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

Riemann extension theorem in several variables

Builds on Holomorphic function of several variables Removable singularity Open set

Statement

Let $D \subseteq \C^n$ be a connected open set. Let $A \subseteq D$ be a closed subset with the following property. Each point $p \in D$ has a connected open neighbourhood $U \subseteq D$ and a holomorphic function $h \not\equiv 0$ on $U$ with $A \cap U \subseteq \set{z \in U : h(z) = 0}$ .

Let $f$ be holomorphic on $D \setminus A$ , and let each point of $A$ have a neighbourhood $V$ with $f$ bounded on $V \setminus A$ . Then $f$ extends uniquely to a holomorphic function on $D$ .
Let $A$ be moreover a complex analytic subset of $D$ of dimension at most $n - 2$ . Then every holomorphic function on $D \setminus A$ extends uniquely to a holomorphic function on $D$ .

Remarks

Under the hypothesis, $D \setminus A$ is dense in $D$ and connected, which gives uniqueness by the identity theorem. For part 1, take a point of $A$ and choose coordinates and a small polydisc around it. They can be chosen so that $h$ has no zeros on the part of the boundary where $\abs{z_n}$ equals the radius. The Cauchy integral in $z_n$ over that circle defines a holomorphic function $F$ . On each line where $z'$ is fixed, $f$ is a bounded holomorphic function of $z_n$ off finitely many points, so by the one-variable removable singularity theorem $F = f$ there. Part 2 is proved with Hartogs figures. Near a point of a set of codimension at least $2$ , one finds Hartogs figures in $D \setminus A$ whose filled-in polydiscs cover a neighbourhood of the point.

Example: every holomorphic function on $\C^2 \setminus \set{0}$ extends to $\C^2$ , by part 2 with $A = \set{0}$ .

The boundedness in part 1 cannot be dropped when $A$ has codimension $1$ . The function $1/z_1$ is holomorphic on $\C^2 \setminus \set{z_1 = 0}$ but unbounded near that line, and it has no holomorphic extension.

Going deeper

For a student

Part 1 is used constantly when holomorphic functions are defined only off an analytic set, for example off the singular points of an analytic variety. Part 2 is a form of the Hartogs phenomenon: a holomorphic function cannot have a singular set that is an analytic set of codimension two or more. Part 1 is also the reason why a continuous function on $D$ that is holomorphic off such a set $A$ is holomorphic on all of $D$ .

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Holomorphic function of several variables concept
Open set concept
Removable singularity concept
Riemann extension theorem in several variables statement

Tag	`5VW718YJQK`
URI	`m2://5VW718YJQK`
Kind	statement, theorem
Writing	W0 stub
MSC	32D20
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About	Holomorphic function of several variables, Removable singularity
Uses	Holomorphic function of several variables, Removable singularity, Open set