Let $X$ be a non-singular compact complex surface such that the anticanonical line bundle admits a smooth Hermitian metric with semi-positive curvature. For a non-singular hypersurface $Y$ which defines an anticanonical divisor, we investigate the $\overline{\partial}$ cohomology group $H^1(M, \mathcal{O}_M)$ of the complement $M=X\setminus Y$.