On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type

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. This serves as partial confirmation of seminal work of Candelas, Derrick, and Parkes describing a cubic 7-fold as a mirror to a rigid CY 3-fold.

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 X₃, the fivefold quartic double solid X₄, and the fivefold intersection of a quadric and a cubic X_2.3.We settle the two remaining cases, following Voisin’s method to demonstrate that the Griffiths group for a general complete intersection FCY manifolds, X4 and X2.3, is also infinitely generated.

In the case of X₄, we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for X_2.3there is a noncommutative CY 3-fold, B, such that the Griffiths group of X_2.3surjects on to the Griffiths group of B. We finish by discussing some examples of noncommutative covers which relate our noncommutative CYs back toq honest algebraic varieties such as products of elliptic curves and K3-surfaces.