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 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:

\underline {w}=(s_{{i_{1}}},s_{{i_{2}}},\ldots ,s_{{i_{l}}})

so that $w=s_{{i_{1}}}s_{{i_{2}}}\cdots s_{{i_{l}}}$ . (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 $Z_{{\underline {w}}}$ be the quotient:

Z_{{\underline {w}}}=P_{{i_{1}}}\times \cdots \times P_{{i_{l}}}/B^{l}

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

(b_{1},\ldots ,b_{l})\cdot (p_{1},\ldots ,p_{l})=(p_{1}b_{1}^{{-1}},b_{1}p_{2}b_{2}^{{-1}},\ldots ,b_{{l-1}}p_{l}b_{l}^{{-1}})

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

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

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

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

Notes

↑ Gorodski & Thorbergsson (2002).
↑ 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, MR 0105694, 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, MR 2143072, arXiv:math/0410240 , doi:10.1007/3-7643-7342-3_2 .
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, MR 1896478, doi:10.1023/A:1014911422026 .
Hansen, H. C. (1973), "On cycles in flag manifolds", Mathematica Scandinavica, 33: 269–274 (1974), MR 0376703, doi:10.7146/math.scand.a-11489 .
Vakil, Ravi (2006), "A geometric Littlewood-Richardson rule", Annals of Mathematics, Second Series, 164 (2): 371–421, MR 2247964, arXiv:math.AG/0302294 , doi:10.4007/annals.2006.164.371 .

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