This might be a silly/obvious question. I know that we have the removable singularity theorem (of Riemann) on the complex line, and we also have the generalization of this to algebraic curves. (Namely: if $U$ is an open subset of a Riemann surface and $a\in U$, and $f \in \mathcal{O}(U-a)$ is bounded in some neighborhood of a, then f can be extended uniquely to a function $F\in \mathcal{O}(U)$.) In particular, if a function on a curve vanishes over a divisor, we can extend over it uniquely.


What do we know about extending of functions over divisors/hyperplanes in higher dimensions?



Divisors, extensions of functions
The very same result holds in arbitrary dimensions: a locally bounded holomorphic function 
defined in the complement of a divisor extends. In many textbooks (like Gunning's) this is also called Riemmann's removable singularity Theorem. 


If you know more about your divisor you can do even better. For instance, if your ambient is a surface and you have a holomorphic function defined on the complement of a smooth and compact rational curve of self-intersection $-1$ then your function extends to the whole surface since
the curve is contractible and after contraction you obtain a smooth surface. More generally, holomorphic functions defined on the complement of compact contractible divisors can also be extended if, after contraction, you obtain a normal variety. 


