Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function 
$$f\colon X \to Y $$
and assume that it is bijective at the level of $\mathbb{C}$-points. Moreover, assume that for every point $x$ of $X$ the differential
$$df(x)\colon T\_xX \to T\_{f(y)}Y $$
is an isomorphism, where the tangent spaces are Zariski tangent spaces.


Can we conclude that $X$ and Y are isomorphic? Do we need any assumption on the singularities? In particular, should $Y$ be normal?


In other words, I am asking under which conditions on the singularities the implicit function theorem holds. I remember some notes by Kollar, where this issue was related with Canonical singularities, but I could not find them anymore.


Any reference or example is welcome.


thanks



Implicit Function Theorem on Singular Varieties
In Joe Harris' book, Algebraic Geometry, this is theorem 14.9.


