Bott–Samelson resolution

In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is independently due to and.

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 $$R^i \pi_* \mathcal{O}_{Z_{\underline{w}}} = 0, \, i \ge 1.$$ In other words, $$X_w$$ has rational singularities.

There are also some other constructions; see, for example,.