Single-Valued Killing Fields of a Meromorphic Affine Connection and Classification

TL;DR

This paper establishes a geometric condition ensuring that Killing vector fields of a meromorphic affine connection are single-valued, linking the connection's poles, geodesics, and automorphisms, and extends previous classification results.

Contribution

It introduces a new geometric criterion for single-valued Killing fields in meromorphic affine connections and extends existing classification results to this broader context.

Findings

Derived a geometric condition based on poles and geodesics

Linked meromorphic affine connections to Cartan geometries

Extended classification results to meromorphic affine connections

Abstract

We give a geometric condition on a meromorphic affine connection for its Killing vector fields to be single valued. More precisely, this condition relies on the pole of the connection and its geodesics, and defines a subcategory. To this end, we use the equivalence between these objects and meromorphic affine Cartan geometries. The proof of the previous result is then a consequence of a more general result linking the distinguished curves of meromorphic Cartan geometries, their poles and their infinitesimal automorphisms, which is the main purpose of the paper. This enables to extend the classification result from [Biswas I., Dumitrescu S., McKay B., \'Epijournal G\'eom. Alg\`ebrique 3 (2019), 19, 10 pages, arXiv:1804.08949] to the subcategory of meromorphic affine connection described before.