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 singularpoints of $V$ and have $U=V\setminus W$ as we can extend $f$ to all thenonsingular points of $V$.

Usually we denote by ${𝒪}^w(V)$ the ring of weakly holomorphic functionsthrough $V$. 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 usuallydenote by ${𝒪}_p^w(V)$ the ring of germs at $p$ of weakly holomorphicfunctions.