Classification of surfaces with the bounded cohomology property

Classify all smooth projective surfaces that satisfy the bounded cohomology property, namely, those for which there exists a constant c_X>0 such that h^1(O_X(C)) is at most c_X h^0(O_X(C)) for every curve C on X.

Background

The bounded cohomology property requires a uniform linear bound on the first cohomology of the line bundle associated with every curve in terms of the dimension of its global sections. The paper states that a general classification of smooth projective surfaces satisfying this property is open.

The paper contributes classification results for particular classes of surfaces, including surfaces with Picard number two and certain minimal surfaces with rational polyhedral Mori cones, but leaves the general classification unresolved.