For a complex analytic subvariety $V$ and $p \in V$ a regular point, let $\dim_p V$ denote the complex dimension of $V$ near the point $p.$

The condition that $\dim_p V > m$ for all regular $p$ is the same as saying that all the irreducible components of $V$ are of dimension strictly greater than $m.$ To show that the restriction on the dimension of $S$ is ``sharp,'' consider the following example where the dimension of $V$ equals the dimension of $S$ . Let $(z,w) \in {\mathbb C}^2$ be our coordinates and let $V$ be defined by $w = e^{1/z}$ in ${\mathbb C}^2 \setminus S,$ where $S$ is defined by $z = 0.$ The closure of $V$ in ${\mathbb C}^2$ cannot possibly be analytic. To see this look for example at $W = \overline{V} \cap \{ w = 1 \}.$ If $\overline{V}$ is analytic then $W$ ought to be a zero dimensional complex analytic set and thus a set of isolated points, but it has a limit point $(0,1)$ by Picard's theorem.

Finally note that there are various generalizations of this theorem where the set $S$ need not be a variety, as long as it is of small enough dimension. Alternatively, if $V$ is of finite volume, we can weaken the restrictions on $S$ even further.

Bibliography

1

Klaus Fritzsche, Hans Grauert. From Holomorphic Functions to Complex Manifolds, Springer-Verlag, New York, New York, 2002.

2

Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.