Hodge moduli algebras and complete invariants of singularities

Stephen S.-T. Yau (Beijing Institute of Mathematical Sciences and Applications and Department of MathematicalSciences, Tsinghua University, Beijing, China)

Huaiqing Zuo (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Abstract

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.

Keywords

isolated singularity, Hodge ideals

2010 Mathematics Subject Classification

14B05, 32S05

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Received 22 February 2023

Accepted 14 June 2024

Published 7 August 2024