Consider a smooth, projective, algebraic surface S defined over
the field of complex numbers. Suppose that the surface is equipped with an
involution (an automorphism of order 2). Then the generalised Bloch
conjecture predicts that, if the involution acts identically at the level
of cohomology then it must act identically on the Chow group of zero
cycles of the surface.
In this talk we will survey some recent progress in this direction and
discuss the proof of the conjecture for some special class of surfaces of
general type with geometric genus zero.