On the Griffiths Groups of Fano Manifolds of Calabi-Yau Hodge Type

Abstract: A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a noncommutative CY 3-fold which has an isomorphic Griffiths group. Similarly, one can consider other examples of Fano manifolds with with the same type of variation of Hodge structure as a Calabi-Yau threefold (FCYs). Among the complete intersections in weighted projective spaces, there are only three classes of smooth FCY manifolds; the cubic 7-fold, the fivefold quartic double solid, and the fivefold intersection of a quadric and a cubic. We settle the two remaining cases, following Voisin's method to demonstrate that the Griffiths group for a smooth general complete intersection FCY manifolds, is also infinitely generated. In the case of the fivefold quartic double solid, we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for the fivefold intersection of a quadric and a cubic there is a noncommutative CY 3-fold, such that the Griffiths group of the intersection surjects on to the Griffiths group of noncommutative 3-fold. We finish by discussing some examples of noncommutative covers which relate our noncommutative CYs back to honest algebraic varieties such as products of elliptic curves and K3-surfaces.