The Bloch conjecture for surfaces asserts that the Albanese homomorphism
is an isomorphism for every smooth connected projective complex surface with . Here the Chow group
has integral coefficients, is the geometric genus, and is the irregularity. The Albanese homomorphism
records the Abelian-variety image of a degree-zero zero-cycle.
When also ,
the Albanese variety is a point and the assertion
becomes
being an isomorphism. In this case any two complex
points are rationally equivalent.
Auel, A. and Bernardara, M. "Cycles, Derived Categories, and Rationality." In Surveys on Recent Developments in Algebraic Geometry
(Ed. I. Coskun, T. de Fernex, and A. Gibney). Providence, RI:
Amer. Math. Soc., pp. 199-266, 2017. https://doi.org/10.1090/pspum/095/01641.Bloch,
S.; Kas, A.; and Lieberman, D. "Zero Cycles on Surfaces with ." Compositio Math.33, 135-145, 1976.
https://www.numdam.org/item/CM_1976__33_2_135_0/.
with 
. Here the Chow group
has integral coefficients, 
is the geometric genus, and 
is the irregularity. The Albanese homomorphism
records the Abelian-variety image of a degree-zero zero-cycle.
,
the Albanese variety is a point and the assertion
becomes
." Compositio Math. 33, 135-145, 1976.