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. 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 X 3 , 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, X 4 and X 2.3 , is also infinitely generated.Received December 11, 2012. David Favero, Atanas Iliev and Ludmil KatzarkovIn 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, B, such that the Griffiths group of X 2.3 surjects 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.

Double Solids, Categories and Non-Rationality

“…The paper is based on examples we have analysed in [3,11,24,25,27]. All these suggest a direct connection between the monodromy of Landau-Ginzburg models, spectra and wall crossings in the moduli space of stability conditions, which was partially explored in [17].…”

A category of kernels for equivariant factorizations, II: Further implications

“…We also believe that in the case of a curve, graded Knörrer periodicity should be related to the classical Prym variety construction of D. Mumford [Mum74] for Jacobians of conic bundles by work of A. Kuznetsov in [Kuz08, Kuz09]. Building on the seminal work of C. Voisin, see [Vos00], this picture is also readily applicable to the study of Griffiths groups, see [FIK12].…”

“…We also believe that in the case of a curve, graded Knörrer periodicity should be related to the classical Prym variety construction of D. Mumford [Mum74] for Jacobians of conic bundles by work of A. Kuznetsov in [Kuz08, Kuz09]. Building on the seminal work of C. Voisin, see [Vos00], this picture is also readily applicable to the study of Griffiths groups, see [FIK12].In the context of homological algebra and triangulated categories, the precise relationship between the quotients arising in the HMS picture above can be expressed in the language of orbit categories. Inspired further by the ideas of B. Keller, D. Murfet, and M. Van den Bergh in [KMV11], we provide this description in Section 4.This categorical/geometric picture is readily applicable to the notions of Rouquier dimension, Orlov spectra, and generation time -an impetus of this work.…”

Nested varieties of K3 type

“…Fano varieties of K3 type have recently been investigated because of their potential relations with hyperKähler manifolds [10,13,19]. More generally, Fano varieties of Calabi-Yau type are endowed with special Hodge structures which can sometimes be mapped, through adequate correspondences, to auxiliary manifolds, or, more generally, used to obtain geometrical information on the variety, either of cycle-theoretical nature (see [16] for cubic fourfolds and [14] for Griffiths groups) or on moduli spaces (see [10]). In some cases these manifolds are genuine K3 surfaces or Calabi-Yau manifolds.…”

“…We do this in the most general setting possible, and then specialize to the case of Gr(3, n) to relate their hyperplane sections to congruences of planes and lines. We describe the details for the K3 case, that is, Gr (3,10), in Section 4, building upon the results from previous sections and diagram (14), separating Hodge theoretical and categorical construction. Some technical results as the calculation of normal bundle of special loci are postponed to the last subsection of Section 4.…”