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

TL;DR

This paper demonstrates that the Griffiths groups of certain Fano manifolds of Calabi-Yau Hodge type are infinitely generated by extending Voisin's method and exploring noncommutative Calabi-Yau 3-folds.

Contribution

It proves the infinite generation of Griffiths groups for all three classes of smooth FCY manifolds, including the two remaining cases, using noncommutative geometry techniques.

Findings

Griffiths groups of cubic 7-folds are infinitely generated.

Griffiths groups of fivefold quartic double solids are infinitely generated.

Existence of noncommutative CY 3-folds with isomorphic Griffiths groups for these manifolds.

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.