Birational Geometry of symplectic resolutions of nilpotent orbits
Birational Geometry of symplectic resolutions of nilpotent orbits
复制标题
DOI：
10.2969/aspm/04510075
复制
发表时间：
2004-04
期刊：
影响因子：
--
通讯作者：
Y. Namikawa
中科院分区：
文献类型：
--
作者：
Y. Namikawa
本文证明了经典单李代数中幂零轨道闭包的任意两个（射影）辛分解都由A型或D型Mukai触发器的局部平凡族图的有限序列连接。A型Mukai触发器与我们以前的论文“Mukai触发器和派生的第二类”中的分层Mukai触发器相同。另一方面，D型向井触发器来自so（2k）中的幂零轨道。设G^+和G^-是正交Grassmannian G_{iso}（k，2k）的两个连通分支.那么这个轨道的闭包通过余切线丛T^*G^+和T^*G^-有两个Springer归结，这就是D型Mukai翻转。我们的结果澄清了Spaleau和Hesselink结果的几何意义。为了说明这一点，将给出三个例子。本文的另一个目的是在幂零轨道的情况下，对“奇异分解与辛奇点的唯一性”一文中的一个猜想给出一个肯定的回答。一个关键的想法是由Borho和Kraft研究的“Dixlobal sheet”的概念。
In this paper, we shall prove that any two (projective) symplectic resolutions of a nilpotent orbit closure in a classical simple Lie algebra are connected by a finite sequence of diagrams which are locally trivial families of Mukai flops of type A or of type D. A Mukai flop of type A is the same as a stratified Mukai flop in our previous paper "Mukai flops and derived categories II". On the other hand, a Mukai flop of type D comes from a nilpotent orbit in so(2k). Let G^+ and G^- be two connected components of the orthogonal Grassmannian G_{iso}(k,2k). Then the closure of this orbit has two Springer resolutions via the cotanegent bundles T^*G^+ and T^*G^-, which is the Mukai flop of type D. Our result would clarify the geometric meaning of the results of Spaltenstein and Hesselink. To illustrate this, three examples will be given. Another purpose of this paper is to give an affimative answer to a conjecture in the paper "Uniqueness of crepant resolutions and symplectic singularities" math.AG/0306091, for nilpotent orbit cases. A key idea is the notion of a "Dixmier sheet" studied by Borho and Kraft.