Finite-Dimensional Motives and the Conjectures of Beilinson and Murre

Guletskiĭ, Vladimir; Pedrini, Claudio

doi:10.1023/b:kthe.0000019787.69435.89

Cited by 31 publications

(23 citation statements)

References 20 publications

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“…The purpose of the present paper is to investigate some interesting links between motivic finite-dimensionality and representability of algebraic cycles on three-dimensional varieties over an arbitrary algebraically closed ground field k. In particular, we will show that the motive of a smooth projective threefold X over k can be decomposed into a sum of Lefschetz and Abelian motives if and only if zero-cycles are representable on X (Theorem 8). To some extent, this result can be considered as a most possible analog of Theorem 7 from [12]. Notice that representability of zero-cycles on Fano threefolds was proved uniformly by Kollár in [16], which was a generalization of the results of Bloch and Murre, see a survey in [23].…”

Section: Introductionmentioning

confidence: 79%

“…Finite-dimensional motives, which had been introduced by S.Kimura in [15], shed a new light on the motivic picture of intersection theory. If X is a surface over an algebraically closed field with algebraic second cohomology group, the Chow group of zero cycles on X is representable if and only if the motive M(X) is finite-dimensional, see [12], Theorem 7. The purpose of the present paper is to investigate some interesting links between motivic finite-dimensionality and representability of algebraic cycles on three-dimensional varieties over an arbitrary algebraically closed ground field k. In particular, we will show that the motive of a smooth projective threefold X over k can be decomposed into a sum of Lefschetz and Abelian motives if and only if zero-cycles are representable on X (Theorem 8).…”

Section: Introductionmentioning

confidence: 99%

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Motives and representability of algebraic cycles on threefolds over a field

Gorchinskiy

Guletskiĭ

2011

J. Algebraic Geom.

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Abstract. We study 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 Griffiths' 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 representability of zero-cycles on several classes of threefolds fibred by surfaces with algebraic H 2 . This gives another new examples of three-dimensional varieties whose motives are finite-dimensional.

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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Motives and representability of algebraic cycles on threefolds over a field

Gorchinskiy

Guletskiĭ

2011

J. Algebraic Geom.

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“…Hence, the above notions apply. In the commutative world these finiteness notions were extensively studied by André-Kahn, Guletskii, Guletskii-Pedrini, Kimura, Mazza, and others; see [1,2,13,14,19,31]. For instance, Guletskii and Pedrini proved that given a smooth projective surface S (over a field of characteristic zero) with p g (S) = 0, the Chow motive M (S) is Kimura-finite if and only if Bloch's conjecture on the Albanese kernel for S holds.…”

Section: Proposition 726 the Functor (−)mentioning

confidence: 99%

Noncommutative Artin motives

Marcolli

Tabuada

2013

Sel. Math. New Ser.

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Abstract. In this article we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k) Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives we then show that the image of this full embedding, which we call the category NAM(k) Q of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group Gal(k/k) from the Tannakian formalism applied to NAM(k) Q . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change we then construct a short exact sequence relating Gal(k/k) with the noncommutative motivic Galois groups of k and k. Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne-Milne. In the mixed world, we introduce the triangulated category NMAM(k) Q of noncommutative mixed Artin motives and construct a faithful functor from the classical category MAM(k) Q of mixed Artin motives to it. When k is a finite field this functor is an equivalence. On the other hand, when k is of characteristic zero NMAM(k) Q is much richer than MAM(k) Q since its higher Ext-groups encode all the (rationalized) higher algebraic K-theory of finité etale k-schemes. In the appendix we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne-Milne's short exact sequence.

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“…Недавно было обнаружено, что для поверхности X с p g = 0 гипотеза Блоха верна тогда и только тогда, когда мотив M (X) конечномерен в смысле Кимуры (см. [4], [5]). …”

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О Непрерывной Части Алгебраических Циклов Коразмерности 2 На Трехмерных Многообразиях

Guletskiĭ¹

2009

Матем. сб.

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Finite-Dimensional Motives and the Conjectures of Beilinson and Murre

Cited by 31 publications

References 20 publications

Motives and representability of algebraic cycles on threefolds over a field

Motives and representability of algebraic cycles on threefolds over a field

Noncommutative Artin motives

О Непрерывной Части Алгебраических Циклов Коразмерности 2 На Трехмерных Многообразиях

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