On the Griffiths group of the cubic sevenfold

Abstract

For X a smooth complex projective variety, one of the important questions in algebraic geometry is the existence and the behaviour of non trivial subvarieties on X, i.e., subvarieties that are not the intersection of X with a hypersurface. In general, the approach is as follows: we let Zd(X) be the free abelian group generated by the irreducible subvarieties of codimension d in X, and introduce various equivalence relation on Zd(X). The problem is then to understand the quotients of zd(x) under these relations. The first, and most obvious, is homological equivalence, but there are other interesting relations, namely algebraic and rational equivalence. The so-called classical cases, when d = 1 (divisors) or d = dimX (0-cycles) are well understood, but little is known in general. In these cases, homological and algebraic equivalence coincide, and the group of cycles algebraically equivalent to zero modulo rational equivalence is isomorphic to an abelian variety. One of the first result for general d is due to Griffiths [8], who showed that homological and algebraic equivalence do not coincide if d # 1,dimX. The group of cycles on X of codimension d homologous to zero modulo algebraic equivalence is called the d-th Griffiths group of X and is denoted by Gd(X); we will mainly consider the Q-vector space G6(X) = Gd(X) Q, or equivalently, the Griffiths group modulo torsion. Clemens [5] then showed that G6(x) can have infinite dimension over Q. Clemens' example is given by a general quintic hypersurface in p4, and this opened the way to the construction of other examples (see [11, [21, [131, [17]).