$\bar{\mathrm{\partial }}$ cohomology of the complement of a semi-positive anticanonical divisor of a compact surface

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 $\bar{\mathrm{\partial }}$ cohomology group $H^1(M,{\mathcal{O}}_M)$ of the complement $M=X\setminus Y$.