# Remmert-Stein theorem

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 $p.$

###### Theorem (Remmert-Stein).

Let $U\mathrm{\subset}{\mathrm{C}}^{n}$ be a domain (http://planetmath.org/Domain2) and let $S$ be a complex analytic subvariety of $U$ of
dimension^{} $$ Let $V$ be a complex analytic subvariety of $U\mathrm{\backslash}S$ such that ${\mathrm{dim}}_{p}\mathit{}V\mathrm{>}m$ for all
regular points $p\mathrm{\in}V\mathrm{.}$ Then the closure^{} of $V$ in $U$ is an analytic variety in $U\mathrm{.}$

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 $(z,w)\in {\u2102}^{2}$ be our coordinates and let $V$ be defined by $w={e}^{1/z}$ in ${\u2102}^{2}\setminus S,$ where $S$ is defined by $z=0.$ The closure of $V$ in ${\u2102}^{2}$ 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.

## References

- 1 Klaus Fritzsche, Hans Grauert. , Springer-Verlag, New York, New York, 2002.
- 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | Remmert-Stein theorem |
---|---|

Canonical name | RemmertSteinTheorem |

Date of creation | 2013-03-22 15:04:55 |

Last modified on | 2013-03-22 15:04:55 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 9 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 32A60 |

Classification | msc 32C25 |

Synonym | Remmert-Stein extension theorem |

Related topic | ChowsTheorem |