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