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 $f\mathrm{:}U\mathrm{\backslash \backslash }V\mathrm{\to }\mathrm{C}$ is holomorphic. Then there exists a unique holomorphicextention of $f$ to all of $U$.

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