The geometric genus of hypersurface singularities

Abstract

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...

Lattice cohomology and rational cuspidal curves

“…(Alternative form of Theorem 1.3.7) For a link L = S 3 −d (K) of a superisolated surface singularity corresponding to a rational cuspidal projective plane curve of degree d we have:eu H 0 can S 3 −d (K) = d(d − 1)(d − 2)/6. This form is also present in the recent article[20] (in Example 2.4.3 (a) and Section 3). Next, using (3.1.14) we give an equivalent formulation of Conjecture 1.2.9 in terms of lattice cohomology (cf.…”

A note on 3-manifolds and complex surface singularities

“…These structures play a significant role in low dimensional manifolds and in the theory of surface singularities, see for instance [15,16]. I believe that the E-invariant, and Looijenga's Riemann-Roch defect, must have a deep relation with the Seiberg-Witten invariant of the link with its canonical Spin c -structure.…”

Analytic singularities supported by a specific integral homology sphere link

“…-superisolated hypersurface singularities [20], -isolated hypersurface Newton-nondegenerate singularities [20], -rational singularities [19], -Gorenstein elliptic singularities [19].…”