Analytic lattice cohomology of surface singularities

TL;DR

This paper introduces an analytic lattice cohomology for complex surface singularities, linking it to geometric genus and topological invariants, and explores its properties and potential connections to deformation theory.

Contribution

It constructs the analytic lattice cohomology for surface singularities, providing a new analytic invariant that extends topological lattice cohomology and relates to deformation theory.

Findings

Analytic lattice cohomology categorifies the geometric genus.

In some cases, analytic and topological cohomologies coincide.

Analytic cohomology is sensitive to the singularity's analytic structure.

Abstract

We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is the analytic analogue of the topological lattice cohomology, associated with the link of the germ whenever it is a rational homology sphere. This topological lattice cohomology is the categorification of the Seiberg--Witten invariant, and conjecturally it is isomorphic with the Heegaard Floer cohomology. We compare the two lattice cohomologies: in some simple cases they coincide, but in general, the analytic cohomology is sensitive to the analytic structure. We expect a deep connection with deformation theory. We provide several basic properties and key examples, and we formulate several conjectures and problems. This is the initial article of a series, in which we develop the analytic lattice cohomology of singularities.