Krdytkyu

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 $mathbb{C}$?

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 $mathbb{R}$, so the corresponding long exact sequence of homotopy groups yields $$pi_1(U) = pi_1(E) = mathbb{Z} oplus mathbb{Z}.$$

In fact, quite the opposite tends to be true. Mumford [1] showed that for $(X,0)$ the germ of a normal surface singularity (over $mathbf{C}$), $U=Xsetminus 0$, one has $pi_1(U)={1}$ if and only if $X$ is smooth. At the same time, $pi_1(X) = {1}$ since $0to X$ is a homotopy equivalence.

If $X$ is smooth, this is true (the etale variant is called "Zariski-Nagata purity").

EDIT. To address Francesco's comment: of course the example is not projective. The easiest projective example was given by Francesco in his comment: $X$ is the (projective) cone over an elliptic curve $E$ and $U$ the complement of the vertex. Then $pi_1(U)= pi_1(E) = mathbb{Z}^2$ and $pi_1(X) = {1}$.

[1] Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math., Inst. Hautes Étud. Sci. 9, 5-22 (1961). ZBL0108.16801.