Bott–Samelson resolution

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.^[1] 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 ${\displaystyle w\in W=N_{G}(T)/T.}$ Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:

${\displaystyle {\underline {w}}=(s_{i_{1}},s_{i_{2}},\ldots ,s_{i_{\ell }})}$

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

${\displaystyle Z_{\underline {w}}=P_{i_{1}}\times \cdots \times P_{i_{\ell }}/B^{\ell }}$

with respect to the action of ${\displaystyle B^{\ell }}$ by

${\displaystyle (b_{1},\ldots ,b_{\ell })\cdot (p_{1},\ldots ,p_{\ell })=(p_{1}b_{1}^{-1},b_{1}p_{2}b_{2}^{-1},\ldots ,b_{\ell -1}p_{\ell }b_{\ell }^{-1}).}$

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

${\displaystyle \pi :Z_{\underline {w}}\to X_{w}}$

is a resolution of singularities called the Bott–Samelson resolution. ${\displaystyle \pi }$ has the property: ${\displaystyle \pi _{*}{\mathcal {O}}_{Z_{\underline {w}}}={\mathcal {O}}_{X_{w}}}$ and ${\displaystyle R^{i}\pi _{*}{\mathcal {O}}_{Z_{\underline {w}}}=0,\,i\geq 1.}$ In other words, ${\displaystyle X_{w}}$ has rational singularities.^[2]

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

Notes

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

References

• Bott, Raoul; Samelson, Hans (1958), "Applications of the theory of Morse to symmetric spaces", American Journal of Mathematics, 80: 964–1029, doi:10.2307/2372843, MR 0105694.
• Brion, Michel (2005), "Lectures on the geometry of flag varieties", Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, pp. 33–85, arXiv:math/0410240, doi:10.1007/3-7643-7342-3_2, MR 2143072.
• Demazure, Michel (1974), "Désingularisation des variétés de Schubert généralisées", Annales Scientifiques de l'École Normale Supérieure (in French), 7: 53–88, MR 0354697.
• Gorodski, Claudio; Thorbergsson, Gudlaugur (2002), "Cycles of Bott-Samelson type for taut representations", Annals of Global Analysis and Geometry, 21 (3): 287–302, arXiv:math/0101209, doi:10.1023/A:1014911422026, MR 1896478.
• Hansen, H. C. (1973), "On cycles in flag manifolds", Mathematica Scandinavica, 33: 269–274 (1974), doi:10.7146/math.scand.a-11489, MR 0376703.
• Vakil, Ravi (2006), "A geometric Littlewood-Richardson rule", Annals of Mathematics, Second Series, 164 (2): 371–421, arXiv:math.AG/0302294, doi:10.4007/annals.2006.164.371, MR 2247964.