A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles

Li Guo, Sylvie Paycha and Bin Zhang

Vol. 307 (2020), No. 1, 159–196

DOI: 10.2140/pjm.2020.307.159

Abstract

We develop a geometric approach using convex polyhedral cones to build Laurent expansions for multivariate meromorphic germs with linear poles, which naturally arise in various contexts in mathematics and physics. We express such a germ as a sum of a holomorphic germ and a linear combination of special nonholomorphic germs called polar germs. In analyzing the supporting cones — cones that reflect the pole structure of the polar germs — we obtain a geometric criterion for the nonholomorphicity of linear combinations of polar germs. For any given germ, the above decomposition yields a Laurent expansion which is unique up to suitable subdivisions of the supporting cones. These Laurent expansions lead to new concepts on the space of meromorphic germs, such as a generalization of Jeffrey–Kirwan’s residue and a filtered residue, all of which are independent of the choice of the specific Laurent expansion.

Keywords

meromorphic function, convex cone, Laurent expansion, residue, Jeffrey–Kirwan residue

Mathematical Subject Classification 2010

Primary: 32A20, 32A27, 52A20, 52C07

Milestones

Received: 26 April 2019

Revised: 20 February 2020

Accepted: 23 February 2020

Published: 8 August 2020