Line bundles associated with normal surface singularities

Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$ associated with the canonical'' $spin^c$ structure of $M$. Since the Seiberg-Witten theory of the link $M$ provides a rational number for any $spin^c$ structure it was a natural challenge to search for a complete set of conjecturally valid identities, which involve all the Seiberg-Witten invariants (giving an analytic -- i.e. singularity theoretical -- interpretation of them). The formulation of this set of identities is one of the goals of the present article. In fact, we formulate conjecturally valid inequalities which became equalities in special rigid situations. In this way, the Seiberg-Witten invariants determine optimal topological upper bounds for the dimensions of the first sheaf-cohomology of line bundles living on the resolution. Moreover, for $\Q$-Gorenstein singularities and some natural’’ line bundles equality holds. The first part of the article constructs these ``natural’’ holomorphic line bundles. The line-bundle construction is compatible with abelian covers. This allows us to reformulate the conjecture in its second version which relates the echivariant geometric genus (associated with the universal abelian cover of the singular germ) with the Seiberg-Witten invariants of the link $M$. In the last section we verify the conjecture for rational singularities.