---
tag: 67RJ0S8F74
kind: statement
role: theorem
name: Andreotti–Grauert theorem
msc: [32F10, 32C35]
uses: [VBHR4R5W7F, DSJET02WPR, PQ6Q143ARJ, JSGWAN2RJV]
about: [VBHR4R5W7F]
sources:
  - url: https://en.wikipedia.org/wiki/Andreotti%E2%80%93Grauert_theorem
same_as:
  wikidata: Q57463
  wikipedia: Andreotti–Grauert theorem
---

## Statement

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$, let $1 \le q \le n$, and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.

1. If $X$ is [[$q$-convex|VBHR4R5W7F]], then the [[sheaf cohomology|PQ6Q143ARJ]] group $H^r(X, \mathcal{F})$ is a finite-dimensional complex vector space for every $r \ge q$.
2. If $X$ is $q$-complete, then $H^r(X, \mathcal{F}) = 0$ for every $r \ge q$.

## 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.

## Explanations

### L1 human

The cohomology groups of a coherent sheaf measure the obstructions to solving holomorphic problems on a complex manifold, such as finding a function with prescribed local behaviour.
The theorem says that convexity of the manifold kills these obstructions in high degrees.
The more convex the manifold is, the more degrees are controlled: a Stein manifold has no obstructions at all in positive degree.

### L2 human

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.