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arXiv · 2507.12173

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

Bita Hajebi·Pooya Hajebi
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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.

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Bita Hajebi, Pooya Hajebi. 2025-07-16. Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture. https://arxiv.org/abs/2507.12173

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