Let $X$ be a smooth projective surface and the closed Mori cone $\overline{\mathrm{NE}}(X)=\sum\mathbb R_{\ge0}[C_i]$ with some curves $C_i$ on $X$. An interesting question is asked that whether $X$ satisfies the bounded cohomology property, which is that there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every curve $C$ on $X$. We prove $X$ satisfies the bounded cohomology property if each curve $C_i$ has either $C_i^2<0$ or the Iitaka dimension $\kappa(X,C_i)=1$. When the Picard number $\rho(X)=2$, we prove that $X$ satisfies the bounded cohomology property if either (i) the Kodaira dimension $\kappa(X)\le1$ or (ii) $\kappa(X,C_1)=1$ and $C_2^2<0$.PDF: Bounded cohomology property on smooth projective surfaces with arbitrary Picard number.pdf
