coniveau filtration
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Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial

AbstractFor a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.


2018 ◽  
Vol 62 (3) ◽  
pp. 565-576
Author(s):  
Eoin Mackall

AbstractWe produce an isomorphism $E_{\infty }^{m,-m-1}\cong \text{Nrd}_{1}(A^{\otimes m})$ between terms of the $\text{K}$-theory coniveau spectral sequence of a Severi–Brauer variety $X$ associated with a central simple algebra $A$ and a reduced norm group, assuming $A$ has equal index and exponent over all finite extensions of its center and that $\text{SK}_{1}(A^{\otimes i})=1$ for all $i>0$.


2018 ◽  
Vol 19 (3) ◽  
pp. 891-918 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial

We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.


2018 ◽  
Vol 167 (01) ◽  
pp. 123-131 ◽  
Author(s):  
JEFFREY D. ACHTER ◽  
SEBASTIAN CASALAINA-MARTIN ◽  
CHARLES VIAL

AbstractA conjecture of Orlov predicts that derived equivalent smooth projective varieties over a field have isomorphic Chow motives. The conjecture is known for curves, and was recently observed for surfaces by Huybrechts. In this paper we focus on threefolds over perfect fields, and unconditionally secure results, which are implied by Orlov's conjecture, concerning the geometric coniveau filtration, and abelian varieties attached to smooth projective varieties.


Author(s):  
Massimo Bertolini ◽  
Henri Darmon ◽  
Kartik Prasanna ◽  
Brian Conrad

AbstractWe give evidence for the refined version of the Beilinson–Bloch conjecture involving coniveau filtrations, by studying several infinite families of CM motives (indexed by the integers


2015 ◽  
Vol 16 (1) ◽  
pp. 189-222 ◽  
Author(s):  
Ashay A. Burungale

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$. The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$. We prove the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$. The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$.


2007 ◽  
Vol 50 (2) ◽  
pp. 161-171 ◽  
Author(s):  
Donu Arapura ◽  
Su-Jeong Kang

AbstractIt is shown that the coniveau filtration on the cohomology of smooth projective varieties is preserved up to shift by pushforwards, pullbacks and products.


2002 ◽  
Vol 45 (2) ◽  
pp. 204-212 ◽  
Author(s):  
Najmuddin Fakhruddin

AbstractWe compute the rational Chow groups of supersingular abelian varieties and some other related varieties, such as supersingular Fermat varieties and supersingular K3 surfaces. These computations are concordant with the conjectural relationship, for a smooth projective variety, between the structure of Chow groups and the coniveau filtration on the cohomology.


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