Bloch-Type Conjectures and an Example a Three-Fold of General Type

Peters, Chris

doi:10.1142/s0219199710003932

Cited by 4 publications

(5 citation statements)

References 17 publications

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“…This way the second result of [27] (quintic hypersurfaces with involutions) and the main application of [23] (3-dimensional complete intersection in weighted projective space) are reproved : in both cases we are reduced to prove the generalized Hodge conjecture for the coniveau 1 Hodge structure on their cohomology of degree 3.…”

Section: Complete Intersections With Group Actionmentioning

confidence: 99%

“…Section 3 will provide a number of other geometric applications. For example, we will show how to recover the results of [27], or [23]. We will get more generally results for many complete intersections X b endowed with the action of a finite group G. In this case, the method applies as well to the χ-invariant part of CH(X b ) where χ : G → {1, −1} is a character.…”

Section: Introductionmentioning

confidence: 97%

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

Voisin¹

2013

Ann. Sci. École Norm. Sup.

195

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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

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Section: Complete Intersections With Group Actionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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

Voisin¹

2013

Ann. Sci. École Norm. Sup.

195

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“…A threefold of general type with finite dimensional motive. In [39] one of the authors investigated a quasi-smooth threefold X which is a complete intersection of three degree 6 hypersurfaces in the weighted projective space P = P(2 4 , 3 3 ) and showed that A 0 (X) = Q. Let us check that this example can also be treated within the present framework.…”

Section: Moreover In This Rangementioning

confidence: 93%

On complete intersections in varieties with finite-dimensional motive

Laterveer

Nagel

Peters

2018

The Quarterly Journal of Mathematics

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Let X be a complete intersection inside a variety M with finite dimensional motive and for which the Lefschetz-type conjecture B(M ) holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps A k (X) → H 2k (X) are injective for k < c. We give variants involving group actions which lead to several new examples with finite dimensional Chow motives.

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“…In this way, the second result of [98] (quintic hypersurfaces with involutions) and the main application of [82] (three-dimensional complete intersections in weighted projective space) are re-proved; in both cases we are reduced to proving the generalized Hodge conjecture for the coniveau 1 Hodge structure on their cohomology of degree 3.…”

Section: Complete Intersections With Group Actionmentioning

confidence: 94%