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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.

Edit:

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.

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  • $\begingroup$ Have you heard of the Beauville-Bogomolov decomposition theorem? I'm not sure this theorem gives you exactly what you want, but it might get you started over $\mathbb{C}$. $\endgroup$ – Andrew Aug 7 '18 at 19:48
  • $\begingroup$ With this definition of a CY variety, there are non-simply connected CY varieties. $\endgroup$ – Ariyan Javanpeykar Aug 7 '18 at 21:54
  • $\begingroup$ What kind of things would you like to know about these groups? $\endgroup$ – Asal Beag Dubh Aug 8 '18 at 8:41
  • $\begingroup$ Thank you for all the comments! I have re-edited my question according to them. $\endgroup$ – user31480 Aug 8 '18 at 12:04
  • $\begingroup$ @user31480 Just a short comment, as I think there is a little bit of confusion. If $X$ is an algebraic variety with infinite (etale) fundamental group, then every finite etale cover of $X$ has an infinite (etale) fundamental group. However, if $X$ is an algebraic variety with finite (topological) fundamental group, it could happen that $X$ has a trivial etale fundamental group. $\endgroup$ – Ariyan Javanpeykar Aug 8 '18 at 19:32

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