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 $\mathbb{C}$? 



Fundamental group of an open subscheme of a normal scheme
Let me expand my comment into an answer. 


Take as $X$ the cone of vertex $v$ over an elliptic curve $E$. Then $X$ is simply connected (this is a general property of projective cones). However, $U = X-\{v\}$ is *not* simply connected: in fact, the projection $\pi \colon U \to E$ onto the basis gives $X$ the structure of a topological fibration with fiber homeomorphic to $\mathbb{C}$, so the corresponding long exact sequence of homotopy groups yields $$\pi\_1(U) = \pi\_1(E) = \mathbb{Z} \oplus \mathbb{Z}.$$ 


