# Hartogs extension theorem

###### Theorem.

Suppose $V$ is an analytic variety in an open set $U\subset{\mathbb{C}}^{n}$ ($n\geq 2$) of dimension at most $n-2$ and suppose that $f\colon U\backslash V\to{\mathbb{C}}$ is holomorphic. Then there exists a unique holomorphic extention of $f$ to all of $U$.

Note that when $V$ is 0 dimensional (a point) then this is just a special case of the Hartogs’ phenomenon. Also note the similarity to Riemann’s removable singularity theorem in several variables, where however we also assume that $f$ is locally bounded.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
• 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
Title Hartogs extension theorem HartogsExtensionTheorem 2013-03-22 15:34:54 2013-03-22 15:34:54 jirka (4157) jirka (4157) 4 jirka (4157) Theorem msc 32H02