# normal complex analytic variety

Let $V$ be a local complex analytic variety (or a complex analytic space). A point $p\in V$ is if and only if every weakly holomorphic function through $V$ extends to be holomorphic in $V$ near $p.$

In particular, if $V\subset {\u2102}^{n}$ is a complex analytic subvariety, it is normal at $p$ if and only if every weakly holomorphic function through $V$ extends to be holomorphic in a neighbourhood of $p$ in ${\u2102}^{n}$.

To see that this definition is equivalent to the usual one, that is, that $V$ is normal
at $p$ if and only if ${\mathcal{O}}_{p}$ (the ring of germs of holomorphic functions at $p$)
is integrally closed^{}, we need the following theorem. Let ${\mathcal{M}}_{p}$ be the total quotient ring of ${\mathcal{O}}_{p}$, that is, the ring of germs of meromorphic functions.

###### Theorem.

Let $V$ be a local complex analytic variety. Then ${\mathrm{O}}_{p}^{w}\mathit{}\mathrm{(}V\mathrm{)}$ is the integral
closure^{} of ${\mathrm{O}}_{p}\mathit{}\mathrm{(}V\mathrm{)}$ in ${\mathrm{M}}_{p}\mathrm{.}$

## References

- 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | normal complex analytic variety |

Canonical name | NormalComplexAnalyticVariety |

Date of creation | 2013-03-22 17:41:48 |

Last modified on | 2013-03-22 17:41:48 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 14M05 |

Classification | msc 32C20 |

Synonym | normal analytic variety |

Related topic | WeaklyHolomorphic |

Defines | normal complex analytic space |

Defines | normal complex analytic subvariety |

Defines | normal analytic space |

Defines | normal analytic subvariety |