Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $D$ an effective divisor in $X$. Is it true that $H^1(\mathcal{O}\_{X\backslash D})=0$ (we know that $H^1(\mathcal{O}\_X)=0$)? If not true in general are there examples of $D$ (other than hyperplane sections of $X$) for which this is true?