View source for Bott–Samelson resolution

In [[algebraic geometry]], the '''Bott–Samelson resolution''' of a [[Schubert variety]] is a [[resolution of singularities]]. It was introduced by {{harvtxt|Bott|Samelson|1958}} in the context of [[compact Lie group]]s.{{sfnp|Gorodski|Thorbergsson|2002}} The algebraic formulation is independently due to {{harvtxt|Hansen|1973}} and {{harvtxt|Demazure|1974}}.

== Definition ==
Let ''G'' be a connected [[reductive group|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 a Weyl element|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 variety|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]].<ref>{{harvtxt|Brion|2005|loc=Theorem 2.2.3.}}</ref>

There are also some other constructions; see, for example, {{harvtxt|Vakil|2006}}.

== Notes ==
{{reflist}}

== References ==
*{{citation
 | last1 = Bott | first1 = Raoul | author1-link = Raoul Bott
 | last2 = Samelson | first2 = Hans | author2-link = Hans Samelson
 | doi = 10.2307/2372843
 | journal = American Journal of Mathematics
 | mr = 0105694
 | pages = 964–1029
 | title = Applications of the theory of Morse to symmetric spaces
 | volume = 80
 | year = 1958}}.
*{{citation
 | last = Brion | first = Michel
 | arxiv = math/0410240
 | contribution = Lectures on the geometry of flag varieties
 | doi = 10.1007/3-7643-7342-3_2
 | mr = 2143072
 | pages = 33–85
 | publisher = Birkhäuser, Basel
 | series = Trends Math.
 | title = Topics in cohomological studies of algebraic varieties
 | year = 2005}}.
*{{citation
 | last = Demazure | first = Michel | authorlink = Michel Demazure
 | journal = Annales Scientifiques de l'École Normale Supérieure
 | language = French
 | mr = 0354697
 | pages = 53–88
 | title = Désingularisation des variétés de Schubert généralisées
 | url = http://www.numdam.org/item?id=ASENS_1974_4_7_1_53_0
 | volume = 7
 | year = 1974}}.
*{{citation
 | last1 = Gorodski | first1 = Claudio
 | last2 = Thorbergsson | first2 = Gudlaugur
 | doi = 10.1023/A:1014911422026
 | issue = 3
 | journal = Annals of Global Analysis and Geometry
 | mr = 1896478
 | pages = 287–302
 | title = Cycles of Bott-Samelson type for taut representations
 | arxiv = math/0101209
 | volume = 21
 | year = 2002}}.
*{{citation
 | last = Hansen | first = H. C.
 | doi = 10.7146/math.scand.a-11489
 | journal = Mathematica Scandinavica
 | mr = 0376703
 | pages = 269–274 (1974)
 | title = On cycles in flag manifolds
 | volume = 33
 | year = 1973| doi-access = free
 }}.
*{{citation
 | last = Vakil | first = Ravi | authorlink = Ravi Vakil
 | arxiv = math.AG/0302294
 | doi = 10.4007/annals.2006.164.371
 | issue = 2
 | journal = Annals of Mathematics
 | mr = 2247964
 | pages = 371–421
 | series = Second Series
 | title = A geometric Littlewood-Richardson rule
 | volume = 164
 | year = 2006}}.

{{DEFAULTSORT:Bott-Samelson resolution}}
[[Category:Algebraic geometry]]
[[Category:Singularity theory]]