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

TL;DR

This paper introduces a geometric and categorical framework that reduces the Hodge Conjecture to the algebraicity of limits, proving it unconditionally for smooth complete intersections using derived categories.

Contribution

It presents a novel deformation theoretic approach that unifies Hodge theory, deformation, and derived categories to address the Hodge Conjecture.

Findings

Proves the Hodge Conjecture for smooth complete intersections unconditionally.

Reduces the general case of the conjecture to the algebraicity of limits.

Provides a new framework linking deformation theory and derived categories.

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.