A Conditional Reduction of the Rational Hodge Conjecture for Threefolds and Deformation-Theoretic Verifications in Several Families
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| Format: | Preprint |
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2025
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| _version_ | 1866918122618880000 |
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| author | Mansour, Karim |
| author_facet | Mansour, Karim |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Conditional Reduction of the Rational Hodge Conjecture for Threefolds and Deformation-Theoretic Verifications in Several Families Mansour, Karim Algebraic Geometry 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. |
| title | A Conditional Reduction of the Rational Hodge Conjecture for Threefolds and Deformation-Theoretic Verifications in Several Families |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2508.08321 |