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

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