Geometric invariant theory for graded unipotent groups and applications GEOMETRIC INVARIANT THEORY FOR GRADED UNIPOTENT GROUPS
Geometric invariant theory for graded unipotent groups and applications GEOMETRIC INVARIANT THEORY FOR GRADED UNIPOTENT GROUPS
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分级单能群的几何不变理论及其应用 分级单能群的几何不变理论
DOI：
10.1112/topo.12075
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发表时间：
2018
影响因子：
1.1
通讯作者：
Bérczi G
中科院分区：
文献类型：
--
作者：
Bérczi G
在复数上设一个渐变的单幂群，在这个意义上，它有一个被乘法群所扩展，使得乘法群通过共轭作用在李代数上的所有权值都是严格正的。给定任何行动巴的射影varietyextending行动处为线性对一个充足的线包上,然后提供一个愿意将线包替换为一个张量力量和扭曲操作的线性化不乘一个合适的(理性的)字符,并提供一个额外的条件是满足条件的模拟在经典几何不变量理论(GIT)应该没有严格semistable点行动,我们证明了‐不变量形成一个有限生成的分级代数；的半稳定子集到包络商的自然模射是满射的，包络商表示为半稳定子集的几何商。应用这个结果与投影线的乘积，我们得到了一个投影变分，它是与仿射线的乘积的一个不变开子集的几何商，并且作为一个开子集包含了与的作用的一个不变开子集的几何商。此外，这些开放子集及其与仿射线的乘积可以用类似经典GIT中的Hilbert-Mumford准则来描述。
Letbe a graded unipotent group over the complex numbers, in the sense that it has an extensionby the multiplicative group such that the action of the multiplicative group by conjugation on the Lie algebra ofhas all its weights strictly positive. Given any action ofon a projective varietyextending to an action ofwhich is linear with respect to an ample line bundle on, then provided that one is willing to replace the line bundle with a tensor power and to twist the linearisation of the action ofby a suitable (rational) character, and provided an additional condition is satisfied which is the analogue of the condition in classical geometric invariant theory (GIT) that there should be no strictly semistable points for the action, we show that the‐invariants form a finitely generated graded algebra; moreover, the natural morphism from the semistable subset ofto the enveloping quotient is surjective and expresses the enveloping quotient as a geometric quotient of the semistable subset. Applying this result withreplaced by its product with the projective line gives us a projective variety which is a geometric quotient byof an invariant open subset of the product ofwith the affine line and contains as an open subset a geometric quotient of a‐invariant open subset ofby the action of. Furthermore, these open subsets ofand its product with the affine line can be described using criteria similar to the Hilbert–Mumford criteria in classical GIT.