### Finite dimensional motives and the conjectures of Beilinson and Murre, by Vladimir Guletskii and Claudio Pedrini

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
of X. We show (Th. 14) that finite dimensionality of M(X) implies uniqueness,
up to isomorphism, of Murre's decomposition of M(X). Conversely (Th.17),
Murre's Conjecture for a suitable n-fold product of X by itself implies
finite dimensionality of M(X). We also show (Th.27) that, for a surface X
with p_g=0, the motive M(X) is finite dimensional if and only if Bloch's
conjecture on Albanese kernel holds for X.

Vladimir Guletskii <guletskii@im.bas-net.by>

Claudio Pedrini <pedrini@dima.unige.it>