Vol. 307, No. 1, 2020

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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

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.

meromorphic function, convex cone, Laurent expansion, residue, Jeffrey–Kirwan residue
Mathematical Subject Classification 2010
Primary: 32A20, 32A27, 52A20, 52C07
Received: 26 April 2019
Revised: 20 February 2020
Accepted: 23 February 2020
Published: 8 August 2020
Li Guo
Department of Mathematics and Computer Science
Rutgers University at Newark
Newark, NJ
United States
Sylvie Paycha
Université Clermont-Auvergne
Institute of Mathematics
Universität Potsdam
Bin Zhang
Yangtze Center for Mathematics
Sichuan University