# weakly holomorphic

Let $V$ be a local complex analytic variety. A function $f\colon U\subset V\to\mathbb{C}$ (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 ${\mathcal{O}}^{w}(V)$ 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 WeaklyHolomorphic 2013-03-22 17:41:46 2013-03-22 17:41:46 jirka (4157) jirka (4157) 4 jirka (4157) Definition msc 32C15 msc 32C20 w-holomoprhic NormalComplexAnalyticVariety