Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916863503499264 |
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| author | Mounda, Badre |
| author_facet | Mounda, Badre |
| contents | We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $α\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a one-parameter family of quartic $K3$'s acquiring at most ten $A_1$-nodes. On the central fibre $\widetilde{X}_0$, the class $α$ specializes to a $\mathbb{Q}$-linear combination of the hyperplane class and the exceptional $(-2)$-curves coming from the blow-ups of the nodes. Using the Clemens--Schmid sequence together with Picard--Lefschetz theory, we identify $\Gr^W_2 H^2_{\lim}\cong H^2(\widetilde{X}_0)$ and transport this combination back to the original smooth surface as an algebraic divisor. This yields an explicit, finite-step procedure that realizes any rational $(1,1)$-class by an algebraic cycle. We also formulate an equivariant extension for $(2,2)$-classes on Calabi--Yau threefolds, indicating how the same strategy might apply in higher dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations Mounda, Badre Algebraic Geometry Complex Variables 14D07 (Primary), 14J28, 32G20, 14C30 (Secondary) We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $α\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a one-parameter family of quartic $K3$'s acquiring at most ten $A_1$-nodes. On the central fibre $\widetilde{X}_0$, the class $α$ specializes to a $\mathbb{Q}$-linear combination of the hyperplane class and the exceptional $(-2)$-curves coming from the blow-ups of the nodes. Using the Clemens--Schmid sequence together with Picard--Lefschetz theory, we identify $\Gr^W_2 H^2_{\lim}\cong H^2(\widetilde{X}_0)$ and transport this combination back to the original smooth surface as an algebraic divisor. This yields an explicit, finite-step procedure that realizes any rational $(1,1)$-class by an algebraic cycle. We also formulate an equivariant extension for $(2,2)$-classes on Calabi--Yau threefolds, indicating how the same strategy might apply in higher dimension. |
| title | Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations |
| topic | Algebraic Geometry Complex Variables 14D07 (Primary), 14J28, 32G20, 14C30 (Secondary) |
| url | https://arxiv.org/abs/2507.18999 |