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

$\underset{\_}{w}=(s_{i_1},s_{i_2},\dots ,s_{i_{\ell }})$

so that $w=s_{i_1}{s}_{i_2}\cdots s_{i_{\ell }}$. (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 $Z_{\underset{\_}{w}}$ be the quotient:

$Z_{\underset{\_}{w}}=P_{i_1}\times \cdots \times P_{i_{\ell }}/B^{\ell }$

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

$(b_1,\dots ,b_{\ell })\cdot (p_1,\dots ,p_{\ell })=(p_1{b}_1^{-1},b_1{p}_2{b}_2^{-1},\dots ,b_{\ell -1}{p}_{\ell }{b}_{\ell }^{-1}).$

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

$\pi :Z_{\underset{\_}{w}}\to X_w$

is a resolution of singularities called the Bott–Samelson resolution. $\pi$ has the property: ${\pi }_{*}{{𝒪}}_{Z_{\underset{\_}{w}}}={{𝒪}}_{X_w}$ and $R^i{\pi }_{*}{{𝒪}}_{Z_{\underset{\_}{w}}}=0,i\ge 1.$ In other words, $X_w$ 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 .