Universal Abelian covers of rational surface singularities and multi-index filtrations

Guseĭn-Zade, S. M.; Delgado, F.; Campillo, A.

doi:10.1007/s10688-008-0013-7

Cited by 18 publications

(9 citation statements)

References 6 publications

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“…The point is that the topological candidate of P.t/ is exactly Z.t/ from the previous subsection; they agree for several singularities; see eg [25], [36] and [39]. 3 Equivariant multivariable Ehrhart theory…”

Section: The Analytic Motivation: Multivariable Hilbert Series Of DIVmentioning

confidence: 80%

Ehrhart theory of polytopes and Seiberg–Witten invariants of plumbed 3–manifolds

Tamás¹,

Némethi

2014

Geom. Topol.

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Let M be a rational homology sphere plumbed 3-manifold associated with a connected negative-definite plumbing graph. We show that its Seiberg-Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial. For this, we construct the corresponding polytope from the plumbing graph together with an action of H 1 .M; Z/ and we develop Ehrhart theory for them. At an intermediate level we define the 'periodic constant' of multivariable series and establish their properties. In this way, one identifies the Seiberg-Witten invariant of a plumbed 3-manifold, the periodic constant of its 'combinatorial zeta function' and a coefficient of the associated Ehrhart polynomial. We make detailed presentations for graphs with at most two nodes. The two node case has surprising connections with the theory of affine monoids of rank two.

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Section: The Analytic Motivation: Multivariable Hilbert Series Of DIVmentioning

confidence: 80%

Ehrhart theory of polytopes and Seiberg–Witten invariants of plumbed 3–manifolds

Tamás¹,

Némethi

2014

Geom. Topol.

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“…[CDG08,N08b,N08c]). In this way, for such singularities, one gets a topological characterization of the constant terms from (2.2.2).…”

Section: Motivation Of Theorem A: Hilbert Seriesmentioning

confidence: 99%

“…The series Z(t) was used in several articles studying invariants of surface singularities [CDG04,CDG08,CHR04,N08b,N08c]. Theorem A puts the results of these articles in a new light.…”

Section: Introductionmentioning

confidence: 99%

The Seiberg–Witten invariants of negative definite plumbed 3-manifolds

Némethi

2011

J. Eur. Math. Soc.

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Abstract. Assume that is a connected negative definite plumbing graph, and that the associated plumbed 3-manifold M is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg-Witten invariant of M. The first one is the constant term of a 'multivariable Hilbert polynomial', it reflects in a conceptual way the structure of the graph , and emphasizes the subtle parallelism between these topological invariants and the analytic invariants of normal surface singularities. The second formula realizes the Seiberg-Witten invariant as the normalized Euler characteristic of the lattice cohomology associated with , supporting the conjectural connections between the Seiberg-Witten Floer homology, or the Heegaard-Floer homology, and the lattice cohomology.

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“…Remark 2. For the case when (S, 0) was a rational surface singularity and v i were divisorial valuations corresponding to all the components of the exceptional divisor of a resolution of (S, 0), the series P W {v i } (t) was defined in [4] and [5] and used in [6]; see also [14].…”

Section: The Weil-poincaré Seriesmentioning

confidence: 99%

Weil-Poincaré series and topology of collections of valuations on rational double points

Campillo¹,

Delgado²,

Guseĭn-Zade³

2021

Preprint

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Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over the space of divisors on the E 8 -singularity.Here we consider a similar integral for rational double point surface singularities over the space of Weil divisors called the Weil-Poincaré series. We show that, except a few explicitly described cases the Weil-Poincaré series of a collection of curve valuations on a rational double point surface singularity determines the topology of the corresponding link. We give analogous statements for collections of divisorial valuations.

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Universal Abelian covers of rational surface singularities and multi-index filtrations

Cited by 18 publications

References 6 publications

Ehrhart theory of polytopes and Seiberg–Witten invariants of plumbed 3–manifolds

Ehrhart theory of polytopes and Seiberg–Witten invariants of plumbed 3–manifolds

The Seiberg–Witten invariants of negative definite plumbed 3-manifolds

Weil-Poincaré series and topology of collections of valuations on rational double points

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