Logarithmic geometry and Infinitesimal Hodge Theory
Mounir Nisse
TL;DR
The paper develops a unified framework tying logarithmic geometry to infinitesimal variations of Hodge structure in singular and equisingular settings. By showing that logarithmic vector fields along a singular locus govern equisingular deformations and that residue calculus governs the nontrivial Hodge variation, it recasts the IVHS problem in terms of Jacobian-type algebras after factoring out these trivial directions. The approach recovers and extends Green–Voisin’s picture to singular complete intersections and connects to Severi varieties and Jacobian rings, providing a global, geometric interpretation of Torelli-type phenomena for singular moduli. This yields explicit residue-based formulas for IVHS, clarifies the global deformation spaces, and elucidates the Jacobian-ring description of Hodge-theoretic variation in the presence of singularities.
Abstract
This paper develops a systematic approach to infinitesimal variations of Hodge structure for singular and equisingular families by means of logarithmic geometry and residue theory. The central idea is that logarithmic vector fields encode precisely those deformation directions that preserve singularities and act trivially on Hodge structures, while the effective variation is entirely governed by residue calculus. This viewpoint provides a conceptual reinterpretation of classical results of Griffiths, Green, and Voisin, and extends them to settings involving singular varieties and equisingular deformations. The resulting framework yields a geometric explanation for the appearance of Jacobian rings in infinitesimal Hodge theory and clarifies the structure of deformation spaces underlying Severi varieties and related moduli problems.