Analytic lattice cohomology of surface singularities, II (the equivariant case)

TL;DR

This paper develops an equivariant analytic lattice cohomology for complex surface singularities with rational homology sphere links, linking analytic, topological, and categorification concepts in singularity theory.

Contribution

It introduces a new equivariant analytic lattice cohomology that categorifies the geometric genus and relates to topological invariants like Seiberg--Witten and Heegaard Floer cohomology.

Findings

Constructed the equivariant analytic lattice cohomology for surface singularities.

Established the categorification of the geometric genus.

Linked the cohomology to topological invariants such as Seiberg--Witten and Heegaard Floer cohomology.

Abstract

We construct the equivariant analytic lattice cohomology associated with the analytic type of a complex normal surface singularity whenever the link is a rational homology sphere. It is the categorification of the equivariant geometric genus of the germ. This is the analytic analogue of the topological lattice cohomology, associated with the link of the germ, and indexed by the spin$^c$--structures of the link (which is a categorification of the Seiberg--Witten invariant and conjecturally it is isomorphic with the Heegaard Floer cohomology).