# Fundamental groups of degree 2 covers of projective spaces

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.

• 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.
– abx
Sep 18 at 9:14

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.