A Conditional Reduction of the Rational Hodge Conjecture for Threefolds and Deformation-Theoretic Verifications in Several Families

TL;DR

This paper proposes a geometric approximation hypothesis that links Hodge classes on threefolds to complete-intersection curves, proving it implies the rational Hodge conjecture for threefolds and verifying it in specific families using deformation theory.

Contribution

It introduces Hypothesis~BB connecting Hodge classes to complete-intersection curves and provides deformation-theoretic criteria to verify this hypothesis in various threefold families.

Findings

Hypothesis~BB implies the rational Hodge conjecture for threefolds.

Deformation criteria are sufficient to verify Hypothesis~BB in specific cases.

Macaulay2 scripts facilitate explicit cohomology and splitting condition checks.

Abstract

We formulate a concrete geometric approximation hypothesis (Hypothesis~BB) asserting that codimension-$2$ Hodge classes on a smooth projective threefold can be realized as specializations of families whose general members are complete-intersection curves. We prove that Hypothesis~BB implies the (rational) Hodge conjecture for the threefold. We then give deformation-theoretic sufficient criteria (cohomology-vanishing and surjectivity conditions) which imply Hypothesis~BB, and we prove these criteria hold for the class of a line on a \emph{general} quintic threefold containing that line. We further formulate and prove several propositions showing that, under natural Noether--Lefschetz and unobstructedness hypotheses, Hypothesis~BB holds \emph{generically} in families of Calabi--Yau and Fano threefolds; these propositions reduce the problem to checkable conditions (normal-bundle cohomology, surjectivity of restriction maps). Finally, we include a Macaulay2 appendix with scripts to verify the required cohomology and splitting conditions in explicit examples.