Let X be a smooth projective variety and G^d(X) be the group of codimension-d cycles onX which are homologous to zero modulo those algebraically equivalent to 0. Set Griff^d(X,Q) = Gd(X)_Q. Griffiths showed first that G^d(X) contains an infinite cyclic group for X a general quintic threefold and d = 2; later on H. Clemens proved that in the same setting Griff^22(X,Q) is infinite-dimensional over Q . Since these two basic results were proved, a certain number of other examples of both kinds have been produced. In particular, we note here C. Voisin’s proof of Clemens’ theorem for the depth of her geometric insight and the new method of proof. In this paper the authors prove that when X is a general smooth cubic sevenfold, Griff^4(X,Q) is not finitely generated over Q. Their proof follows by applying Voisin’s technique of her paper quoted above, the result due to Steenbrink that the Hodge (3, 3) conjecture holds for any smooth cubic sixfold, and some ad hoc computations on the rational cohomology of the Fermat cubic sixfold. As a nice consequence of their main result the authors show that if D_1 and D_2 are two general hypersurfaces of sufficiently high degree in P^8 and Y = X \cap D_1 \cap D_2, Y_ 0 = X \cap D1 then Griff^4(Y,Q) and Griff^4(Y_0,Q) are not finitely generated over Q. This follows from their main theorem and M. V. Nori’s theorem ; it is interesting because the intermediate Jacobian J^4(Y ) is trivial.( review by C. Voisin )
On the Griffiths group of the cubic sevenfold.
ALBANO, Alberto;COLLINO, Alberto
1994-01-01
Abstract
Let X be a smooth projective variety and G^d(X) be the group of codimension-d cycles onX which are homologous to zero modulo those algebraically equivalent to 0. Set Griff^d(X,Q) = Gd(X)_Q. Griffiths showed first that G^d(X) contains an infinite cyclic group for X a general quintic threefold and d = 2; later on H. Clemens proved that in the same setting Griff^22(X,Q) is infinite-dimensional over Q . Since these two basic results were proved, a certain number of other examples of both kinds have been produced. In particular, we note here C. Voisin’s proof of Clemens’ theorem for the depth of her geometric insight and the new method of proof. In this paper the authors prove that when X is a general smooth cubic sevenfold, Griff^4(X,Q) is not finitely generated over Q. Their proof follows by applying Voisin’s technique of her paper quoted above, the result due to Steenbrink that the Hodge (3, 3) conjecture holds for any smooth cubic sixfold, and some ad hoc computations on the rational cohomology of the Fermat cubic sixfold. As a nice consequence of their main result the authors show that if D_1 and D_2 are two general hypersurfaces of sufficiently high degree in P^8 and Y = X \cap D_1 \cap D_2, Y_ 0 = X \cap D1 then Griff^4(Y,Q) and Griff^4(Y_0,Q) are not finitely generated over Q. This follows from their main theorem and M. V. Nori’s theorem ; it is interesting because the intermediate Jacobian J^4(Y ) is trivial.( review by C. Voisin )
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