The analytic lattice cohomology of isolated singularities

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.