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 X4, and the fivefold intersection of a quadric and a cubic X2.3.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 X2.3there is a noncommutative CY 3-fold, B, such that the Griffiths group of X2.3surjects 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-55 |
| Number of pages | 55 |
| Journal | Pure and Applied Mathematics Quarterly |
| Volume | 10 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2014 |
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Keywords
- Algebraic cycles
- Calabi-Yau geometries
- Derived categories
- Hodge theory
ASJC Scopus subject areas
- Mathematics(all)
Cite this
On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type. / Favero, David; Iliev, Atanas; Katzarkov, Ludmil.
In: Pure and Applied Mathematics Quarterly, Vol. 10, No. 1, 2014, p. 1-55.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type
AU - Favero, David
AU - Iliev, Atanas
AU - Katzarkov, Ludmil
PY - 2014
Y1 - 2014
N2 - 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 X2.3.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 X2.3there is a noncommutative CY 3-fold, B, such that the Griffiths group of X2.3surjects 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.
AB - 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 X2.3.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 X2.3there is a noncommutative CY 3-fold, B, such that the Griffiths group of X2.3surjects 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.
KW - Algebraic cycles
KW - Calabi-Yau geometries
KW - Derived categories
KW - Hodge theory
UR - http://www.scopus.com/inward/record.url?scp=84907294602&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84907294602&partnerID=8YFLogxK
U2 - 10.4310/PAMQ.2014.v10.n1.a1
DO - 10.4310/PAMQ.2014.v10.n1.a1
M3 - Article
AN - SCOPUS:84907294602
VL - 10
SP - 1
EP - 55
JO - Pure and Applied Mathematics Quarterly
JF - Pure and Applied Mathematics Quarterly
SN - 1558-8599
IS - 1
ER -