The analytic lattice cohomology of isolated singularities

Abstract

We associate (under a minor assumption) to any analytic isolated singularity of dimension $n\ge 2$ the `analytic lattice cohomology' ${\mathbb{H}}_{an}^{\ast }={\oplus }_{q\ge 0}{\mathbb{H}}_{an}^q$. Each ${\mathbb{H}}_{an}^q$ 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 $\tilde{X}$ of the singularity $(X,o)$. Then we prove the independence of the choice of the resolution, and we show that the Euler characteristic of ${\mathbb{H}}_{an}^{\ast }$ is $h^{n-1}({\mathcal{O}}_{\tilde{X}})$. In the case of a hypersurface weighted homogeneous singularity we relate it to the Hodge spectral numbers of the first interval.

arXiv admin note: text overlap with arXiv:2108.12429, arXiv:2108.12294