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 $\mathbb{C}$?

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    In codimension $2$, is it not a cone over an elliptic curve a counterexample? – Francesco Polizzi Nov 19 at 16:05
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    @FrancescoPolizzi Is cone over an elliptic curve reduced? (sorry for asking a stupid question). Actually, I want to know the result for X normal projective integral scheme over $\mathbb{C}$. – Anonymous Nov 19 at 16:53
  • Of course it is reduced – Francesco Polizzi Nov 19 at 17:02
up vote 4 down vote accepted

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 $\mathbf{C}$), $U=X\setminus 0$, one has $\pi_1(U)=\{1\}$ if and only if $X$ is smooth. At the same time, $\pi_1(X) = \{1\}$ since $0\to X$ 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.

  • Well, strictly speaking, $X$ is not projective in your example. – Francesco Polizzi Nov 19 at 17:04

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