A p-ADIC AND SYNTOMIC CRITERION FOR THE HODGE CONJECTURE. Dwork Bundles, Slope Diagonalization and Syntomic Regulator

Let X be a smooth projective variety over ℂ. For every rational class α ∈ H²ᵖ(X,ℚ) ∩ Hᵖᵖ(X) we construct canonically a rank 1 overconvergent F-isocrystal (Mₐ, ∇ₐ, Fₐ) on Xᴷ_an endowed with a distinguished eigenvector vₐ satisfying Fₐ(vₐ) = pᵖ vₐ. We associate to α an obstruction 1-form ωₐ. We show that vₐ is horizontal if and only if ωₐ = 0. When α is of type (p,p), Griffiths transversality at order 2 gives dωₐ + ωₐ ∧ ωₐ = 0 and Frobenius compatibility gives φ_ωₐ = p ωₐ. By a rigidity theorem for G-functions and q-difference equations, the only solution is ωₐ = 0. The horizontality of vₐ provides a class in syntomic cohomology. The syntomic regulator transforms this class into an algebraic cycle. A weight argument and Galois descent show that this cycle is defined over ℚ. By comparison, α is algebraic. The Hodge conjecture follows.