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. 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:

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

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 $Z_{\underline {w}}$ be the quotient:

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

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

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

It is a smooth projective variety. Writing $X_{w}={\overline {BwB}}/B=(P_{i_{1}}\cdots P_{i_{\ell }})/B$ 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.

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

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.