MOTIVIC DECOMPOSITION AND THE HODGE CONJECTURE: A PROOF VIA K3 SURFACES AND LOGICAL CALABI-YAU MANIFOLDS
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2026
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| _version_ | 1866901101335281664 |
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| author | Valeri Vukolov |
| author_facet | Valeri Vukolov |
| contents | <p>We prove the Hodge conjecture for all smooth projective complex manifolds. The proof unifies three fundamental streams:</p> <p> (1) a logical-geometric construction based on the principle I2C = −Id that produces a class of Calabi-Yau manifolds M2n;</p> <p>(2) Shioda’s 1974 theorem for K3 surfaces; and</p> <p>(3) the theory of motives and the Minimal Model Program.<br>The logical principle I2C = −Id arises from minimizing deviation in cycles of interpretations between formal systems, providing a complex structure on an extended type space ˜X ∼= Ei × K. From this geometry we construct explicit projective manifolds M2n<br>and prove, via a variational principle for torsion and the identification of gauge fields with harmonic forms, that every rational Hodge class on M2n is algebraic — represented by geometric cycles and by worldvolumes of BPS solitons.<br>Using the Minimal Model Program and motivic methods, we prove a universal decomposition theorem: every projective manifold X admits a motivic decomposition into motives of K3 surfaces, M2n manifolds, and Tate motives. The Hodge realization functor is additive, so any Hodge class on X decomposes into Hodge classes on these building<br>blocks. By Shioda’s theorem and our result for M2n, each such class is algebraic. Hence every Hodge class on X is algebraic.<br>This completes the proof of the Hodge conjecture, one of the seven Millennium Prize Problems. The proof is constructive: algebraic cycles are built from divisors on K3 surfaces and from BPS solitons on M2n manifolds, providing a physical interpretation<br>of Hodge classes as worldvolumes of topological solitons.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18987554 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | MOTIVIC DECOMPOSITION AND THE HODGE CONJECTURE: A PROOF VIA K3 SURFACES AND LOGICAL CALABI-YAU MANIFOLDS Valeri Vukolov Hodge Conjecture, Motivic Decomposition, K3 Surfaces, Calabi-Yau Manifolds, Logical Imaginary Unit, BPS Solitons, Minimal Model Program, Chow Motives, Algebraic Cycles, Torsion, Gauge Fields, Yang-Mills Equations, Instantons, Monopoles, Elliptic Curves, Noncommutative Geometry, Göttingen Circle, Millennium Prize Problems, Shioda's Theorem, Twisted Product, Group Actions, Projective Varieties, Hodge Classes, Lefschetz Theorem, Bogomol'nyi Equations, Harmonic Forms, Chern Classes, Moduli Spaces, String Theory, Quantum Geometry <p>We prove the Hodge conjecture for all smooth projective complex manifolds. The proof unifies three fundamental streams:</p> <p> (1) a logical-geometric construction based on the principle I2C = −Id that produces a class of Calabi-Yau manifolds M2n;</p> <p>(2) Shioda’s 1974 theorem for K3 surfaces; and</p> <p>(3) the theory of motives and the Minimal Model Program.<br>The logical principle I2C = −Id arises from minimizing deviation in cycles of interpretations between formal systems, providing a complex structure on an extended type space ˜X ∼= Ei × K. From this geometry we construct explicit projective manifolds M2n<br>and prove, via a variational principle for torsion and the identification of gauge fields with harmonic forms, that every rational Hodge class on M2n is algebraic — represented by geometric cycles and by worldvolumes of BPS solitons.<br>Using the Minimal Model Program and motivic methods, we prove a universal decomposition theorem: every projective manifold X admits a motivic decomposition into motives of K3 surfaces, M2n manifolds, and Tate motives. The Hodge realization functor is additive, so any Hodge class on X decomposes into Hodge classes on these building<br>blocks. By Shioda’s theorem and our result for M2n, each such class is algebraic. Hence every Hodge class on X is algebraic.<br>This completes the proof of the Hodge conjecture, one of the seven Millennium Prize Problems. The proof is constructive: algebraic cycles are built from divisors on K3 surfaces and from BPS solitons on M2n manifolds, providing a physical interpretation<br>of Hodge classes as worldvolumes of topological solitons.</p> |
| title | MOTIVIC DECOMPOSITION AND THE HODGE CONJECTURE: A PROOF VIA K3 SURFACES AND LOGICAL CALABI-YAU MANIFOLDS |
| topic | Hodge Conjecture, Motivic Decomposition, K3 Surfaces, Calabi-Yau Manifolds, Logical Imaginary Unit, BPS Solitons, Minimal Model Program, Chow Motives, Algebraic Cycles, Torsion, Gauge Fields, Yang-Mills Equations, Instantons, Monopoles, Elliptic Curves, Noncommutative Geometry, Göttingen Circle, Millennium Prize Problems, Shioda's Theorem, Twisted Product, Group Actions, Projective Varieties, Hodge Classes, Lefschetz Theorem, Bogomol'nyi Equations, Harmonic Forms, Chern Classes, Moduli Spaces, String Theory, Quantum Geometry |
| url | https://doi.org/10.5281/zenodo.18987554 |