The analytic lattice cohomology of isolated singularities

TL;DR

This paper introduces the concept of analytic lattice cohomology for isolated singularities of dimension at least two, extending previous work on surface singularities and linking it to topological and Hodge-theoretic invariants.

Contribution

It defines a new invariant called analytic lattice cohomology for higher-dimensional isolated singularities, proves its resolution independence, and relates its Euler characteristic to Hodge numbers.

Findings

Analytic lattice cohomology is well-defined and resolution-independent.

Euler characteristic of the cohomology equals the Hodge number $h^{n-1}({ m O}_{ ilde{X}})$.

For weighted homogeneous hypersurface singularities, it connects to Hodge spectral numbers.

Abstract

We associate (under a minor assumption) to any analytic isolated singularity of dimension $n\geq 2$ the `analytic lattice cohomology' ${\mathbb H}^*_{an}=\oplus_{q\geq 0}{\mathbb H}^q_{an}$. Each ${\mathbb H}^q_{an}$ is a graded ${\mathbb Z}[U]$--module. It is the extension to higher dimension of the `analytic lattice cohomology' defined for a normal surface singularity with a rational homology sphere link. This latest one is the analytic analogue of the `topological lattice cohomology' of the link of the normal surface singularity, which conjecturally is isomorphic to the Heegaard Floer cohomology of the link. The definition uses a good resolution $\widetilde{X}$ of the singularity $(X,o)$. Then we prove the independence of the choice of the resolution, and we show that the Euler characteristic of ${\mathbb H}^*_{an}$ is $h^{n-1}({\mathcal O}_{\widetilde{X}})$. In the case of a hypersurface weighted homogeneous singularity we relate it to the Hodge spectral numbers of the first interval.