surfaces of general type with $H^q(S,\Omega^p_S)=0$, for $q\neq p$

Question

Is it possible to say what surfaces occur as minimal models of surfaces of general type with $K_0(S)$ finitely generated and $H^q(S,\Omega^p_S)=0$, for $q\neq p$?

Answer 1

According to Bloch's conjecture, all surfaces of general type $S$ with $H^0(S,\Omega^2_S)=0$ should have $CH(S)$ finitely generated, hence also $K_0(S)$. This is probably a very difficult conjecture, but it has been checked in a number of examples : see for instance this preprint and the references therein.