Invariants of Newton non-degenerate surface singularities

Braun, Gábor; Némethi, András

doi:10.1112/s0010437x07002941

Cited by 18 publications

(29 citation statements)

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“…Therefore, even for hypersurface singularities the two values eu(H * ) and min γ eu(H 0 (γ)) might be different. Here we wish to recall that in [9] it is shown that for such germs from M one can recover the Newton diagram of the equation, hence the equisingularity type of the germ too. Hence, in principle, p g can be recovered from M ; however this statement does not indicate any topological candidate for p g .…”

Section: Some Conjectures Results and Examplesmentioning

confidence: 97%

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The geometric genus of hypersurface singularities

Némethi¹,

Sigurðsson²

2016

J. Eur. Math. Soc.

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Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.A. Némethi and B. Sigurðsson fixed resolution, at each step increasing only by a base element, and connecting the trivial cycle with the anticanonical cycle. For such a path γ one defines a path lattice cohomology H 0 (γ), and one takes its normalized rank eu(H 0 (γ)). Then one shows that for any analytic type one has p g ≤ min γ eu(H 0 (γ)), hence it provides a natural topological upper bound for the geometric genus. (The authors do not know if min γ eu(H 0 (γ)) can be defined by any other construction, say, using gauge theory or low dimensional topology.)The main result of the article is the following.Theorem 1.1.1. Assume that (X, 0) is a normal surface singularity whose link is rational homology sphere. Then the identity p g = min γ eu(H 0 (γ)) is true in the following cases: (a) if H q (M ) = 0 for q ≥ 1 and the singular germ satisfies the SWIC Conjecture (in particular, for all weighted homogeneous and minimally elliptic singularities);(b) superisolated singularities (with arbitrary number of cusps); (c) singularities with non-degenerate Newton principal part. Moreover, since the conjecture is stable with respect to equisingular deformation of hypersurfaces, the conjecture remains valid for such deformations of any of the above cases.The next section contains all the necessary definitions, main guiding examples, and status quo of the problem. Section 3 contains the proof for superisolated germs. In this case the link is a surgery 3-manifold. The proof has two non-trivial ingredients, already present in the recent literature: the first one provides the lattice cohomology of surgery 3-manifolds [44], the other is an application of d-invariant vanishing result of certain L-space surgery 3-manifolds in Heegaard Floer knot theory [8]. The next sections contain the proof of the Newton non-degenerate case: it involves deeply the very specific combinatorics of the associated toric resolution, and a lattice point counting. Geometric genus formulae, conjectures, guiding examples2.1. Preliminaries: the geometric genus. Let us fix a complex analytic normal surface singularity (X, 0). Let M be its link, the oriented smooth 3-manifold which is the boundary of a convenient small representative X of the germ. Since the real cone over M is homeomorphic to X, M characterizes completely the local topology of (X, 0). If we consider a resolution of X with dual resolution graph G, then M can be realized as a plumbed 3-manifold associated with G, and, in fact, it contains the same information as G itself (cf.[45]). The topological invariants of the germ (X, 0) are read either from the topology of M , or from the combinatorics of G.The analytic invariants of (X, 0) are a priori associated with the analytic structure of (X, 0) read e.g. from...

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Section: Some Conjectures Results and Examplesmentioning

confidence: 97%

“…If a ∈ A corresponds to △ ∈ F \ F c then N a = N △ . All the other vectors are determined in a unique way by the next identities (see [9]):…”

Section: (The Algorithm)mentioning

confidence: 99%

The geometric genus of hypersurface singularities

Némethi¹,

Sigurðsson²

2016

J. Eur. Math. Soc.

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“…According to this, for such a germ one defines the Newton polytope N using the nontrivial monomials of the defining equation of the germ, and one proves that several invariants of the germ can be recovered from N ; see eg Braun and Némethi [10]. For example, by a result of Merle and Teissier [30], the geometric genus p g equals the number of lattice points in ..Z >0 / 3 \ N /.…”

Section: A 'Classical' Connection Between Polytopes and Gauge Invariamentioning

confidence: 96%

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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“…The purely combinatorial nature of the description we make of the manifold ∂F opens the way for the computation of a great number of examples through a future implementation in a computer program. But it also calls for more theoretic work, such as for example extending what is done in [BN07], where Braun & Némethi do the opposite work, retrieving a possible Newton polyhedron of a function f : C 3 → C having a given graph manifold as boundary of its Milnor fiber, under the hypothesis that this manifold is a rational homology sphere. Another use of this method would be to answer the widely open question of which manifolds can appear as boundaries of Milnor fibers of non degenerate surface singularities.…”

Section: Introductionmentioning

confidence: 99%