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

[math]\displaystyle{ \underline{w} = (s_{i_1}, s_{i_2}, \ldots, s_{i_\ell}) }[/math]

so that [math]\displaystyle{ w = s_{i_1} s_{i_2} \cdots s_{i_\ell} }[/math]. (ℓ is the length of w.) Let [math]\displaystyle{ P_{i_j} \subset G }[/math] be the subgroup generated by B and a representative of [math]\displaystyle{ s_{i_j} }[/math]. Let [math]\displaystyle{ Z_{\underline{w}} }[/math] be the quotient:

[math]\displaystyle{ Z_{\underline{w}} = P_{i_1} \times \cdots \times P_{i_\ell}/B^\ell }[/math]

with respect to the action of [math]\displaystyle{ B^\ell }[/math] by

[math]\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}). }[/math]

It is a smooth projective variety. Writing [math]\displaystyle{ X_w = \overline{BwB} / B = (P_{i_1} \cdots P_{i_\ell})/B }[/math] for the Schubert variety for w, the multiplication map

[math]\displaystyle{ \pi: Z_{\underline{w}} \to X_w }[/math]

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

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

References

• "Applications of the theory of Morse to symmetric spaces", American Journal of Mathematics 80: 964–1029, 1958, doi:10.2307/2372843 .
• Brion, Michel (2005), "Lectures on the geometry of flag varieties", Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, pp. 33–85, doi:10.1007/3-7643-7342-3_2 .
• "Désingularisation des variétés de Schubert généralisées" (in French), Annales Scientifiques de l'École Normale Supérieure 7: 53–88, 1974, http://www.numdam.org/item?id=ASENS_1974_4_7_1_53_0 .
• Gorodski, Claudio; Thorbergsson, Gudlaugur (2002), "Cycles of Bott-Samelson type for taut representations", Annals of Global Analysis and Geometry 21 (3): 287–302, doi:10.1023/A:1014911422026 .
• Hansen, H. C. (1973), "On cycles in flag manifolds", Mathematica Scandinavica 33: 269–274 (1974), doi:10.7146/math.scand.a-11489 .
• "A geometric Littlewood-Richardson rule", Annals of Mathematics, Second Series 164 (2): 371–421, 2006, doi:10.4007/annals.2006.164.371 .