I'm interested in the etale fundamental group of a Calabi-Yau variety. The definition of a Calabi-Yau variety that I use, is a smooth projective variety over field $k=\bar k$ with trivial canonical bundle plus $H^i(X, \mathcal O_X)=0$ for $i=1,...,dim X -1$. I assume for the moment that the base field is of characteristic 0. If necessary, I could also assume $k=\mathbb C$.
Thank you in advance.
I met (a version of) Beauville-Bogomolov decomposition theorem in Thm 6.1 On the geometry of hyoersurfaces of low degrees in the projective space by Debarre. It says:
For a smooth projective variety $X$ over $\mathbb C$ with vanishing first $\mathbb R$-Chern class, there exists a finite etale cover which is isomorphic to the product of
- abelian varieties, or
- simply connected variety $Y$ of dim $\ge3$ with $H^i(Y,\mathcal O_Y)=0$ for $0<i<n$,
- simply connected holomorphic sympletic varieties.
(Note that the definition there for a CY variety is different from the one I take). But because of my weak background in complex geoemtry, I could not deduce anything special for $X$ being a Calabi-Yau variety... In particular, I would guess "Any Calabi–Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi–Yau manifold" (from Wikipedia) is a consequence of this decomposition theorem, but I could not work it out.
As I was looking through the literature (they might have different definitions for a Calabi-Yau!), I am aware that under such a definition, a Calabi-Yau might not be simply connected. And at the same time, most of the examples that people are concerned have finite or even trivial (topological) fundamental group (for a 3-fold with infinite fundamental group, see Oguiso-Sakurai and Kanazawa). So my question is what does the finite etale covers of a Calabi-Yau look like. More precisely,
- Is there an if and only if criterion for a Calabi-Yau to be simply connected? In our definition, dimension 2 are precisely the K3 surfaces, which are always simply connected. So the next is Calabi-Yau 3-fold. Can people say something at least for some special types of Calabi-Yau 3-fold?
- If the fundamental group is non-trivial, what could be the possible finite etale covers? In the decomposition theorem stated above, it only asserts the existence part of a good decomposition. In other words, not every finite etale cover is of that form. Some questions on this aspects include: what is the criterion for a finite etale cover of a Calabi-Yau to be still a Calabi-Yau? What can be the quotient of a Calabi-Yau by a finite group? For the latter , I found this Kollar-Larsen paper, but our definition for Calabi-Yau is stronger than the one they use. Can people say something more with our stronger definition?
I apologize if I missed out any results that are already written in the literature above. Different definitions of Calabi-Yau make me dizzy and maybe it's always better to get a thread from experts.
In the end, I would be glad to know any introductory notes to Calabi-Yau that use more the algebraic language.