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