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.

  • 2
    $\begingroup$ 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

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,


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.