Infinitesimal invariants of mixed Hodge structures II: Log Clemens conjecture and log connectivity

Abstract: Following previous work, we continue the study of infinitesimal methods in mixed Hodge theory. In the first part, inspired by the deformation theory of curves on Calabi-Yau threefolds, we study deformations of smooth $\mathbb{Q}$-log Calabi-Yau pairs $(X,Y)$. We prove unobstructedness results for these pairs under Fano hypotheses. We define families of infinitesimal Abel-Jacobi maps associated with these deformation problems and show that they control the first-order deformations of smooth curves embedded in the pair. Crucially, for the $\frac{1}{2}$-log Calabi-Yau case, we establish an exact duality between deformations and obstructions, recovering the symmetry found in the absolute Calabi-Yau setting. We apply this framework to the cubic threefold, proposing a relative generalization of the Clemens conjecture regarding the injectivity of the infinitesimal Abel-Jacobi map, and establishing a criterion for its non-vanishing. In the second part, we define infinitesimal invariants for normal functions using extension classes and the log-Leray filtration. Relying on the theory of generalized Jacobian rings developed by Asakura and Saito, we prove a logarithmic Nori connectivity theorem for the universal family of open hypersurfaces, we also deduce a sharp algebraic criterion for the properness of the Hodge loci for open hypersurfaces, generalizing the proof of Carlson-Green-Griffiths-Harris.