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 $R^i \pi_* \mathcal{O}_{Z_{\underline{w}}} = 0, \, i \ge 1.$ 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)