Derived Hodge-to-de Rham Degeneration and Quasihomogeneous Curve Singularities

Abstract

We study the Hodge-completed derived Hodge-to-de Rham spectral sequence of integral projective complex curves. We prove E2-degeneration whenever every singularity is analytically quasihomogeneous, without a local complete intersection (lci) hypothesis. An explicit lci counterexample shows that the converse fails in general. For lci curves, E2-degeneration bounds the total Milnor-Tjurina defect by the dimension of the space of global Kahler one-forms modulo torsion, and hence by the genus of the normalization. This yields a converse for lci curves whose global Kahler one-forms are all torsion, including those with rational normalization. We also establish the converse for locally planar curves. An example with elliptic normalization shows that the numerical defect bound alone is not sufficient for E2-degeneration, even when equality holds.

arXiv comment43 pages. Substantially revised. A counterexample disproves the general lci converse claimed in v1; corrected converse statements are established under additional hypotheses. The locally planar characterization remains valid, with a revised proof. The Hochschild-to-cyclic claims have been removed. Junxiao Wang has joined as coauthor, and the title and abstract have been revised