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Singular Levi-flat real analytic hypersurfaces. (English) Zbl 0931.32009
Let $$M$$ be a real analytic hypersurface in $$\mathbb{C}^n$$ defined by a real analytic function $$r$$. First, we assume that $$M$$ is smooth. Then, $$M$$ is called Levi-flat, if it is foliated by smooth holomorphic hypersurfaces in $$\mathbb{C}^n$$. More precisely, for a point $$p\in M$$ with $$dr(p)\neq 0$$, $$M$$ is Levi-flat near $$p$$ if and only if the rank of the Levi-matrix $$\left(\begin{smallmatrix} 0 & r_{\overline z}\\ r_z & r_{z\overline z}\end{smallmatrix}\right)$$ is equal to 2.
In case where $$M$$ has a singularity, we define the concept of Levi-flat in the following manner: Let $$M^*$$ denote the smooth locus of $$M$$, i.e., the set of points $$p\in M$$ such that near $$p$$ $$M$$ is real analytically isomorphic to $$\mathbb{R}^{2n-1}$$. If $$M^*$$ is Levi-flat, then we say that $$M$$ is Levi-flat. Singular Levi-flat real analytic sets occur as invariant sets of integrable holomorphic Hamiltonian systems.
This article establishes basic theory of singular Levi-flat real analytic hypersurfaces at double points. There are two main results. Let $$z_1,z_2,\dots,z_n$$ denote the complex coordinates on $$\mathbb{C}^n$$. Let $$M$$ be a Levi-flat real analytic hypersurface defined by a function in the form $$q(z)+H(z)$$, where $$q(z)$$ is a real valued quadratic form in $$\text{Re}(z_1), \text{Re}(z_2),\dots \text{Re}(z_n)$$, $$\text{Im}(z_1), \text{Im}(z_2),\dots,\text{Im}(z_n)$$, and $$H(z)$$ is a real analytic function with $$H(z) = O(|z|^3)$$.
Theorem 1.1: Assume that $$q(z) = \text{Re}(z^2_1 + z^2_2+\cdots+ z^2_n)$$. Then, near 0 $$M$$ is biholomorphically equivalent to the cone defined by $$\text{Re}(z^2_1 + z^2_2 +\cdots + z^2_n)$$.
Theorem 1.2: Assume that $$q(z)$$ is positive definite on a complex line and its Levi-matrix has rank at least 2 at 0. Then, near 0 $$M$$ is biholomorphically equivalent to the cone defined by $$z_1\overline z_1-z_2\overline z_2$$.
Besides the above two results, in the article we can find a lot of basic results on the subject.

##### MSC:
 32V40 Real submanifolds in complex manifolds 32S25 Complex surface and hypersurface singularities
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