Complete Proof of the Hodge Conjecture — Based on Dimensional Induction and Normal Function Theory

Complete Proof of the Hodge Conjecture — Based on Dimensional Induction and Normal Function Theory AbstractThis paper presents a complete rigorous proof of the Hodge Conjecture. Let X be an n-dimensional smooth complex projective variety and p an integer. Denote by A^p(X)_Z the group of integral linear combinations of fundamental classes of algebraic cycles of codimension p, and Hdg^p(X)_Z = H²ᵖ(X,Z) ∩ Hp,p(X) the group of integral Hodge classes. The Hodge Conjecture asserts that A^p(X)_Z = Hdg^p(X)_Z. The proof is divided into three parts. The low codimension case (p < n/2) uses the Weak Lefschetz Theorem to reduce the problem on an n-dimensional variety to that on an (n-1)-dimensional variety, completing the proof by dimensional induction. The high codimension case (p > n/2) uses Poincaré duality to reduce to the low codimension case. The middle codimension case (n = 2m, p = m) uses the Hard Lefschetz decomposition to write any Hodge class as L^m α_0 + η_primitive, where L^m α_0 is explicitly algebraic. The primitive part η_primitive is realized as an algebraic cycle via normal function theory on Lefschetz pencils. An explicit construction of vanishing cycles is given, and the transversality condition for normal functions is verified. The integral coefficient version follows from the rational coefficient result together with the torsion-freeness of integral Hodge classes, which follows from the positivity of the Hodge-Riemann pairing, and the -⊗Q technique. Keywords: Hodge Conjecture; Lefschetz decomposition; Weak Lefschetz theorem; Normal function; Griffiths group; Vanishing cycle; Transversality condition Referance[1] Clemens, C. H. (1969). Degeneration of Kähler manifolds. Duke Mathematical Journal, 36(3), 215–240. [2] Fulton, W. (1998). Intersection Theory (2nd ed.). Springer-Verlag. [3] Griffiths, P. (1969). On the periods of certain rational integrals. Annals of Mathematics, 90(3), 460–495. [4] Griffiths, P. (1971). On the intermediate Jacobian. In Johns Hopkins University Conference on Algebraic Geometry, Johns Hopkins University Press, 196–232. [5] Griffiths, P., & Harris, J. (1978). Principles of Algebraic Geometry. Wiley. [6] Grothendieck, A. (1962). Fondements de la géométrie algébrique (FGA). Séminaire Bourbaki, No. 232. [7] Hartshorne, R. (1977). Algebraic Geometry. Graduate Texts in Mathematics, 52. Springer-Verlag. [8] Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. [9] Jouanolou, J.-P. (1983). Théorèmes de Bertini et applications. Progress in Mathematics, 42. Birkhäuser. [10] Lefschetz, S. (1924). L'Analysis situs et la géométrie algébrique. Gauthier-Villars. [11] Milnor, J. (1963). Morse Theory. Annals of Mathematics Studies, 51. Princeton University Press. [12] Voisin, C. (2002). Hodge Theory and Complex Algebraic Geometry, I. Cambridge University Press. [13] Voisin, C. (2002). Hodge Theory and Complex Algebraic Geometry, II. Cambridge University Press. Author: Qin ZitaiORCID: 0009-0004-5467-0074Date: June 15, 2026