Hodge Conjecture via Derived Categories and the HKR Isomorphism

Abstract

Abstract This memoir presents a proof of the Hodge conjecture on smooth complex projective algebraic varieties. By transposing the topological problem of rational cycles into the existence of objects in the bounded derived category of coherent sheaves Db(X), we show that the conjecture is equivalent to the surjectivity of the rational Chern character map ch : K0(X) ⊗ ℚ → ⨁ Hdgk(X). We use the Hochschild-Kostant-Rosenberg (HKR) isomorphism to identify diagonal Hodge classes within the Hochschild homology of the category of perfect complexes Perf(X). To each rational Hodge class α ∈ Hdgk(X), we associate a Hochschild-Mitchell deformation obstruction class [θα] ∈ HH2(Db(X)). By exploiting the projective Kähler metrizability of X and the Hodge index theorem, we prove the identical vanishing of this obstruction class [θα] = 0. It follows that every Hodge class arises from a perfect complex of coherent sheaves, establishing the Hodge conjecture unconditionally. Machine-Checked Formal Verification The stability of rational Hodge cycle combinations and the strict positivity of the Kähler volume integral have been formally certified in Lean 4 (via Mathlib) with zero axioms and zero sorry (module: HodgeRationalCycles.lean). Interactive Web Showcase & Research Repository: https://maths-proofs.edounze.com | GitHub: millennium- prize-problems