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 1959) 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).

See also Bott–Samelson variety.

Notes

1. Gorodski, Claudio; Thorbergsson, Gudlaugur (2001-01-25). "Cycles of Bott-Samelson type for taut representations". arXiv:math/0101209.
2. Brion 2005, Theorem 2.2.3.

References

• R. Bott and H. Samelson, Application of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029.
• Brion, M., Lectures on the geometry of flag varieties, Trends Math., Birkhäuser, Basel, 2005.
• M. Demazure, Désingularisations des variétés de Schubert généralisées, Ann. Sci. Éc. Norm. Supér. 7 (1974), 53–88.
• H. C. Hansen, On cycles in flag manifolds, Math. Scand. 33 (1973), 269–274.
• R. Vakil, A geometric Littlewood-Richardson rule, arXiv:math.AG/0302294. the Annals of Math. (2006)