Let $X$ and $Y$ be two algebraic varieties (reduced schemes of finite type) over $\mathbb{C}$. Let $f : X \to Y$ be a morphism of schemes. Let $X^{an}$, $Y^{an}$ and $f^{an}$ the corresponding analytified objects.
Is it true that $f$ is a finite morphism of schemes if and only if $f^{an}$ is a finite morphism of complex analytic varieties ? (i.e. $f^{an}$ sends closed subset to closed subset and each fiber $f^{-1}(y)$, $y \in f^{an}(X^{an})$ is a finite set) ?