TITLE: Fundamental group of an open subscheme of a normal scheme
QUESTION [6 upvotes]: Let $X$ be an irreducible normal projective scheme over $\mathbb{C}$. Let $U$ be the open subscheme of smooth points of $X$. Consider the closed subscheme $Z = X \setminus U$. Suppose that the codimension of $Z$ in $X$ is at least $2$. Is it true that the fundamental group of $U$ and $X$ are isomorphic? 
Edit: Is it true for $X$ an integral normal projective scheme over $\mathbb{C}$?

REPLY [3 votes]: Although $X$ and $U$ need not have isomorphic fundamental group (as the accepted answer shows), the induced map from the inclusion $U\hookrightarrow X$ is generally $\pi_1$-surjective.
Precisely, if $X$ is a normal projective variety, and $A\subset X$ is a proper closed subvariety, then the natural homomorphism $\pi_1(X-A) \to \pi_1(X)$ is surjective.
You can find this result in On the fundamental groups of normal varieties by Donu Arapura, Alexandru Dimca, Richard Hain.
Interestingly, although natural inclusions only give $\pi_1$-surjections, it turns out that natural projections do give $\pi_1$-isomorphisms.
Precisely, let $G$ be a connected reductive algebraic affine group over an algebraically closed field $k$ (arbitrary characteristic).  Assume $G$ is acting on a smooth connected projective variety $M$ (and there is an appropriate ample line bundle $\mathcal{L}$).  Then the homomorphism (induced by the GIT projection) of algebraic fundamental groups $\pi_1(M)\to \pi_1(M/\! /_{\mathcal{L}}G)$ is an isomorphism.  If $k = \mathbb{C}$, then there is also an isomorphism between the topological fundamental groups.
You can find this result in Fundamental group of a geometric invariant theoretic quotient by Indranil Biswas, Amit Hogadi, A. J. Parameswaran.