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) 中复维度为 n 的孤立奇点，令 M 为该 V 与实超球面 S^2N-1_E(o) 的交集，以原点 o 为中心，半径为 ε 。然后，在 M 上，从 V 导出一个 CR 结构，并且该 CR 结构决定了孤立奇点 V（罗西定理）。建立了CR结构的等价性，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方面非常有用（。例如，CalabI Yau 流形）。因此，尝试在孤立奇点中获得类似的结果似乎很自然。设 {(M,^<φ(t)>T''),t ∈ M} 为通用族。 (M,^<o>T'') 的 CR 结构，构建了我们之前的论文在构造通用族时，我们必须处理二阶微分算子（因此，相应的拉普拉斯算子必须是四阶算子。对于标量值微分形式，这种现象发生在中维次（在我们的例子中，n和n-1）实际上，中维次微分形式的调和空间是由四阶偏微分方程确定的，并且其解空间有一个特定的子空间，它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，平面坐标是由K. Saito找到的。我的第一个动机是：可能与Saito有关系。我最近在 Al 奇点和一些 Hilzebruch-Jung 奇点的研究中研究了平坐标和上述 4 阶偏微分方程。