Cartan's theorem B

Remarks

The converse holds: a complex space on which $H^1(X, \mathcal{F}) = 0$ for every coherent analytic sheaf $\mathcal{F}$ is Stein, so theorem B characterizes Stein spaces. The proof first treats compact polydiscs and analytic polyhedra. There, vanishing follows from Cousin-type gluing and from the local theory of coherent sheaves. Then it passes to the limit over an exhaustion of $X$ by holomorphically convex compact sets, using approximation to control the inverse limit. Example: on a Stein manifold, $H^1(X, \mathcal{O}) = 0$, so the first Cousin problem is always solvable. Example: for an analytic subset $Y$ with ideal sheaf $\mathcal{I}$, the exact sequence $0 \to \mathcal{I} \to \mathcal{O}_X \to \mathcal{O}_X / \mathcal{I} \to 0$ and $H^1(X, \mathcal{I}) = 0$ show that every holomorphic function on $Y$ extends to a holomorphic function on $X$. The Stein hypothesis is needed: on the projective line, $H^1(\mathbb{P}^1, \mathcal{O}(-2)) \cong \C$ is not zero; and on $\C^2 \setminus \set{0}$, the group $H^1(X, \mathcal{O})$ is infinite-dimensional.

Going deeper

For a student

Theorem B, with the exponential sequence, solves the Cousin problems: the additive one always, and the multiplicative one exactly when a topological obstruction in $H^2(X, \Z)$ vanishes. It gives extension of holomorphic functions from closed analytic subsets, interpolation on discrete sets, and the embedding of Stein manifolds into $\C^N$. It is the analytic analogue of Serre's vanishing theorem for quasi-coherent sheaves on affine schemes.