Categorification of the plurigenera of Gorenstein normal surface singularities

Consider a complex normal surface singularity and its three plurigenera, the m-th \(L^2\) $L^2$ –plurigenus of Watanabe, the m-th plurigenus of Knöller and the m-th log-plurigenus of Morales. For any of these invariants we construct a double graded \(\mathbb {Z}[U]\) $\mathbb{Z}[U]$ –module, whose Euler characteristic is the chosen plurigenus. The three outputs are compared with the analytic lattice cohomology of the germ, whose Euler characteristic is the classical geometric genus.