Bloch's Conjecture for surfaces, which predicts the converse to Mumford's famous necessary condition for finite-dimensionality of the Chow group of 0-cycles,has been a source of inspiration in
the theory of algebraic cycles ever since its formulation in 1975. It
is known for surfaces not of general type, and for some example of surfaces of general type. The results of
S.I. Kimura and Guletskii-Pedrini have shown that Bloch's Conjecture
for a complex surface of general type without regular 2-forms is equivalent to its motive being finite dimensional, in the category of Chow Motives. On the other hand this last condition is equivalent to the motive being equal to its image in the
semisimple category of numerical motives.
The main purpose of the talk is to illustrate joint work with
B. Kahn and J. Murre, where we introduce, for any algebraic surface
a birational invariant, the transcendental
part of the motive. We compute the group of its endomorphisms and we relate this group to the Conjectures of Bloch, Beilinson and Murre.