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

Rodolfo Aguilar

TL;DR

The paper develops infinitesimal methods for mixed Hodge structures in the logarithmic setting, extending Abel–Jacobi theory to smooth $\mathbb{Q}$-log Calabi--Yau pairs and establishing dualities that recover Calabi--Yau symmetry in the $m=2$ case. It proves unobstructedness of certain deformations under Fano hypotheses, analyzes relative Clemens-type injectivity via log Abel–Jacobi maps on cubic threefolds, and introduces logarithmic invariants for families of pairs through the log Leray filtration. In the second part, it proves a logarithmic Nori connectivity theorem for the universal open hypersurfaces and provides algebraic criteria for the properness of log Hodge loci via generalized Jacobian rings, enriching the link between deformation theory, log geometry, and Hodge theory. Together these results tie the geometry of curves and open hypersurfaces to infinitesimal Hodge-theoretic data, with potential consequences for Clemens-type conjectures and the structure of Hodge loci in log settings.

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.

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

TL;DR

The paper develops infinitesimal methods for mixed Hodge structures in the logarithmic setting, extending Abel–Jacobi theory to smooth -log Calabi--Yau pairs and establishing dualities that recover Calabi--Yau symmetry in the case. It proves unobstructedness of certain deformations under Fano hypotheses, analyzes relative Clemens-type injectivity via log Abel–Jacobi maps on cubic threefolds, and introduces logarithmic invariants for families of pairs through the log Leray filtration. In the second part, it proves a logarithmic Nori connectivity theorem for the universal open hypersurfaces and provides algebraic criteria for the properness of log Hodge loci via generalized Jacobian rings, enriching the link between deformation theory, log geometry, and Hodge theory. Together these results tie the geometry of curves and open hypersurfaces to infinitesimal Hodge-theoretic data, with potential consequences for Clemens-type conjectures and the structure of Hodge loci in log settings.

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 -log Calabi-Yau pairs . 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 -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.
Paper Structure (41 sections, 42 theorems, 211 equations)

This paper contains 41 sections, 42 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Sections
  5. Section \ref{['S:infAJ']}
  6. Section \ref{['S:Dualities']}
  7. Section \ref{['S:infAJCri']}
  8. Section \ref{['S:NorFun']}
  9. Section \ref{['S:logConnect']}
  10. Section \ref{['S:RelNL']}
  11. Infinitesimal log Abel-Jacobi
  12. Projective case
  13. Restriction
  14. Kodaira-Spencer type map
  15. Dualities for threefolds
  16. ...and 26 more sections

Key Result

Theorem 1.1

Suppose $X$ is Fano. Then: (A) If $s_j\ge 1$ (i.e. $\mathscr{O}_X(s_jY)$ is ample), then (B) If one of the equalities holds, then for the corresponding $j\in\{1,2,3\}$ In particular, in all cases $H^2(X,S_j)=0$, which implies that the deformations associated with $S_j$ are unobstructed.

Theorems & Definitions (74)

  • Theorem 1.1: Theorem \ref{['thm:vanishing']}
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Theorem \ref{['thm:duality']}
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7: Zariski thm of Lefschetz type
  • Theorem 1.8: Log Connectivity
  • Theorem 1.9
  • Proposition 3.1
  • ...and 64 more