Bounded cohomology property on a smooth projective surface with Picard number two

Abstract

We say a smooth projective surface $X$ satisfies the bounded cohomology property if there exists a positive constant $c_X$ such that $h^1({\mathcal{O}}_X(C))\le c_X{h}^0({\mathcal{O}}_X(C))$ for every prime divisor $C$ on $X$. Let the closed Mori cone $\mathrm{NE}(X)={\mathbb{R}}_{\ge 0}[C_1]+{\mathbb{R}}_{\ge 0}[C_2]$ such that $C_1$ and $C_2$ with $C_2^2<0$ are some curves on $X$. If either (i) the Kodaira dimension $\hat{I}º(X)\le 1$ or (ii) $\hat{I}º(X)=2$, the irregularity $q(X)=0$ and the Iitaka dimension $\hat{I}º(X,C_1)=1$, then we prove that $X$ satisfies the bounded cohomology property.

9 pages, to appear in Communications in Algebra, minor revision