Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture

TL;DR

This paper introduces a new symbolic analytic framework using the Hermitian spectral fingerprint to connect the vanishing of certain invariants with the algebraicity of Hodge classes, aiming to resolve the Hodge Conjecture.

Contribution

It develops a refined invariant and proof strategy linking the vanishing of this invariant to absolute Hodge classes, offering a novel approach to the Hodge Conjecture.

Findings

The Hermitian spectral fingerprint vanishes for rational classes of type (k,k).

Vanishing of the refined fingerprint across all realization functors implies the class is absolute Hodge.

The framework provides a new criterion for detecting algebraic cycles.

Abstract

This paper presents a novel symbolic analytic framework to address the Hodge Conjecture, utilizing a refined invariant called the Hermitian spectral fingerprint. We modify the fingerprint functional to specifically exclude $(k,k)$ components, demonstrating its vanishing for rational classes of type $(k,k)$. Critically, we develop a comprehensive proof strategy to establish the converse: the vanishing of this refined fingerprint across all realization functors (de Rham and $\ell$adic) implies the class is absolute Hodge. By fundamental theorems in arithmetic algebraic geometry, absolute Hodge classes of type $(k,k)$ are equivalent to algebraic cycles. This framework offers a new, robust criterion for detecting algebraic cycles, reformulating the conjecture into a problem of establishing the exhaustive spanning properties of GaussManin derivatives and Galois actions within their respective cohomology spaces. While building upon established deep results, this approach provides a fresh perspective and a pathway towards a complete resolution.