权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

On the construction of the Seiberg-Witten equation over CR

CR上Seiberg-Witten方程的构造

基本信息

批准号：
12640219
负责人：
AKAHORI Akao
金额：
$ 0.77万
依托单位：
Himejj Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640219/
关键词：
isolated singularity CR structure 有理特異点

项目摘要

Let (V, o) be an isolated singularity with complex dimension n, in a complex euclidean space (C^N/,o). Let M be the intersection of this V and the real hyperspere S^2N-1_E(o), centered at the origin o with radius ε. Then, over M, a CR structure is induced from V, and this CR structure determines the isolated singularity V(Rossi's theorem). So, with this in mind and with Kuranishi equivalence for CR structures, the deformation theory of CR structures is established and the versal family of CR structures is constructed. Related to mathematical physics, it is found that: the Seiberg-Witten invariant is quite useful in studying the geometry of the moduli space (for example, CalabI Yau manifolds).Therefore it seems natural to try to obtain a similar result in isolated singularities. Let {(M,^<φ(t)>T''),t ∈ M} be the versal family of CR structures of (M,^<o>T''), constructed our former paper. In the construction of the versal family, we have to handle a second order diferential operator(so, the corresponding Laplace operator must be a 4th-order differential operator). For scalar valued differential forms, this phenomenon occurs in the middle dimension degree (in our case, n and n - 1). In fact, the harmonic space of differential forms of the middle dimension degree is determined by fourth order partial differential equations, and its solution space has a particular subspace, which is determined by second order partial differential equations.While in algebraic geometry, for Al singularities and their moduli spaces, flat coordinates are found by K. Saito. My first motivation is that: there might be a relation with Saito flat coordinate and the above 4-th order partial differential equations. This relation is studied in my recent research, in Al singularities and some Hilzebruch-Jung singularities.

令 (V, o) 为复欧几里得空间 (C^N/,o) 中具有复维度 n 的孤立奇点。令 M 为该 V 与真实超球面 S^2N-1_E(o) 的交集，以原点 o 为中心，半径为 ε。然后，在M上，由V导出一个CR结构，这个CR结构决定了孤立奇点V（罗西定理）。因此，考虑到这一点并利用 CR 结构的 Kuranishi 等价，建立了 CR 结构的变形理论并构建了 CR 结构的通用族。与数学物理相关，发现：Seiberg-Witten 不变量在研究模空间的几何形状（例如 CalabI Yau 流形）时非常有用。因此，尝试在孤立奇点中获得类似的结果似乎是很自然的。令 {(M,^<φ(t)>T''),t ∈ M} 为 (M,^<o>T'') 的 CR 结构的通用族，构建了我们之前的论文。在构造Versal族时，我们必须处理一个二阶微分算子（因此，相应的拉普拉斯算子必须是一个四阶微分算子）。对于标量值微分形式，这种现象发生在中间维度（在我们的例子中，n 和 n - 1）。事实上，中维次微分形式的调和空间是由四阶偏微分方程确定的，其解空间有一个特定的子空间，该子空间是由二阶偏微分方程确定的。而在代数几何中，对于Al奇点及其模空间，K. Saito找到了平面坐标。我的第一个动机是：斋藤平坐标和上面的四阶偏微分方程可能有关系。我最近在 Al 奇点和一些 Hilzebruch-Jung 奇点的研究中研究了这种关系。