Spectral Analysis of Hodge Cycles: A Novel Approach to the Hodge Conjecture via Generalized Moments
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916878741405696 |
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| author | Hajebi, Bita Hajebi, Pooya |
| author_facet | Hajebi, Bita Hajebi, Pooya |
| contents | The Hodge Conjecture, posits a profound connection between the topology and algebraic geometry of complex algebraic varieties. It asserts that Hodge cycles, specific elements in the cohomology of a Kähler variety with rational properties, originate from algebraic subvarieties. This paper introduces a novel approach to investigate this conjecture by generalizing the concept of Zernike moments through the lens of harmonic analysis and spectral geometry. Our core idea involves defining a ``characteristic form'' $η_Z$ for a Hodge cycle $Z$ within a Kähler variety $X$, and expanding this form in terms of the eigenfunctions of the Laplace-Beltrami operator on $Z$. We hypothesize that for algebraic Hodge cycles, the coefficients of this spectral expansion (termed ``spectral fingerprints'') will exhibit specific algebraic patterns, such as being rational numbers, algebraic numbers, or algebraic functions of moduli parameters. We illustrate the computational methodology with a simplified ``toy model'' using a characteristic function on a torus, demonstrating how such coefficients can indeed be rational in a controlled setting. We then outline a conceptual framework for applying this approach to more complex scenarios, specifically K3 surfaces, by leveraging the theory of variations of Hodge structures and moduli spaces to define ``dynamic characteristic forms'' and analyze the algebraic nature of their coefficients. This framework promises to open new avenues in understanding the Hodge Conjecture by translating a deep geometric problem into a question about the algebraic properties of spectral data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Analysis of Hodge Cycles: A Novel Approach to the Hodge Conjecture via Generalized Moments Hajebi, Bita Hajebi, Pooya Algebraic Geometry The Hodge Conjecture, posits a profound connection between the topology and algebraic geometry of complex algebraic varieties. It asserts that Hodge cycles, specific elements in the cohomology of a Kähler variety with rational properties, originate from algebraic subvarieties. This paper introduces a novel approach to investigate this conjecture by generalizing the concept of Zernike moments through the lens of harmonic analysis and spectral geometry. Our core idea involves defining a ``characteristic form'' $η_Z$ for a Hodge cycle $Z$ within a Kähler variety $X$, and expanding this form in terms of the eigenfunctions of the Laplace-Beltrami operator on $Z$. We hypothesize that for algebraic Hodge cycles, the coefficients of this spectral expansion (termed ``spectral fingerprints'') will exhibit specific algebraic patterns, such as being rational numbers, algebraic numbers, or algebraic functions of moduli parameters. We illustrate the computational methodology with a simplified ``toy model'' using a characteristic function on a torus, demonstrating how such coefficients can indeed be rational in a controlled setting. We then outline a conceptual framework for applying this approach to more complex scenarios, specifically K3 surfaces, by leveraging the theory of variations of Hodge structures and moduli spaces to define ``dynamic characteristic forms'' and analyze the algebraic nature of their coefficients. This framework promises to open new avenues in understanding the Hodge Conjecture by translating a deep geometric problem into a question about the algebraic properties of spectral data. |
| title | Spectral Analysis of Hodge Cycles: A Novel Approach to the Hodge Conjecture via Generalized Moments |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.04089 |