Smooth projective surfaces with bounded cohomology property II

Abstract

Let $X$ be a smooth projective surface with Picard number two, where either $X$ is a geometrically ruled surface or the closed Mori cone is rational polyhedral. In this paper, we characterize $X$ with the bounded cohomology property, i.e., there exists a constant $c_X>0$ such that $h^1({\mathcal{O}}_X(C))\le c_X{h}^0({\mathcal{O}}_X(C))$ for every curve $C$ on $X$.

10 pages, comments are welcome!