Andreotti–Grauert theorem

Remarks

Here $q$-convexity is taken in the convention of Andreotti and Grauert: there is a smooth exhaustion function whose Levi form has at least $n - q + 1$ positive eigenvalues outside a compact set. Andreotti and Grauert proved the theorem in 1962, for complex spaces as well as for manifolds. The proof extends Grauert's bumping method. The cohomology on a sublevel set $\set{\varphi < c}$ is compared with the cohomology on a slightly larger one. Local vanishing results near strongly $q$-convex boundary points control the restriction maps in degrees $r \ge q$. The $1$-complete manifolds are the Stein manifolds. For them, part 2 says that $H^r(X, \mathcal{F}) = 0$ for $r \ge 1$, which is Cartan's theorem B. For $q = 1$, part 1 gives Grauert's finiteness theorem for strongly pseudoconvex manifolds. On $\C^2 \setminus \set{0}$, which is $2$-complete but not $1$-complete, the group $H^1(X, \mathcal{O})$ is infinite-dimensional, so the bound $r \ge q$ cannot be lowered.

Going deeper

For a student

The theorem puts Cartan's theorem B and Grauert's finiteness theorem into one framework, indexed by $q$. Applied to the sheaf of holomorphic $p$-forms and combined with the Dolbeault isomorphism, part 2 says that on a $q$-complete manifold every $\bar\partial$-closed smooth $(p, r)$-form with $r \ge q$ is $\bar\partial$-exact.