Semicomplete meromorphic vector fields on complex surfaces
type: Item
instance of: scholarly article
title: Semicomplete meromorphic vector fields on complex surfaces
zbMATH Open document ID: 1250.32023
DOI: 10.1515/CRELLE.2011.127
author: Adolfo Guillot
author: Julio C. Rebelo
publication date: 2012-07-12T00:00:00Z
Mathematics Subject Classification ID: 32M25
Mathematics Subject Classification ID: 37F75
zbMATH DE Number: 6055137
zbMATH Keywords: semicomplete meromorphic vector field
zbMATH Keywords: curve of poles
zbMATH Keywords: vector field on a compact complex Kähler surface
MaRDI profile type: Publication
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full work available at URL: crelle.2011.127
OpenAlex ID: W1999568408
published in: Journal für die Reine und Angewandte Mathematik
review text: The authors study meromorphic vector fields on complex surfaces such that the vector field obtained when they are restricted to the complement of their locus of poles is semicomplete. Semicomplete means that, viewed as ordinary differential equations, their solutions are single-valued in their maximal definition domain. The main result in the paper can be thought as a local minimal model statement for semicomplete meromorphic vector fields. In the paper it is proved that for such vector fields, up to a birational transformation, a compact connected component of the curve of poles is either a rational or an elliptic curve whose self-intersection vanishes or it has the combinatorics of a singular fiber of an elliptic fibration. This result is also globalized by proving that, up to a birational transformation, a semicomplete meromorphic vector field on a compact complex Kähler surface must satisfy at least one of the following conditions: it is globally holomorphic, it has a non-trivial meromorphic first integral or it preserves a fibration.
reviewed by: Carlos Galindo
description: scientific article; zbMATH DE number 6055137