  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.
