Let $X$ be a complex algebraic variety. We can ask if $X$ is *normal* as an algebraic variety, but also, if its analytification is normal as a complex analytic space. Is there a relationship between the two? 


Do we have
$$\text{algebraic normality} \implies\text{analytic normality}$$
or
$$\text{analytic normality} \implies\text{algebraic normality}$$
or both or neither?



Algebraic vs analytic normality
Over $\mathbf{C}$, algebraic normalization and analytic normalization are equivalent concepts. See 


N. Kuhlmann: [Die Normalisierung komplexer Räume](http://dx.doi.org/10.1007/BF01451331), *Math. Ann.* **144** (1961), 110-125, [ZBL0096.27801](https://zbmath.org/?q=an:0096.27801).


Quoting directly from Satz 4, p. 122 of the aforementioned paper:



> 
> Es sei $Y$ eine irreduzible algebraische Varietät über dem Grundkörper $\mathbf{C}$
>  der komplexen Zahlen, $Y^\*$ der zugeordnete komplexe Raum, $Y'$ die Normalisierung
>  von $Y$ (im Sinne der algebraischen Geometrie). Dann ist der $Y'$ zugeordnete
>  komplexe Raum die Normalisierung von $Y^\*$ (im analytischen Sinne).
> 
> 
> 


The english translation is straightforward:



> 
> Let $Y$ be an irreducible algebraic variety over the field $\mathbb{C}$ of complex numbers, $Y^\*$ the associated complex space, $Y'$ the normalization of $Y$ (in the sense of algebraic geometry). Then the complex space associated with $Y'$ is the normalization of $Y^\*$ (in the analytic sense).
> 
> 
> 


In particular, $Y$ is normal (in the algebraic sense) if and only if its associated complex space $Y^\*$ is normal (in the analytic sense).


