Surface Singularities, Seiberg–Witten Invariants of Their Links and Lattice Cohomology

The present note aims to focus on certain topological and analytical invariants of complex normal surface singularities and wishes to analyse their interferences. The first preliminary part introduces the needed notations, definitions and terminologies: e.g. resolutions, universal abelian coverings, natural line bundles on resolutions, links, spinc structures on the links. Here we also recall certain vanishing theorems and statements connected with Serre’s and Laufer’s dualities. The next part presents two multivariable series, a topological one (associated with a dual resolution graph) and an analytic one (associated with the divisorial filtration), then we compare them. Then we introduce several topological invariants, as the Casson and Casson–Walker invariants, Turaev’s torsion, the Seiberg–Witten invariant. By the ‘Seiberg–Witten Invariant Conjecture’ they are compared with the cohomology of the natural line bundles. In this discussion certain ‘additivity formulae’ will also be crucial. After a preparation (introduction of the weighted cubes) we continue with the presentation of the (topological) lattice cohomology and of the (topological) graded roots associated with rational homology sphere singularity links. They are exemplified by links of superisolated singularities, when the theory is also connected with the classification of irreducible rational cuspidal projective plane curves.