Complete holomorphic vector fields on C^2 whose underlying foliation is polynomial

TL;DR

This paper extends the classification of complete holomorphic vector fields on C^2 with polynomial underlying foliation, broadening previous results from purely polynomial to include certain non-polynomial cases.

Contribution

It provides a comprehensive classification of complete holomorphic vector fields on C^2 with polynomial foliation, including non-polynomial vector fields, building on Brunella's earlier polynomial case.

Findings

Extended classification to non-polynomial vector fields

Identified conditions under which holomorphic vector fields have polynomial foliation

Enhanced understanding of the structure of holomorphic vector fields on C^2

Abstract

We extend the classification of complete polynomial vector fields on C^2 given by Marco Brunella (Topology 43(2): 433-445, 2004) to cover the case of holomorphic (non-polynomial) vector fields whose underlying foliation is however still polynomial.