Logarithmic geometry and Infinitesimal Hodge Theory

TL;DR

This paper introduces a logarithmic geometric framework to study infinitesimal Hodge variations in singular and equisingular families, offering new insights into deformation theory and classical Hodge results.

Contribution

It develops a systematic approach using logarithmic geometry and residue calculus to analyze infinitesimal Hodge structures in singular settings, extending classical theories.

Findings

Logarithmic vector fields encode deformation directions preserving singularities.

Residue calculus governs the effective variation of Hodge structures.

Provides geometric explanations for Jacobian rings in Hodge theory.

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.