Brill-Noether problem on splice quotient singularities and duality of topological Poincar\'e series

TL;DR

This paper extends the Brill-Noether problem to normal surface singularities, determining maximal cohomology values for line bundles in specific singularity cases and introducing virtual cohomology numbers with duality properties.

Contribution

It introduces the concept of virtual cohomology numbers for line bundles on surface singularities and generalizes duality formulas for Seiberg-Witten invariants in this context.

Findings

Maximal $h^1$ values determined for certain singularities.

Definition of virtual cohomology numbers for all Chern classes.

Generalization of Seiberg-Witten duality formulas.

Abstract

In this manuscript we investigate the analouge of the Brill-Noether problem for smooth curves in the case of normal surface singularities. We determine the maximal possible value of $h^1$ of line bundles without fixed components in the Picard group $\pic^{l'}(\tX)$ in the following cases: for some special Chern classes $l'$ if $\tX$ is a resolution of a splice quotient singularity $(X, 0)$ and for arbitrary Chern classes in the case of weighted homogenous singularities. Motivated by this problem, we define the \emph{virtual cohomology numbers} $h^1_{virt}(l')$ for all Chern classes $l'$ such that $h^1_{virt}(0)$ is the canonical normalized Seiberg-Witten invariant and we generalize the duality formulae of Seiberg-Witten invariants obtained by the authors and A. N\'emethi in \cite{LNNdual}, for the virtual cohomology numbers.