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22
Projectors on the intermediate algebraic Jacobians
, 2009
"... Let X be a smooth projective variety over an algebraically closed field k ⊂ C. Under mild assumption, we construct projectors modulo rational equivalence onto the last step of the coniveau filtration on the cohomology of X. We obtain a “motivic” description of the AbelJacobi maps to the algebraic p ..."
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Cited by 14 (8 self)
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Let X be a smooth projective variety over an algebraically closed field k ⊂ C. Under mild assumption, we construct projectors modulo rational equivalence onto the last step of the coniveau filtration on the cohomology of X. We obtain a “motivic” description of the AbelJacobi maps to the algebraic part of the intermediate Jacobians. As an application, this enables us to relate the injectivity of the total AbelJacobi map ⊕iCHi(XC)hom ⊗ Q → ⊕iJi(XC) ⊗ Q to finite dimensionality for the motive of X.
BIRATIONAL MOTIVES, I: PURE BIRATIONAL MOTIVES
"... 2. Pure birational motives 8 ..."
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Niveau and coniveau filtrations on cohomology groups and Chow groups
 Proc. Lond. Math. Soc
"... The Bloch–Beilinson–Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology – i.e. the cohomology – of smooth projective var ..."
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Cited by 10 (8 self)
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The Bloch–Beilinson–Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology – i.e. the cohomology – of smooth projective varieties. In this paper, given a smooth projective complex variety X, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of X has an incidence on the Chow groups of X. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in lowdimensional cases or when a variety is known to have small Chow groups. For instance, we give a new example of a fourfold of general type with trivial Chow group of zerocycles and we prove Murre’s conjectures for threefolds dominated by a product of curves, for threefolds rationally dominated by the product of three curves, for rationally connected fourfolds and for complete intersections of low degree. The BBM conjectures are closely related to Kimura–O’Sullivan’s notion of finitedimensionality. Assuming the standard conjectures on algebraic cycles the former is known to imply the latter. We show that the missing ingredient for finitedimensionality to imply the BBM conjectures is the coincidence of a certain niveau filtration with the coniveau filtration on Chow groups.
Tikhomirov Algebraic cycles on quadric sections of cubics in P4 under the action of symplectomorphisms, arXiv:1109.5725v1
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Finite dimensional motives and applications following Kimura, O’Sullivan and others
"... This survey paper is an expanded version of a lecture given in July 2006 at the École d’été FrancoAsiatique de géométrie algébrique et de théorie des nombres (IHÉSUniversité Paris 11). It provides an overview of the notion of finite dimensionality introduced by KimuraO’Sullivan and expla ..."
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Cited by 4 (0 self)
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This survey paper is an expanded version of a lecture given in July 2006 at the École d’été FrancoAsiatique de géométrie algébrique et de théorie des nombres (IHÉSUniversité Paris 11). It provides an overview of the notion of finite dimensionality introduced by KimuraO’Sullivan and explains some of the stricking implications of this idea.
Murre’s conjectures for certain product varieties
, 2005
"... Let X be a smooth projective variety over C of dimension d. Let ∆ ⊂ X × X be the diagonal. There is a cohomology class cl(∆) ∈ H2d (X × X). In this paper we use Betti cohomology with rational coefficients. There is the Künneth ..."
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Cited by 2 (0 self)
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Let X be a smooth projective variety over C of dimension d. Let ∆ ⊂ X × X be the diagonal. There is a cohomology class cl(∆) ∈ H2d (X × X). In this paper we use Betti cohomology with rational coefficients. There is the Künneth
SOME SURFACES OF GENERAL TYPE FOR WHICH BLOCH’S CONJECTURE HOLDS
, 2013
"... We give many examples of surfaces of general type with pg = 0 for which Bloch’s conjecture holds, for all values of K2 != 9. Our surfaces are equipped with an involution. ..."
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We give many examples of surfaces of general type with pg = 0 for which Bloch’s conjecture holds, for all values of K2 != 9. Our surfaces are equipped with an involution.