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

By Xianghong Gong and Laurent Stolovitch

Abstract

126 pagesWe study a germ of real analytic $n$-dimensional submanifold of ${\mathbf C}^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck transformations, we study its transformation to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We first conjugate formally its associated reversible map $\sigma$ to suitable normal forms and show that all these normal forms can be divergent. If the singularity is {\it abelian}, we show, under some assumptions on the linear part of $\sigma$ at the singularity, that the real submanifold is holomorphically equivalent to an analytic normal form. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we prove that, in general, there exists a complex submanifold of positive dimension in ${\mathbf C}^n$ that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a CR singularity of the {\it complex type}

Topics: Local analytic geometry, CR singularity, normal form, integrability, reversible mapping, linearization, small divisors, hull of holomorphy, 32V40, 37F50, 32S05, 37G05, [MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]
Publisher: HAL CCSD
Year: 2014
OAI identifier: oai:HAL:hal-01001831v1
Provided by: HAL-UNICE

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