On the construction of the Seiberg-Witten equation over CR

设（V，o）是复欧氏空间（C^N/，o）中的一个复维数为n的孤立奇点。设M是这个V与真实的超空间S^2N-1_E（o）的交点，中心在原点o，半径为ε。然后，在M上，从V导出CR结构，并且该CR结构确定孤立奇点V（罗西定理）。因此，考虑到这一点，并与Kuranishi等效的CR结构，CR结构的变形理论，CR结构的变形族的构造。与数学物理相关的是，Seiberg-Witten不变量在模空间（如Calab-Yau流形）的几何研究中非常有用，因此在孤立奇点中得到类似的结果似乎是很自然的。设{（M，^&lt;φ（t）&gt; T“），t ∈ M}是（M，^T”）的CR结构的集合族<o>，这是我们在前一篇文章中所构造的.在构造这类算子族时，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）。实际上，中维阶微分形式的调和空间是由四阶偏微分方程确定的，其解空间有一个特殊的子空间，这个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.齐藤我的第一个动机是：这可能与Saito平面坐标和上述四阶偏微分方程有关系。在我最近的研究中，在Al奇点和一些Hilzebruch-Jung奇点中研究了这种关系。

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.