Constructive Degenerations and the Algebraicity of Limiting Hodge
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918098907430912 |
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| author | Mounda, Badre |
| author_facet | Mounda, Badre |
| contents | We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth projective variety becomes algebraic in the limit of a semi-stable degeneration. We provide examples where vanishing cycles and monodromy explicitly generate new algebraic classes, and propose a general principle: every rational (p, p) class arises as the limit of algebraic cycles under controlled geometric degenerations. This viewpoint opens a new path toward an effective formulation of the Hodge conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructive Degenerations and the Algebraicity of Limiting Hodge Mounda, Badre Algebraic Geometry We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth projective variety becomes algebraic in the limit of a semi-stable degeneration. We provide examples where vanishing cycles and monodromy explicitly generate new algebraic classes, and propose a general principle: every rational (p, p) class arises as the limit of algebraic cycles under controlled geometric degenerations. This viewpoint opens a new path toward an effective formulation of the Hodge conjecture. |
| title | Constructive Degenerations and the Algebraicity of Limiting Hodge |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.15012 |