The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Abstract

We prove an unconditional (but slightly weakened) version of the main result of [13], which was, starting from dimension 4, conditional to the Lefschetz standard conjecture. Let X be a variety with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on CH(X) Q ). We prove that if the cohomology of a general hypersurface Y in X is "parameterized by cycles of dimension c", then the Chow groups CHi(Y ) Q are trivial for i ≤ c − 1.Let X be a smooth projective variety. We will say that X has geometric coniveau ≥ c if the transcendental cohomology of X, that is, the orthogonal with respect to Poincaré duality of the "algebraic cohomology" of X generated by classes of algebraic cycles,According to the generalized Hodge conjecture [7], X has geometric coniveau ≥ c if and only if X has Hodge coniveau ≥ c, where we define the Hodge coniveau of X as the minimum over k of the Hodge coniveaux of the Hodge structures H k (X, Q) tr . Here we recall that the Hodge coniveau of a weight k Hodge structure (L, L p,q ) is the integer c ≤ k/2 such thatwith L k−c,c = 0. As the Hodge coniveau is computable by looking at the Hodge numbers, we know conjecturally how to compute the geometric coniveau.A fundamental conjecture on algebraic cycles is the generalized Bloch conjecture (see [17, Conjecture 1.10]), which was formulated by Bloch [1] in the case of surfaces, and can be stated as follows:Conjecture 0.1. Assume X has geometric coniveau ≥ c. Then the cycle class mapConcrete examples are given by hypersurfaces in projective space, or more generally complete intersections. For a smooth complete intersection Y of r hypersurfaces in P n , the Hodge coniveau of Y is equal to the Hodge coniveau of H n−r (Y, Q) tr , the last space being for the very general member Y , except in a small number of cases, equal to the Hodge coniveau of H n−r (Y, Q) prim . The latter is computed by Griffiths:Theorem 0.2. If Y ⊂ P n is a complete intersection of hypersurfaces of degrees d 1 ≤ . . . ≤ d r , the Hodge coniveau of H n−r (Y, Q) prim is ≥ c if and only if n ≥ i d i + (c − 1)d r .Conjecture 0.1 thus predicts that for such a Y , the Chow groups CH i (Y ) Q are equal to Q for i ≤ c − 1, a result which is essentially known only for coniveau 1 (then Y is a Fano

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

“…Our main result in [114] presented in Section 4.3 is obtained as a consequence of the spreading principle, Theorem 1.7. It proves Conjecture 1.10 for very general complete intersections in ambient varieties with "trivial Chow groups," assuming the Lefschetz standard conjecture.…”

“…The following conjecture (which could have been stated as a standard conjecture) is stated in [114].…”

“…Of course, the two versions are equivalent assuming the generalized Hodge conjecture. In Section 4.3 we will prove this conjecture, following [114], for the geometric coniveau and for very general complete intersections of ample hypersurfaces in a smooth projective variety X with "trivial" Chow groups, that is, having the property that the cycle class map cl : CH * (X) Q → H 2 *…”

“…In the above-mentioned work of Kimura, one can replace "varieties that are dominated by products of curves" by "motives that are a direct summand of the motive of a product of curves." In our paper [114], we can work with a variety X endowed with the action of a finite group G and consider the submotives of G-invariant complete intersections obtained by considering the projectors Γ π ∈ CH(Y × Y ) Q associated via the action of G on Y to projectors π ∈ Q [G].…”

A short note on the weak Lefschetz property for Chow groups

“…Conjecture 4 (Voisin standard conjecture [25]) Let X be a smooth projective variety, and Y ⊂ X closed with complement U . Then the natural sequence…”