Pure and Applied Mathematics Quarterly

Volume 10 (2014)

Number 1

Special Issue: In Memory of Andrey Todorov, Part 2 of 3

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

Pages: 1 – 55

DOI: http://dx.doi.org/10.4310/PAMQ.2014.v10.n1.a1

Authors

David Favero (Fakultät für Mathematik, Universität Wien, Austria)

Atanas Iliev (Department of Mathematics, Seoul National University, Seoul, Korea)

Ludmil Katzarkov (Department of Mathematics, University of Miami, Florida, U.S.A.; Fakultät für Mathematik, Universität Wien, Austria)

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. 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 $X_4$, 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 $X_4$, we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for $X_{2.3}$ there is a noncommutative CY 3-fold, $\mathcal{B}$, such that the Griffiths group of $X_{2.3}$ surjects on to the Griffiths group of $\mathcal{B}$. 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.

Keywords

Hodge theory, algebraic cycles, Calabi-Yau geometries, derived categories

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