Transversely projective structures on smooth foliations on surfaces

TL;DR

This paper characterizes smooth foliations on compact complex surfaces that have singular transversely projective structures but lack regular ones, expanding understanding of foliation structures in complex geometry.

Contribution

It provides a classification of smooth foliations on compact complex surfaces that do not admit regular transversely projective structures, building on recent work by Biswas and Dumitrescu.

Findings

Certain foliations lack regular transversely projective structures

Classification of foliations with only singular structures

Extension of Brunella's classification results

Abstract

Brunella's classification implies that every smooth foliation on a compact complex surface admits a singular transversely projective structure. However, Biswas and Dumitrescu's recent work shows that certain foliations on compact complex surfaces, despite possessing a singular transversely projective structure, do not admit a regular transversely projective structure. Here, we describe the smooth foliations on compact complex surfaces which fail to possess a regular transversely projective structure.