| _version_ | 1866901608336457728 |
|---|---|
| author | Ross, Logan |
| author_facet | Ross, Logan |
| contents | <p>The Hodge conjecture is a treewidth surjection. A smooth projective variety X carries two treewidths: the algebraic intersection graph (shadow, treewidth O(n)) and the Hodge coupling graph (mirror, treewidth Ω(h^{p,p})). The Hodge-Riemann inner product on H^{p,p}(X,R) provides the natural Hilbert space — algebraic classes and transcendental classes are orthogonal complements, and V² + D² = 1 follows from the Pythagorean theorem applied to this decomposition. The Hodge conjecture states that for every rational (p,p)-class, D = 0: every rational Hodge class is algebraic. Two theorems establish the treewidth hierarchy: algebraic cycles give bounded treewidth, Hodge classes give unbounded treewidth, and the Lefschetz coupling graph connects them. The article identifies three open steps and names the computation on Calabi-Yau threefolds that falsifies the treewidth mechanism.</p> <p>See also: "The Equals Sign as a Rotation" (DOI: 10.5281/zenodo.19412721), "The Birch and Swinnerton-Dyer Conjecture as Treewidth Equality" (DOI: 10.5281/zenodo.19448933).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19448950 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Hodge Conjecture as Treewidth Surjection Ross, Logan Treewidth Hodge <p>The Hodge conjecture is a treewidth surjection. A smooth projective variety X carries two treewidths: the algebraic intersection graph (shadow, treewidth O(n)) and the Hodge coupling graph (mirror, treewidth Ω(h^{p,p})). The Hodge-Riemann inner product on H^{p,p}(X,R) provides the natural Hilbert space — algebraic classes and transcendental classes are orthogonal complements, and V² + D² = 1 follows from the Pythagorean theorem applied to this decomposition. The Hodge conjecture states that for every rational (p,p)-class, D = 0: every rational Hodge class is algebraic. Two theorems establish the treewidth hierarchy: algebraic cycles give bounded treewidth, Hodge classes give unbounded treewidth, and the Lefschetz coupling graph connects them. The article identifies three open steps and names the computation on Calabi-Yau threefolds that falsifies the treewidth mechanism.</p> <p>See also: "The Equals Sign as a Rotation" (DOI: 10.5281/zenodo.19412721), "The Birch and Swinnerton-Dyer Conjecture as Treewidth Equality" (DOI: 10.5281/zenodo.19448933).</p> |
| title | The Hodge Conjecture as Treewidth Surjection |
| topic | Treewidth Hodge |
| url | https://doi.org/10.5281/zenodo.19448950 |