Motives and representability of algebraic cycles on threefolds over a field

@article{Gorchinskiy2008MotivesAR,
  title={Motives and representability of algebraic cycles on threefolds over a field},
  author={S. Gorchinskiy and V. Guletskiĭ},
  journal={Journal of Algebraic Geometry},
  year={2008},
  volume={21},
  pages={347-373}
}
We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove… 
View PDF on arXiv
54 Citations
Highly Influential Citations
5
Background Citations
12
Methods Citations
5

54 Citations

Derived categories of coherent sheaves and motives of K3 surfaces

Let X and Y be smooth complex projective varieties. Orlov conjectured that if X and Y are derived equivalent then their motives M(X) and M(Y) are isomorphic in Voevodsky's triangulated category of…

PROJECTORS ON THE INTERMEDIATE ALGEBRAIC JACOBIANS

Let X be a complex smooth projective variety of dimension d. Under some assump- tion on the cohomology of X, we construct mutually orthogonal idempotents in CHd(X X) Q whose action on algebraically…

Étale monodromy and rational equivalence for -cycles on cubic hypersurfaces in

Let be an uncountable algebraically closed field of characteristic , and let be a smooth projective connected variety of dimension , embedded into over . Let be a hyperplane section of , and let and…

From exceptional collections to motivic decompositions

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X) of…

From exceptional collections to motivic decompositions via noncommutative motives

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X) of…

Motives of moduli spaces on K3 surfaces and of special cubic fourfolds

For any smooth projective moduli space M of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive hpMq becomes a direct summand of a…

Integral Chow motives of threefolds with K-motives of unit type

We prove that if a smooth projective algebraic variety of dimension less or equal to three has a unit type integral $K$-motive, then its integral Chow motive is of Lefschetz type. As a consequence,…

INTEGRAL OF THREEFOLDS WITH K -MOTIVES OF UNIT

. We prove that if a smooth projective algebraic variety of dimension less or equal to three has a unit type integral K -motive, then its integral Chow motive is of Lefschetz type. As a consequence,…

K3 surfaces and cubic fourfolds with a motive of Abelian type.

Starting from known 4-dimensional families of K3 surfaces, we construct two families of cubic fourfolds whose motive is of Abelian type. Cubics from the first family are smooth, and their Chow motive…

A family of special cubic fourfolds with motive of abelian type

In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic…
...

References

SHOWING 1-10 OF 43 REFERENCES

On the continuous part of codimension 2 algebraic cycles on three-dimensional varieties

Let  be a nonsingular projective threefold over an algebraically closed field and let  be the group of algebraically trivial codimension 2 algebraic cycles on  modulo rational equivalence with…

Finite dimensional motives and the Conjectures of Beilinson and Murre

We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the Conjectures of Beilinson, Bloch and Murre on the existence of…

Motivic sheaves and filtrations on Chow groups

Grothendieck's motives, as described in [Dem, K12, M a ] are designed as a tool to understand the cohomology of smooth projective varieties and the algebraic cycles modulo homological and numerical…

Torification and factorization of birational maps

Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular…

Algebraic Cycles and Motives: On the Transcendental Part of the Motive of a Surface

Bloch’s conjecture on surfaces [B1], which predicts the converse to Mumford’s famous necessary condition for finite-dimensionality of the Chow group of 0-cycles [Mum1], has been a source of…

On the motive of an algebraic surface.

0.1. The theory of motives has been created by Grothendieck in order to understand better — among other things — the underlying "objects" of the cohomology groups and to explain their common…

Motivic decomposition of abelian schemes and the Fourier transform.

such that the /-adic realization of h (A) is H (A, Qt). Recall that Chow motives are obtained from the category of smooth projective varieties over a field by a construction of Grothendieck using äs…

The Chow motive of the Godeaux surface

Let X be the Godeaux surface obtained as a quotient of the Fermat quintic in P3C under the appropriate action of Z/5. We show that its Chow motive h(X) splits as 1 ⊕ 9L ⊕ L where L = (P, [P × pt]) is…

K-cohomology of Severi-Brauer Varieties and the norm residue homomorphism

The basic purpose of this paper is to prove bijectivity of the norm residue homomorphism for any field of characteristic prime to . In particular, if , then any central simple algebra of exponent is…

Motivic decomposition and intersection Chow groups, I

For a quasiprojective variety S, we define a category CHM(S) of pure Chow motives over S. Assuming conjectures of Grothendieck and Murre, we show that the decomposition theorem holds in CHM(S). As a…