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Does being a degree 2 cover of a projective space impose restrictions on the fundamental groups of non-singular complex projective varieties? For curves it does not.

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    $\begingroup$ A (smooth) double cover of $\Bbb{P}^n$ is simply connected for $n\geq 2$. This follows from the Fulton-Hansen theorem, though this case was certainly known much before. $\endgroup$
    – abx
    Sep 18 at 9:14
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Here is a general statement.

The main theorem of M. Cornalba, Una osservazione sulla topologia dei rivestimenti ciclici di varietà algebriche, Boll. UMI (5) 18-A (1981), 323-328. Is:

Theorem (Cornalba): Suppose that $f: X \rightarrow Y$ is a branched cover of smooth projective $n$-folds, along an ample divisor $D \subset Y$ Then, the following holds

  1. for each $1 \leq i \leq n-1$, $f_{*}: \pi_{i}(X) \rightarrow \pi_{i}(Y)$ is an isomorphism.

  2. $f_{*}: \pi_{n}(X) \rightarrow \pi_{n}(Y)$ is surjective.

Note the similarity to the to the Lefshetz hyperplane theorem. Cornalba states in the introduction that this is an analogue of the Lefshetz hyperplane theorem,

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