Let $X$ be a smooth, connected, complex analytic variety, and $Y\subset X$ a closed, analytic subvariety of codimension at least 2. Now let $V\subset X\backslash Y$ be a closed, analytic subvariety. Is the closure $\bar{V}$ of $V$ in $X$ also analytic?


**Motivation:** In the case where $X$ is a surface, this seems to follow from Riemanns extension theorem, and perhaps it can even be proven this way. But it'd be nice if there were a reference (or if I'm missing something and this is false in general!)


Thanks!



Can an analytic variety extend along a codimension 2 subvariety?
Yes, by Remmert-Stein's extension theorem. See e.g. Fritsche-Grauert, *From holomorphic functions to complex manifolds*, Theorem 6.9.


