# 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 $V\subset{\mathbb{C}}^{n}$ 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 ${\mathbb{C}}^{n}$.

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 ${\mathcal{O}}_{p}^{w}(V)$ is the integral closure of ${\mathcal{O}}_{p}(V)$ in ${\mathcal{M}}_{p}.$

## 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