Singular Levi-flat real analytic hypersurfaces.

*(English)*Zbl 0931.32009Let \(M\) 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.

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.

Reviewer: Tohsuke Urabe (Ibaraki)