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