Classification of flat pencils of foliations on compact complex surfaces

TL;DR

This paper classifies compact complex surfaces that admit a flat pencil of foliations with an invariant tangency set, extending previous work on foliations with genus one first integrals and isolated singularities.

Contribution

It completes the classification of such surfaces by including cases with an invariant tangency set, building upon Lins Neto's prior classification.

Findings

Classified surfaces with flat pencils and invariant tangency sets

Extended the understanding of foliations with genus one first integrals

Provided a comprehensive classification framework

Abstract

Related to the classification of regular foliations in a complex algebraic surface, we address the problem of classifying the complex surfaces which admit a flat pencil of foliations. On this matter, a classification of flat pencils which admit foliations with a first integral of genus one and isolated singularities was done by Lins Neto. In this work, we complement Lins Neto's work, by obtaining the classification of compact complex surfaces which have a pencil with an invariant tangency set.