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

TL;DR

This paper advances the understanding of infinitesimal invariants in mixed Hodge theory, focusing on log Calabi-Yau pairs, Abel-Jacobi maps, and a log Nori connectivity theorem, with applications to the Clemens conjecture.

Contribution

It introduces new infinitesimal invariants for mixed Hodge structures, proves unobstructedness results for log Calabi-Yau pairs, and establishes a logarithmic Nori connectivity theorem.

Findings

Unobstructedness of log Calabi-Yau pairs under Fano hypotheses

Duality between deformations and obstructions in the log Calabi-Yau setting

A logarithmic Nori connectivity theorem for open hypersurfaces

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.