On the Geometry of the Anti-canonical Bundle of the Bott–Samelson–Demazure–Hansen Varieties  

作　　者：Indranil BISWAS S.Senthamarai KANNAN Pinakinath SAHA 

机构地区：[1]School of Mathematics,Tata Institute of Fundamental Research,1 Homi Bhabha Road,Mumbai 400005,India [2]Chennai Mathematical Institute,Plot H1,SIPCOT IT Park,Siruseri,Kelambakkam,603103,India

出　　处：《Acta Mathematica Sinica,English Series》2024年第8期1920-1940,共21页数学学报（英文版）

基　　金：partially supported by a J.C.Bose Fellowship(Grant No.JBR/2023/000003);The second author would like to thank the Infosys Foundation for the partial financial support。

摘　　要：Let G be a semi-simple simply connected algebraic group over the field C of complex numbers.Let T be a maximal torus of G,and let W be the Weyl group of G with respect to T.Let Z(w,i)be the Bott–Samelson–Demazure–Hansen variety corresponding to a tuple i associated to a reduced expression of an element w∈W.We prove that for the tuple i associated to any reduced expression of a minuscule Weyl group element w,the anti-canonical line bundle on Z(w,i)is globally generated.As consequence,we prove that Z(w,i)is weak Fano.Assume that G is a simple algebraic group whose type is different from A2.Let S={α1,...,αn}be the set of simple roots.Let w be such that support of w is equal to S.We prove that Z(w,i)is Fano for the tuple i associated to any reduced expression of w if and only if w is a Coxeter element and w^(−1)(Σ_(t=1)^(n)α_(t))∈−S.

关 键 词：Bott-Samelson-Demazure-Hansen variety Coxeter element anti-canonical line bundle Fano weak Fano 

分 类 号：O152[理学—数学]