On the classification of nilpotent singularities

Abstract

We deal in this paper with the analytic classification of singular holomorphic foliations defined in a neighborhood of the origin of C2 and having nonzero linear part. In the “semi simple” case this classification which is closely related to the ratio of the eigenvalues of the linear part, is quasi complete. It appears in particular that the moduli are essentially the ones of a holonomy diffeomorphism (see [2,4,6,7,10,11], . . .). Our aim here is to study the analytic classification in the “nilpotent case”, i.e., when both eigenvalues are zero. Let Λ denote the set of germs at 0 ∈ C2 of holomorphic 1-forms for which the origin is an isolated singularity. Let also Diff(C,0) (respectively Diff(C,0)) be the group of germs at 0 ∈ C of holomorphic (respectively formal) diffeomorphisms which fix the origin. Two elements ω1 and ω2 of Λ are said to be holomorphically (respectively formally) conjugated if there exists an element φ of Diff(C2,0) (respectively Diff(C2,0)) such that φ∗ω1 ∧ ω2 = 0. We will write ω1 hol ∼ ω2 (respectively ω for ∼ ω2). The moduli space ωfor of ω is the set