On the construction of the Seiberg-Witten equation over CR

设(V，o)是复欧氏空间(C^N/，o)中的n维孤立奇点。设M是这个V和实超S^2N-1_E(O)的交，中心在半径为ε的原点o。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(Rossi定理)。在此基础上，结合CR结构的Kuranishi等效性，建立了CR结构的变形理论，构造了CR结构的逆族。结合数学物理，我们发现：Seiberg-Witten不变量在研究模空间(例如，Calabi-Yau流形)的几何中是非常有用的。因此，在孤立奇点中尝试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，^&lt；o&gt；T’‘)的CR结构族。在Versal族的构造中，我们必须处理二阶微分算子(因此，对应的拉普拉斯算子一定是四阶微分算子)。对于标量值微分形式，这种现象发生在中维度(在我们的例子中，n和n-1)。实际上，中维微分形式的调和空间是由四阶偏微分方程组决定的，它的解空间有一个特定的子空间，这个子空间由二阶偏微分方程组决定.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：可能与齐藤平面坐标和上面的四阶偏微分方程组有关。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.