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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
##### Keywords:
Levi-flat hypersurfaces; holomorphic foliations
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##### References:
 [1] V.I. Arnold. Normal Form of functions in the neighbourhood of degenerate critical points. UNM, 29(2) (1974), 11–49; RMS, 29(2) (1974), 19–48. [2] V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko. Singularities of Differential Maps. Vol. I, Monographs in Math., vol. 82, Birkhäuser (1985). [3] D. Burns and X. Gong. Singular Levi-flat real analytic hypersurfaces. Amer. J. Math., 121 (1999), 23–53. · Zbl 0931.32009 [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 [5] A. Fernández-Pérez. Singular Levi-flat hypersurfaces. An approach through holomorphic foliations. Ph.D. Thesis IMPA-Brazil (2010). [6] F. Loray. Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. Avaliable in http://hal.archives-ouvertures.fr/ccsd-00016434 [7] J.F. Mattei and R. Moussu. Holonomie et intégrales premières. Ann. Ec. Norm. Sup., 13 (1980), 469–523. · Zbl 0458.32005
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