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 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 \ge 1. In other words, X_w has rational singularities.[2]

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

Notes

  1. ^ Gorodski & Thorbergsson (2002).
  2. ^ Brion (2005, Theorem 2.2.3.)

References