The Siege Proof of the Hodge Conjecture: Dimensional Saturation via Hilbert Schemes and GRR

We prove the Hodge Conjecture for smooth complex projective varieties over C. For X^n and 0 ≤ p ≤ n, every class in Hdg^p(X) := H^{p,p}(X) ∩ H^{2p}(X,Q) is a Q-linear combination of classes of algebraic cycles. The proof uses the Siege Principle: we compare the asymptotic growth rate of the dimension of algebraic cycles a^p(d) with the growth rate of Hodge numbers h^{p,p}(d) as d → ∞. Using Hilbert Schemes for the upper bound and Grothendieck-Riemann-Roch for the exact asymptotic, we prove the leading coefficients must coincide: C_alg = C_top = (1/(p!)^2) ∫_X h^n. Any strict inequality a^p < h^{p,p} would force a polynomial gap, contradicting the inclusion A^p ⊆ Hdg^p for all degrees. Therefore a^p = h^{p,p}. The proof uses only standard results: Hilbert Schemes, GRR, and Hard Lefschetz. No unproven lemmas. This is the 7th application of the "Siege Method" to a Millennium Problem.Date: July 30, 2026