Residues and Infinitesimal Torelli for Equisingular Curves
Mounir Nisse
TL;DR
The paper develops a unified, residue-based framework for infinitesimal Torelli and maximal IVHS in singular and equisingular settings. By representing holomorphic forms as Poincaré residues and translating deformations into Jacobian-ring multiplications, it proves injectivity results for equisingular plane curves of high degree and constructs relative IVHS exact sequences for curves on threefolds. The approach extends the Green–Voisin philosophy to singular contexts and yields explicit, computable descriptions of how Hodge structures vary under constrained embeddings. It also clarifies the interaction between ambient geometry and intrinsic curve deformations via residue sequences and Serre duality, enabling maximal IVHS results across dimensions and codimensions with high geometric relevance.
Abstract
We study infinitesimal Torelli problems and infinitesimal variations of Hodge structure for families of curves arising in singular and extrinsically constrained geometric settings. Motivated by the Green--Voisin philosophy, we develop an explicit approach based on Poincaré residue calculus, allowing a uniform treatment of smooth, singular, and equisingular situations. In particular, we prove infinitesimal Torelli theorems for general equisingular plane curves of sufficiently high degree and construct relative IVHS exact sequences for curves lying on smooth projective threefolds. Our results show that maximal infinitesimal variation of Hodge structure persists even after imposing strong extrinsic conditions, such as fixed degree and prescribed singularities, and in the presence of isolated planar singularities. The methods presented here provide a concrete and geometric realization of Jacobian-type constructions and extend the Green--Voisin philosophy to singular and equisingular settings and provide a unified residue--theoretic framework for Torelli--type problems across dimensions and codimensions.