The Abel map for surface singularities II. Generic analytic structure

TL;DR

This paper develops topological formulas for various discrete analytic invariants of complex normal surface singularities with generic structures, extending the Abel map concept and relating to Brill-Noether theory.

Contribution

It introduces a notion of generic structure for surface singularities and applies the Abel map to derive formulas for analytic invariants in this setting.

Findings

Derived topological formulas for geometric genus and cohomology invariants.

Established the concept of generic analytic structure based on Laufer's work.

Connected the Abel map approach to classical Brill-Noether theory.

Abstract

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants include: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincar\'e series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle. The first part contains the definition of `generic structure' based on the work of Laufer. The second technical ingredient is the Abel map introduced by the authors. The results can be compared with certain parallel statements from the Brill-Noether theory and from the theory of Abel map associated with projective smooth curves, though the tools and machineries are very different.