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On normal forms of singular Levi-flat real analytic hypersurfaces. (English) Zbl 1215.32016
A classical theorem due to E. Cartan states that a real analytic smooth Levi-flat hypersurface \(M\) in \(\mathbb{C}^{n}\) is locally biholomorphic to a hypersurface of the form \(\{\mathcal{R}e(z_{1})=0\}\). D. Burns and X. Gong [Am. J. Math. 121, No. 1, 23–53 (1999; Zbl 0931.32009)] proved that, if \(M=F^{-1}(0)\) is Levi-flat, where \(F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)\), \(n\geq 2\), is a germ of real analytic function such that \[ F(z_{1},\dots,z_{n})=\mathcal{R}e(z_{1}^{2}+\dots+z_{n}^{2})+h.o.t, \]
then \(M\) is locally biholomorphic to a hypersurface of the form \(\{\mathcal{R}e(z_{1}^{2}+\dots+z_{n}^{2})=0\}\).
In the paper under review, the author exhibits similar normal forms in a situation more general. More precisely, the author proves that, if
\[ F(z)=\mathcal{R}e(P(z)) + h.o.t, \]
such that \(M=F^{-1}(0)\), where \(F: (\mathbb C^n,0)\to(\mathbb R,0)\) is a germ of real analytic functions, is Levi-flat at \(0\in\mathbb{C}^{n}\), \(n\geq{2}\), where \(P(z)\) is a homogeneous polynomial of degree \(k\) with an isolated singularity at \(0\in\mathbb{C}^{n}\) and Milnor number \(\mu\), then there exists a holomorphic change of coordinates \(\phi\) such that \(\phi(M)=\{\mathcal{R}e(h)=0\}\), where \(h(z)\) is a polynomial of degree \(\mu+1\) and \(j^{k}_{0}(h)=P\).
The idea of the proof is to study the singular set of the complexification of the Levi \(1\)-form and to apply a result of D. Cerveau and A. Lins Neto [Am. J. Math. 133, No. 3, 677–716 (2011; Zbl 1225.32038)]. The result follows from a generalization of the Morse lemma.

MSC:
32V40 Real submanifolds in complex manifolds
37F75 Dynamical aspects of holomorphic foliations and vector fields
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References:
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[4] D. Cerveau and A. Lins Neto. Local Levi-Flat hypersurfaces invariants by a codimension one holomorphic foliation. To appear in Amer. J. Math. · Zbl 1225.32038
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