Finite dimensional motives and the Conjectures of Beilinson and Murre

By V. Guletskiĭ, C. Pedrini. 2003

65 citations recorded by the source corpus.

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 a filtration on the Chow ring CH(X). We show (Theorem 14) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 17), Murre's Conjecture for a suitable m-fold product of X by itself implies finite dimensionality of M(X). We also show (Theorem 27) that, for a surface X with trivial geometrical genus, the motive M(X) is finite dimensional if and only if the Chow group of 0-cycles of X is finite dimensional in the sense of Mumford, i.e. iff the Bloch Conjecture holds for X.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fd87b0d1b94a7de0f47031f837f9a61415866516

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Gonçalo Tabuada
2. D. Huybrechts
3. Claire Voisin
4. S. Gorchinskiy

In Geometry and Topology

1. D. Huybrechts
2. Claire Voisin
3. Gonçalo Tabuada
4. S. Gorchinskiy
5. V. Guletskiĭ
6. Kieran G. O’Grady
7. E. Markman
8. P. Stellari

In Mathematical Physics

1. D. Huybrechts
2. Claire Voisin
3. Gonçalo Tabuada
4. S. Gorchinskiy
5. V. Guletskiĭ
6. P. Stellari
7. E. Markman
8. Emanuele Macrì

Citations in the corpus

• Curves and cycles on K3 surfaces (2013, background): “Thanks to Burt Totaro for a question that triggered the results in Section 9 and for detailed comments on the first version, to Rahul Pandharipande for bringing [GG03] to my attention, and to François Charles and Davesh Maulik for help with arguments related to Section 8.”
• Motives and representability of algebraic cycles on threefolds over a field (2008, background): “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 [10], Theorem 7.”
• On the finite dimensionality of a K3 surface (2011, unknown): “(see [6]).”
• Chow motives versus non-commutative motives (2011, background): “In the commutative world, the aforementioned general finiteness notions were extensively studied by André–Kahn, Guletskii, Guletskii–Pedrini, Kimura, Mazza, and others; see [1], [3], [24], [25], [32], [40].”