About $\mathcal{C}^\infty$ foliations by holomorphic curves on complex surfaces

TL;DR

This paper investigates real smooth foliations by holomorphic curves on complex surfaces, focusing on their local structure near curves and classifying certain real-analytic cases under specific conditions.

Contribution

It provides restrictions and classifications for such foliations, especially those locally diffeomorphic to line foliations, enhancing understanding of their geometric properties.

Findings

Classified real-analytic foliations near curves under non-degeneracy conditions.

Explored conditions under which foliations are holomorphic or resemble holomorphic families.

Analyzed the local geometry of foliations with holomorphic leaves in complex surfaces.

Abstract

We study those real $\mathcal{C}^\infty$ foliations in complex surfaces whose leaves are holomorphic curves. The main motivation is to try and understand these foliations in neighborhoods of curves: can we expect the space of foliations in a fixed neighborhood to be infinite-dimensional, or are there some contexts under which every such foliation is holomorphic? We give some restrictions and study in more details the geometry of foliations whose leaves belong to a holomorphic family of holomorphic curves. In particular, we classify all real-analytic foliations on neighborhoods of curves which are locally diffeomorphic to foliations by lines, under some non-degeneracy hypothesis.