Riemann’s removable singularity theorem in several variables

Theorem.

Suppose $V$ is a proper analytic variety in an open set $U\mathrm{\subset }{\mathrm{C}}^n$ (that is of dimension at most $n\mathrm{-}\mathrm{1}$) suppose that $f\mathrm{:}U\mathrm{\backslash }V\mathrm{\to }\mathrm{C}$ is holomorphic and further that $f$ is locally bounded in $U$ Then there exists a unique holomorphic extention of $f$ to all of $U$.

If $V$ is of even lower dimension we can in fact even drop the locally bounded requirement, see the Hartogs extension theorem.

Title	Riemann’s removable singularity theorem in several variables
Canonical name	RiemannsRemovableSingularityTheoremInSeveralVariables
Date of creation	2013-03-22 15:34:57
Last modified on	2013-03-22 15:34:57
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	4
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30D30
Classification	msc 32H02
Synonym	Riemann’s extension theorem