An example of a non-algebraizable singularity

TL;DR

This paper constructs a concrete example of a non-algebraizable singularity in holomorphic foliations, advancing understanding beyond previous existential results.

Contribution

It provides the first explicit example of a non-algebraizable singularity, moving beyond prior existence proofs.

Findings

Constructed a specific non-algebraizable singularity

Demonstrated the limitations of algebraization in singular holomorphic foliations

Extended the theoretical understanding of foliation singularities

Abstract

We say that the the germ of a singular holomorphic foliation on $(\mathbb{C}^2,0)$ is algebraizable whenever it is holomorphically conjugate to the singularity of a foliation defined globally on a projective algebraic surface. The object of this work is to construct a concrete example of a non--algebraizable singularity. Previously only existential results were known.