Hodge moduli algebras and complete invariants of singularities

We introduce the Hodge moduli algebras and Hodge moduli sequence associated with an isolated hypersurface singularity. These are new subtle invariants of singularities. We propose several characterization conjectures by using of these invariants. We investigate structural properties and numerical invariants of Hodge ideals naturally associated with isolated hypersurface singularities.In particular, we establish that the analytic isomorphisms class of an isolated two dimensional rational hypersurface singularities is determined by the Hodge moduli algebras and Hodge moduli sequence. As a result, we prove that Hodge moduli algebra together with the geometric genus give complete characterization of such singularities. In the proof, we concretely compute the Hodge ideals and the associated Hodge moduli algebras of these singularities.