Vector fields, separatrices and Kato surfaces

TL;DR

This paper characterizes certain singular complex surfaces with complete holomorphic vector fields, linking their structure to Kato surfaces and classifying singularities based on vector field behavior.

Contribution

It provides a new characterization of singular surfaces related to Kato surfaces and classifies singular points in Stein surfaces with respect to vector field zeroes.

Findings

Surfaces with complete vector fields and no invariant curves are obtained from Kato surfaces.

Singular points with non-accumulating zeroes are quasihomogeneous or cyclic quotient singularities.

Reproves classification results for complex surfaces admitting holomorphic vector fields.

Abstract

We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeroes of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. The proofs rely in a combinatorial description of the vector field on a resolution of the singular point based on previous work of Rebelo and the author. With the same tools, we reprove some facts about the classification of compact complex surfaces admitting holomorphic vector fields.