Journal of Differential Geometry

Real submanifolds of maximum complex tangent space at a CR singular point, II

Xianghong Gong and Laurent Stolovitch

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We study germs of real analytic $n$-dimensional submanifold of $\mathbf{C}^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions, we first classify holomorphically the quadrics having this property. We then study higher order perturbations of these quadrics and their transformations to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We are led to study formal Poincaré–Dulac normal forms (non-unique) of reversible biholomorphisms. We exhibit a reversible map of which the normal forms are all divergent at the singularity. We then construct a unique formal normal form of the submanifolds under a non degeneracy condition.


The research of L. Stolovitch was supported by ANR-FWF grant “ANR-14-CE34-0002-01” for the project “Dynamics and CR geometry”, and by ANR grant “ANR-15-CE40-0001-03” for the project “BEKAM”.

Article information

J. Differential Geom., Volume 112, Number 1 (2019), 121-198.

Received: 15 June 2015
First available in Project Euclid: 8 May 2019

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Zentralblatt MATH identifier

local analytic geometry CR singularity integrability reversible mapping small divisors divergent normal forms


Gong, Xianghong; Stolovitch, Laurent. Real submanifolds of maximum complex tangent space at a CR singular point, II. J. Differential Geom. 112 (2019), no. 1, 121--198. doi:10.4310/jdg/1557281008.

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