Английская Википедия:Bott–Samelson resolution

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

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

Definition

Let G be a connected reductive complex algebraic group, B a Borel subgroup and T a maximal torus contained in B.

Let <math>w \in W = N_G(T)/T.</math> Any such w can be written as a product of reflections by simple roots. Fix minimal such an expression:

<math>\underline{w} = (s_{i_1}, s_{i_2}, \ldots, s_{i_\ell})</math>

so that <math>w = s_{i_1} s_{i_2} \cdots s_{i_\ell}</math>. (ℓ is the length of w.) Let <math>P_{i_j} \subset G</math> be the subgroup generated by B and a representative of <math>s_{i_j}</math>. Let <math>Z_{\underline{w}}</math> be the quotient:

<math>Z_{\underline{w}} = P_{i_1} \times \cdots \times P_{i_\ell}/B^\ell</math>

with respect to the action of <math>B^\ell</math> by

<math>(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}).</math>

It is a smooth projective variety. Writing <math>X_w = \overline{BwB} / B = (P_{i_1} \cdots P_{i_\ell})/B</math> for the Schubert variety for w, the multiplication map

<math>\pi: Z_{\underline{w}} \to X_w</math>

is a resolution of singularities called the Bott–Samelson resolution. <math>\pi</math> has the property: <math>\pi_* \mathcal{O}_{Z_{\underline{w}}} = \mathcal{O}_{X_w}</math> and <math>R^i \pi_* \mathcal{O}_{Z_{\underline{w}}} = 0, \, i \ge 1.</math> In other words, <math>X_w</math> has rational singularities.

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

References

• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Citation.