幂零轨道闭包上的Legendrian直线族
Legendrian families of lines on nilpotent orbit closures
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中文总结 AI 辅助
该研究针对复半单李代数的幂零轨道闭包，分析过一般点的直线切方向空间的不可约分量性质，在两类情形下研究其Legendrian分量，并给出相关推论。
中文摘要 AI 辅助
设𝔤为复半单李代数，$\bar{Z}$为射影空间$\boldsymbol{P}(\boldsymbol{g})$中的一个幂零轨道闭包。我们研究经过$\bar{Z}$上一般点$z$的直线的切方向空间，记为$F(\bar{Z}, z)$。我们首先证明$F(\bar{Z}, z)$的每个不可约分量都是射影切空间中接触超平面的积分子簇。接下来，我们在两种情形下研究$F(\bar{Z}, z)$的Legendrian分量：$\bar{Z}$容许Springer消解的情形；$\bar{Z}$是经典型射影单李代数中的幂零元平方为零的情形。作为应用，我们给出两个推论：根据$F(\bar{Z}, z)$刻画幂零元平方为零的轨道中的Richardson轨道；描述与不可约埃尔米特对称空间相关的分层Mukai flops所对应的$\bar{Z}$的$F(\bar{Z}, z)$。
英文摘要
Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and $\overline{Z}$ a nilpotent orbit closure in the projectivization $\mathbb{P}(\mathfrak{g})$. We investigate the space of tangent directions of lines on $\overline{Z}$ passing through a general point $z$, denoted by $F(\overline{Z},\,z)$. We first prove that every irreducible component of $F(\overline{Z},\,z)$ is an integral subvariety of the contact hyperplane in the projectivized tangent space. Next, we study Legendrian components of $F(\overline{Z},\,z)$ in two cases: the case where $\overline{Z}$ admits a Springer resolution; the case where $\overline{Z}$ is square-zero in a projectivized simple Lie algebra of classical type. As an application, two corollaries are presented: a characterization of Richardson orbits among square-zero orbits in terms of $F(\overline{Z},\,z)$; a description of $F(\overline{Z},\,z)$ for $\overline{Z}$ arising from stratified Mukai flops associated to irreducible Hermitian symmetric spaces.
发表机构
- Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）
- Beijing Postdoctoral Research Foundation（北京博士后科研基金会）
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