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.

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Mathematical Subject Classification 2010

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