# A cohomological variant of the second Riemann's extension theorem

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!

• An algebraic version of this could be found in EGA IV. 4ème partie Remarque 19.9.9. The idea seems to exploit the local cohomology.
– Z. M
Mar 17 at 15:30
• A large part of SGA 2 is devoted to this. There is a long exact sequence involving $r_n$ where the extra terms are $H^{i}_Z(X,\mathscr{O}_X)$ (with $Z:=X\smallsetminus U$), and some criteria for the vanishing of these groups.
– abx
Mar 17 at 15:54