Residues and Infinitesimal Torelli for Equisingular Curves

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.