Holomorphic vector fields with real integral manifolds

TL;DR

This paper classifies singular holomorphic vector fields in complex two-space that admit real-analytic invariant 3-folds, completing the understanding of their symmetries and revealing overlaps with existing classification theories.

Contribution

It provides a complete classification of such vector fields with real integral manifolds, extending the theory of infinitesimal symmetries of Levi-nonflat hypersurfaces.

Findings

Most resonances in Lombardi-Stolovitch theory do not occur with Levi-nonflat integral manifolds.

The classification reveals new overlaps with existing singularity theories.

The work advances understanding of holomorphic vector fields with real-analytic invariants.

Abstract

We classify singular holomorphic vector fields in two-dimensional complex space admitting a (Levi-nonflat) real-analytic invariant 3-fold through the singularity. In this way, we complete the classification of infinitesimal symmetries of real-analytic Levi-nonflat hypersurfaces in complex two-space. The classification of holomorphic vector fields obtained in the paper has very interesting overlaps with the recent Lombardi-Stolovitch classification theory for holomorphic vector fields at a singularity. In particular, we show that most of the resonances arising in Lombardi-Stolovitch theory do not occur under the presence of (Levi-nonflat) integral manifolds.