Normal Surface Singularities

For a fixed resolution of a normal surface singularity we discuss the following objects: the local divisor class group, Q-Cartier divisors and the analytic and topological canonical coverings, natural line bundles, properties of the canonical cycle, the Gorenstein and Q-Gorenstein properties, different vanishing theorems (e.g. the local version of the Kawamata-Viehweg vanishing theorem and Lipman’s vanishing theorem), properties of the cohomology cycle, base point freeness, the structure of topological and analytical monoids, local Zariski decomposition, geometric genus and its reinterpretation in terms of differential forms via Laufer’s duality, upper and lower bounds of the geometric genus, plurigenera. We also add subsections in which we relate resolution invariants with smoothing invariants (formulae of Laufer, Durfee, Wahl), we present classical formulae regarding Milnor number and signature of Brieskorn and suspension hypersurface singularities. We also review some famous conjectures and open problems regarding hypersurface singularities. The last part reviews the theory of spin and spin^c structures for manifolds of dimension 3 and 4.

We define the multivariable Hilbert and Poincaré series associated with a resolution of a normal surface singularity. They are related with a multi-indexed linear subspace arrangement associated with the analytic type of the germ. We consider their topological analogues as well: a multivariable series defined from the resolution graph and also a topological multi-indexed linear subspace arrangement. Using them we can extend the multivariable series to their motivic versions. In certain geometric situation we can reduce the variables of the series in such a way that the reduced series contains essentially the same amount of information. In several classical cases even the one-variable reduction might carry crucial information (e.g., in the case of weighted homogeneous germs). We define the ‘periodic constant’ of one-variable series by ‘regularization of their coefficients’, via the quasipolynomial of the counting function of the coefficient. This numerical invariant serves as ‘correction term’ in several inductive additivity formulae.

In this chapter we examine the family of splice quotient singularities as well, and we compute the most important analytic invariants (Poincaré series, multiplicity, analytic semigroup) topologically from their graph.

We fix the link of a normal surface singularity and we assume that it is a rational homology sphere. Via its plumbing graph we introduce and discuss several topological invariants: the Casson invariant (whenever the link is an integral homology sphere), the Casson-Walker invariant, the Turaev torsion and the Seiberg-Witten invariant. Then we recall a conjecture of Nicolaescu and the author (in the literature known as the Seiberg-Witten Invariant Conjecture), which relates the Seiberg-Witten invariant of the link to the (equivariant) geometric genus. We prove it for several cases (e.g., rational, weighted homogeneous, splice quotient germs), and we provide also counterexamples (certain superisolated germs).

First, we define the lattice cohomology associated with the link of a normal surface singularity, whenever this link is a rational homology sphere. This is done via the lattice of a resolution and well-chosen (Riemann-Roch type) weight functions of the lattice points. We prove that it is independent of all the choices and depends only on the link. The author conjectured that it is isomorphic as a graded Z[U]-module with the Heegaard Floer homology of the link (this fact was verified recently by Zemke). In parallel we define the graded roots as well as improvements of the 0-homology group. We compute this topological lattice cohomology and the graded root in many examples (star shaped graphs, surgery 3-manifolds). Then we discuss its path-version, the path lattice cohomology and its relationship with the geometric genus (of any analytic structure). In the final part we discuss the ‘analytic pair’: the analytic lattice cohomology. We compare the two theories and we test their behavior with respect to certain analytic deformations.