Real analytic manifolds in ℂ n with parabolic complex tangents along a submanifold of codimension one

Abstract

We will classify n-dimensional real submanifolds in ℂ n which have a set of parabolic complex tangents of real dimension n-1. All such submanifolds are equivalent under formal biholomorphisms. We will show that the equivalence classes under convergent local biholomorphisms form a moduli space of infinite dimension. We will also show that there exists an n-dimensional submanifold M in ℂ n such that its images under biholomorphisms (z 1 ,⋯,z n )↦(rz 1 ,⋯,rz n-1 ,r 2 z n ), r>1, are not equivalent to M via any local volume-preserving holomorphic map.