Let $X$ be a connected compact complex manifold, $U$ an open subset of $X$ such that the complement of $U$ in $X$ is an analytic subset of codimension at least 2 in $X$. Let $O_X$ (resp. $O_U$) be the sheaf of holomorphic functions on $X$ (resp. on $U$). If $n$ is a nonnegative integer then there is a natural homomorphism of complex vector spaces

$$r_n: H^n(X,O_X) \to H^n(U,O_U).$$

The second Riemann extension theorem actually asserts that $r_n$ is an isomorphism for $n=0$. I am looking for a reference where it is proven that $r_n$ is an isomorphism, say, for $n=1$ or $2$ (may be, under some additional assumptions). Thanks!