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摘要翻译：
设G是作用于仿射簇V上的（实或复）线性约化代数群，设W是子簇。本文研究了G-轨道如何与W相交，并给出了一个判别准则，用以判定该交点何时可以描述为一个约化子群的轨道的有限并。该判据的条件在实践中易于验证，并用于构造不允许左不变Ricci孤子度量的（非同构）幂零李群的连续族。给出了李群左不变几何的其它应用。注释通过将我们的技术应用于伴随表示来结束。证明了幂零轨道有限的经典结果，并证明了每一个幂零轨道都含有矩映射范数平方的临界点。
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英文标题：
《Detecting orbits along subvarieties via the moment map》
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作者：
Michael Jablonski
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Differential Geometry 微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the criterion are easily verified in practice and are used to construct continuous families of (non-isomorphic) nilpotent Lie groups which do not admit left-invariant Ricci soliton metrics. Other applications to the left-invariant geometry of Lie groups are also given. The note finishes by applying our techniques to the adjoint representation. The classical result of finiteness of nilpotent orbits is reproven and it is shown that each of these orbits contains a critical point of the norm squared of the moment map.
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PDF链接：
https://arxiv.org/pdf/0810.5697
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