle P_{i_{j}}\subset G} be the subgroup generated by B and a representative of s i j {\displaystyle s_{i_{j}}} . Let Z w _ {\displaystyle Z_{\underline {w}}} be the quotient:

Z w _ = P i 1 × ⋯ × P i ℓ / B ℓ {\displaystyle Z_{\underline {w}}=P_{i_{1}}\times \cdots \times P_{i_{\ell }}/B^{\ell }}

with respect to the action of B ℓ {\displaystyle B^{\ell }} 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 ) . {\displaystyle (b_{1},\ldots ,b_{\ell })\cdot (p_{1},\ldots ,p_{\ell })=(p_{1}b_{1}^{-1},b_{1}p_{2}b_{2}^{-1},\ldots ,b_{\ell -1}p_{\ell }b_{\ell }^{-1}).}

It is a smooth projective variety. Writing X w = B w B ¯ / B = ( P i 1 ⋯ P i ℓ ) / B {\displaystyle X_{w}={\overline {BwB}}/B=(P_{i_{1}}\cdots P_{i_{\ell }})/B} for the Schubert variety for w, the multiplication map

π : Z w _ → X w {\displaystyle \pi :Z_{\underline {w}}\to X_{w}}

is a resolution of singularities called the Bott–Samelson resolution. π {\displaystyle \pi } has the property: π ∗ O Z w _ = O X w {\displaystyle \pi _{*}{\mathcal {O}}_{Z_{\underline {w}}}={\mathcal {O}}_{X_{w}}} and R i π ∗ O Z w _ = 0 , i ≥ 1. {\displaystyle R^{i}\pi _{*}{\mathcal {O}}_{Z_{\underline {w}}}=0,\,i\geq 1.} In other words, X w {\displaystyle X_{w}} has rational singularities.

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

Notes

  1. Gorodski & Thorbergsson (2002).
  2. Brion (2005, Theorem 2.2.3.)

References

Categories: