Hartogs extension theorem

Theorem.

Suppose $V$ is an analytic variety in an open set $U\mathrm{\subset }{\mathrm{C}}^n$ ($n\mathrm{\ge }\mathrm{2}$) of dimension at most $n\mathrm{-}\mathrm{2}$ and suppose that $f\mathrm{:}U\mathrm{\backslash }V\mathrm{\to }\mathrm{C}$ 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.

Title	Hartogs extension theorem
Canonical name	HartogsExtensionTheorem
Date of creation	2013-03-22 15:34:54
Last modified on	2013-03-22 15:34:54
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	4
Author	jirka (4157)
Entry type	Theorem
Classification	msc 32H02