Is a flat morphism of complex algebraic varieties open in the analytic topology?

Question

Let $X,Y$ be algebraic varieties over $\mathbb{C}$. A morphism $f:X\to Y$ induces a morphism $f^{an}:X^{an}\to Y^{an}$ between the associated complex analytic spaces (actually, I am interested only in the topological aspects). There are a lot of results relating the properties of $X,Y,f$ and those of $X^{an},Y^{an},f^{an}$, known collectively as GAGA. I am interested in the following:

If $f$ is flat, does it follow that $f^{an}$ is open?

It is known that if $f$ if flat then it is open in the Zariski topology. A similar looking "dual" statement that I am aware of is that if $f$ is proper, then $f^{an}$ is proper in the topological sense and in particular closed.

In general, I would like to know sufficient conditions on $f$, for $f^{an}$ to be open.

Edit:

This is not stated or proved in Serre's GAGA (at least not explicitly). I did read somewhere that it is true, but without a proof or reference to one.

Answer 1

The answer is yes. See this post from mathoverflow.

Blablablabla.....