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 cosntant. 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 the geometric genus of singularities with non-degenerate Newton principal part.
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Acknowledgements
Open access funding provided by MTA Alfréd Rényi Institute of Mathematics (MTA RAMKI). TL was supported by ERCEA Consolidator Grant 615655—NMST and by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. AN was partially supported by ERC Adv. Grant LDTBud of A. Stipsicz at Rényi Institute of Math., Budapest. TL and AN were also supported by NKFIH Grant “Élvonal” (Frontier) KKP 126683. JN was partially supported by NKFIH Grant K119670, JN and AN were partially supported by NKFIH Grant K112735.
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László, T., Nagy, J. & Némethi, A. Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds. Sel. Math. New Ser. 25, 21 (2019). https://doi.org/10.1007/s00029-019-0468-9
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DOI: https://doi.org/10.1007/s00029-019-0468-9
Keywords
- Normal surface singularities
- Links of singularities
- Plumbing graphs
- Rational homology spheres
- Seiberg–Witten invariant
- Poincaré series
- Quasipolynomials
- Surgery formula
- Periodic constant
- Ehrhart polynomials
- Ehrhart–Macdonald–Stanley reciprocity law
- Gorenstein duality