# weakly holomorphic

Let $V$ be a local complex analytic variety.
A function $f:U\subset V\to \u2102$ (where $U$ is open in $V$)
is said to be weakly holomorphic through $U$
if there exists a nowhere dense complex analytic subvariety $W\subset V$
and $W$ contains the singular points^{} of $V$ and $V\setminus W\subset U$,
and such that $f$ is holomorphic on $V\setminus W$ and
$f$ is locally bounded on $V$.

It is not hard to show that we can then just take $W$ to be the set of singular points of $V$ and have $U=V\setminus W$ as we can extend $f$ to all the nonsingular points of $V$.

Usually we denote by ${\mathcal{O}}^{w}(V)$ the ring of weakly holomorphic functions through $V$. Since any neighbourhood of a point $p$ in $V$ is a local analytic subvariety, we can define germs of weakly holomorphic functions at $p$ in the obvious way. We usually denote by ${\mathcal{O}}_{p}^{w}(V)$ the ring of germs at $p$ of weakly holomorphic functions.

## References

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

Title | weakly holomorphic |
---|---|

Canonical name | WeaklyHolomorphic |

Date of creation | 2013-03-22 17:41:46 |

Last modified on | 2013-03-22 17:41:46 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32C15 |

Classification | msc 32C20 |

Synonym | w-holomoprhic |

Related topic | NormalComplexAnalyticVariety |