Chow's theorem for real analytic Levi-flat hypersurfaces

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space \(\mathbb{P}^{n}\), \(n \geq 2\). More specifically, we prove that a real analytic Levi-flat hypersurface \(M \subset \mathbb{P}^{n}\), with singular set of real dime...

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Bibliographic Details
Published in:arXiv.org 2021-12
Main Authors: Fernández-Pérez, Arturo, Mol, Rogério, Rosas, Rudy
Format: Article
Language:English
Subjects:
Algebra
Hyperspaces
Mathematical analysis
Rational functions
Theorems
Online Access:Get full text
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Summary:In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space \(\mathbb{P}^{n}\), \(n \geq 2\). More specifically, we prove that a real analytic Levi-flat hypersurface \(M \subset \mathbb{P}^{n}\), with singular set of real dimension at most \(2n-4\) and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in \(\mathbb{P}^{n}\). As a consequence, \(M\) is a semialgebraic set. We also prove that a Levi foliation on \(\mathbb{P}^{n}\) - a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one - satisfying similar conditions - singular set of real dimension at most \(2n-4\) and all leaves algebraic - is defined by the level sets of a rational function.
ISSN:2331-8422