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

Xianghong Gong (1) , Laurent Stolovitch (2)

1 Department of Mathematics [Madison]
2 LJAD - Laboratoire Jean Alexandre Dieudonné

We 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}.

Mots clés: Local analytic geometry; CR singularity; normal form; integrability; reversible mapping; linearization; small divisors; hull of holomorphy

Domaines: Variables complexes [math.CV]; Systèmes dynamiques [math.DS]

Xianghong Gong, Laurent Stolovitch. Real submanifolds of maximum complex tangent space at a CR singular point. 2014. ⟨hal-01001831⟩