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Mathematics > Algebraic Geometry

arXiv:1805.03457 (math)
[Submitted on 9 May 2018 (v1), last revised 26 Jun 2018 (this version, v2)]

Title:Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds

Authors:Tamás László, János Nagy, András Némethi
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Abstract:Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincaré series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of the coefficients).
We show that the (a Gorenstein type) symmetry of the zeta function combined with Ehrhart-Macdonald-Stanley reciprocity (of Ehrhart theory of polytopes) provide a simple expression for the periodic constant. Using these dualities we also find a multivariable polynomial generalization of the Seiberg-Witten invariant, and we compute it in terms of lattice points of certain polytopes. All these invariants are also determined via lattice point counting, in this way we establish a completely general topological analogue of formulae of Khovanskii and Morales valid for singularities with non-degenerate Newton principal part.
Subjects: Algebraic Geometry (math.AG)
MSC classes: 32S05, 32S25, 32S50, 57M27 (Primary), 14Bxx, 14J80, 57R57 (Secondary)
Cite as: arXiv:1805.03457 [math.AG]
  (or arXiv:1805.03457v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1805.03457

Submission history

From: Andras Nemethi [view email]
[v1] Wed, 9 May 2018 11:14:29 UTC (37 KB)
[v2] Tue, 26 Jun 2018 09:12:05 UTC (38 KB)
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