The Griffiths Group of a General Calabi-yau Threefold Is Not Finitely Generated

Fn−k+1H 2n−2k+1(X)∼= F n−k+1A2n−2k+1(X)c dFn−k+1A2n−2k(X) . If (Zt )t∈C is a flat family of codimension k algebraic cycles on X parametrized by a smooth irreducible curve C, the map t → kX(Zt − Z0) factors through a homomorphism from the Jacobian J (C) to J 2k−1(X), and one can show that the image of this morphism is a complex subtorus of J 2k−1(X) whose tangent space is contained in Hk−1,k(X) ⊂ H 2k−1(X,C)/F H 2k−1(X). Defining the subgroup