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This is a Quantitative Study study.

October 2, 20250 citationsOpen Access

A Deformation Theoretic Reduction of the Hodge Conjecture via Derived Categories and Complete Intersections

KMKarim MansourUniversity Frères Mentouri Constantine 1

Key Points

The proposed approach proves the Hodge conjecture unconditionally for complete intersections, and indicates a method to tackle the general case.
Derived category methods are employed, linking Hodge theory and deformation in a unified framework, which is significant for understanding complex varieties.
Deformation arguments are used to connect the general case to the algebraicity of limits, although this remains an assumption in the study.
This framework provides a concrete path toward resolving the Hodge conjecture, highlighting the importance of geometric and categorical strategies.

Abstract

We propose a geometric and categorical approach to the Hodge Conjecture for all smooth projective complex varieties. By embedding any such variety into a flat family with general fibers smooth complete intersections, we prove the conjecture unconditionally in the complete intersection case using derived category methods. Through deformation arguments, we reduce the general case to the algebraicity of limits. While this remains an assumption, our framework unifies Hodge theory, deformation, and derived techniques, offering a concrete path toward resolving the conjecture.

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Karim Mansour (Mon,) studied this question.

synapsesocial.com/papers/68de5da783cbc991d0a20d45 https://doi.org/https://doi.org/10.48550/arxiv.2507.09934