Finite dimensional motives and the Conjectures of Beilinson and Murre

Abstract

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.

23 pages

References in corpus (1)

• Balanced Varieties

Cited by in corpus (11)

• Bloch's conjecture for the surface of Craighero and Gattazzo
• Bloch's conjecture for surfaces with involutions and of geometric genus zero
• Motives and representability of algebraic cycles on threefolds over a field
• Non-trivial elements in the Abel-Jacobi kernels of higher dimensional varieties
• On the finite dimensionality of a K3 surface
• Transcendence degree of zero-cycles and the structure of Chow motives
• Motives of smooth families and cycles on threefolds
• ${\mathbb{A}}^1$-equivalence of zero cycles on surfaces II
• Ambiguity of certain Chow-Kuenneth projectors
• Explicit Chow-Lefschetz Decomposition for Kummer manifolds
• Chow-Kuenneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles