Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909720391974912 |
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| author | Hajebi, Bita Hajebi, Pooya |
| author_facet | Hajebi, Bita Hajebi, Pooya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12173 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture Hajebi, Bita Hajebi, Pooya Algebraic Geometry 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. |
| title | Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.12173 |