Several complex variables and analytic spaces Extension phenomena and analytic continuation Theorem

Cartan–Thullen theorem

Builds on Domain of holomorphy Holomorphically convex domain Holomorphically convex hull Holomorphic function of several variables

Statement

Let $D \subseteq \C^n$ be a domain, that is, a nonempty connected open set. The following are equivalent.

$D$ is a domain of holomorphy.
$D$ is holomorphically convex: for every compact set $K \subseteq D$ , its holomorphically convex hull $\hat K_D$ is compact.
There is a holomorphic function $f$ on $D$ that cannot be continued beyond $D$ . That is, there are no nonempty open sets $U_1 \subseteq D \cap U_2$ , with $U_2$ connected and $U_2 \not\subseteq D$ , and no holomorphic $g$ on $U_2$ with $g = f$ on $U_1$ .

Remarks

Clause 3 implies clause 1 directly from the definition. For 1 implies 2, the key estimate concerns a domain of holomorphy $D$ . The distance from $\hat K_D$ to the boundary of $D$ equals the distance from $K$ to the boundary, measured with polydiscs. It comes from the Cauchy estimates: the Taylor series at points of $\hat K_D$ of every holomorphic function on $D$ converge on polydiscs of a fixed radius. For 2 implies 3, take a sequence of points of $D$ that accumulates at every boundary point. Holomorphic convexity gives holomorphic functions that are large at chosen points of the sequence. An infinite product or series of them gives a function unbounded near every boundary point.

Example: a convex domain is holomorphically convex, because $\abs{e^{\ell}} = e^{\operatorname{Re} \ell}$ for complex linear $\ell$ shows that $\hat K_D$ lies in the convex hull of $K$ ; so it is a domain of holomorphy.

Going deeper

For a student

Holomorphic convexity is an intrinsic condition, stated without reference to points outside $D$ . This makes it the right condition to generalise to complex manifolds, where it leads to the notion of a Stein manifold. It reduces the Levi problem to showing that pseudoconvex domains are holomorphically convex. For $n = 1$ every domain satisfies all three conditions.

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Domain of holomorphy concept
Holomorphic function of several variables concept
Holomorphically convex domain concept
Holomorphically convex hull concept
Cartan–Thullen theorem statement

Where this leads

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Tag	`1XCBCB3MD7`
URI	`m2://1XCBCB3MD7`
Kind	statement, theorem
Writing	W0 stub
MSC	32D05, 32E05, 32T05
Status	This copy of the site has no build data. Run `m2 status 1XCBCB3MD7`.
About	Domain of holomorphy, Holomorphically convex domain
Uses	Domain of holomorphy, Holomorphically convex domain, Holomorphically convex hull, Holomorphic function of several variables

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