Finite fundamental groups of 3-dimensional Calabi-Yau manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:52:08Z http://mathoverflow.net/feeds/question/72998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72998/finite-fundamental-groups-of-3-dimensional-calabi-yau-manifolds Finite fundamental groups of 3-dimensional Calabi-Yau manifolds Dmitri 2011-08-16T16:52:37Z 2011-08-17T03:29:48Z <p><strong>Question.</strong> Is there an example of a compact $3$-dimensional Calabi-Yau manifold with finite fundamental group $G$ that does not admit a free action on $S^3$?</p> <p>This question is motivated by the following: it is known that many simply-connected Clabi-Yau 3-folds admit a singular Lagrangian torus fibration over $S^3$. I don't know if there are exceptions. On the other hand, if $\pi_1$ is finite and we still have a lagrangean torus fibration, one can expect that the base is a lens space. But in this case probably $\pi_1$ of the CY-manifold will be equal to $\pi_1$ of the base.</p> <p><strong>PS.</strong> As Tony Pantev explains, the answer to this question is YES -- there are such examples. On the other hand, if we assume that a finite group $G$ is acting freely on a CY 3-manifold preserving the volume form and preserving a Lagrangian torus fibration, this should impose some very strong restrictions on $G$. I wonder if anyone bothered to work out what is the restriction :). </p> http://mathoverflow.net/questions/72998/finite-fundamental-groups-of-3-dimensional-calabi-yau-manifolds/73005#73005 Answer by Tony Pantev for Finite fundamental groups of 3-dimensional Calabi-Yau manifolds Tony Pantev 2011-08-16T17:32:21Z 2011-08-17T00:00:27Z <p>This intuition seems to be only loosely right. There are many smooth compact CY threefolds with large fundamental groups. For instance $\mathbb{Z}/3\times \mathbb{Z}/3$, $\mathbb{Z}/8\times \mathbb{Z}/8$, are allowed fundamental groups and I am pretty sure that those do not act freely on $S^{3}$. </p> <p>More to the point - the Calabi-Yau threefolds that have these fundamental groups are explicitly constructed and we have a pretty good idea of the shape of (at least one of) their slag torus fibrations. For instance, in the first case the Calabi-Yau fibers by genus one curves over a rational elliptic surface, and the slag fibration is compatible with the genus one fibration. In the second case the Calabi-Yau fibers by abelian surfaces and again the slag fibration is compatible. So guided by the holomorphic picture you can easily imagine a situation where you group acts freely on the CY, preserves the slag torus fibration, and the induced action on the base of the fibration is <em>not</em> free. The only thing you can conclude really is that the action of the group on any fiber sitting over a fixed point in the base is free. This is possible to arrange on a torus by taking action by translations.</p> <p>So, even if your fundamental group happens to admit some free action on $S^{3}$, this doesn't mean that the action on the base of the slag fibration will be free. And, in general, I don't expect it to be free.</p>