# Riemann’s removable singularity theorem in several variables

###### Theorem.

Suppose $V$ is a proper analytic variety in an open set $U\subset{\mathbb{C}}^{n}$ (that is of dimension at most $n-1$) suppose that $f\colon U\backslash V\to{\mathbb{C}}$ 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.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
• 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
Title Riemann’s removable singularity theorem in several variables RiemannsRemovableSingularityTheoremInSeveralVariables 2013-03-22 15:34:57 2013-03-22 15:34:57 jirka (4157) jirka (4157) 4 jirka (4157) Theorem msc 30D30 msc 32H02 Riemann’s extension theorem