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