On Poincaré series associated with links of normal surface singularities

2015/03/31 by László, Tamás, Szilágyi, Zsolt
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1503.09012

Abstract

We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seiberg--Witten invariant given by the Poincaré series. We also reprove and discuss surgery formulas of Némethi for the counting function, and of Braun and Némethi for the Seiberg--Witten invariant.

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