Analytic lattice cohomology of surface singularities

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.X → X (and in some of the cases below we need to assume that the link is a rational homology sphere).

The analytic lattice cohomology of isolated singularities

“…In this section we review several combinatorial statements regarding the lattice cohomology associated with any weight function with certain combinatorial properties. We follow [1].…”

“…In the topological case of surface singularities the possible choice of V was dictated by combinatorial properties of the Riemann-Roch expression χ (the topological weight function), with a special focus on the topological characterization of rational germs [26,22]. In the analytical case of surface singularities we used certain analytic properties of 2-forms [1]. The present high-dimensional case is a direct generalization of this.…”

“…Then the rational cycle Z K ∈ L ⊗ Q can be defined, it is the (anti)canonical cycle, numerically equivalent with K X (see e.g. [1]). Then again Z coh ≤ ⌊Z K,+ ⌋ (see e.g [1]), hence the very same proof gives the following: if…”

“…1.1.2. Recently, in [1,2] we introduced their analytic analogues, the analytic lattice cohomology H * an = ⊕ q≥0 H q an , associated with a normal surface singularity with a rational homology sphere link. It is constructed from analytic invariants of a good resolution, however it turns out that it is independent of the choice of the resolution.…”

“…In fact, historically, this identity (The Seiberg-Witten Invariant Conjecture of the the second author and Nicolaescu [33,30], valid for 'nice' analytic structures) led to the discovery of H * top . However, if we fix a topological type, and we move the possible analytic structure supported on this topological type, then the analytic lattice cohomologies reflect the modification of the analytic structures, for several examples see [1].…”