The Albanese variety is an Abelian variety canonically attached to an algebraic variety which is the solution to a certain universal problem. The Albanese variety is dual to the Picard variety.
For a smooth connected projective complex
surface 
,
the Albanese homomorphism from the integral
degree-zero Chow group to 
is surjective. Bloch's
conjecture predicts that it is an isomorphism
when 
.
When also 
,
the Albanese variety is a point and the conjecture predicts
that the degree map identifies the integral zero-cycle Chow
group with 
(Auel and Bernardara 2017, p. 232).
