weakly holomorphic

Let $V$ be a local complex analytic variety. A function $f:U\subset V\to ℂ$ (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 ${𝒪}^w(V)$ 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 ${𝒪}_p^w(V)$ the ring of germs at $p$ of weakly holomorphic functions.

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