Fundamental groups of Calabi-Yau varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:07:12Z http://mathoverflow.net/feeds/question/65208 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65208/fundamental-groups-of-calabi-yau-varieties Fundamental groups of Calabi-Yau varieties ulrich 2011-05-17T07:35:45Z 2011-05-17T19:01:20Z <p>By a Calabi-Yau variety I mean a smooth projective variety over the complex numbers with numerically trivial canonical divisor. </p> <blockquote> <p>For each postive integer \$n\$ does there exist a finite group \$G\$ (possibly depending on \$n\$) which is not the fundamental group of a Calabi-Yau variety of dimension \$n\$?</p> </blockquote> <p>For \$n=1,2\$ this follows easily from the classification of Calabi-Yau varieties of these dimensions, the only non-trivial finite fundamental group being \$\mathbb{Z}/2\mathbb{Z}\$ (for Enriques surfaces). For \$n=3\$ some finite non-abelian groups are known to occur as fundamental groups but I do not know of any non-existence results.</p> <p>If there are only finitely many families of Calabi-Yau varieties of a given dimension then the question would clearly have a positive answer. However, this is far from being known so I am interested in other possible approaches.</p> http://mathoverflow.net/questions/65208/fundamental-groups-of-calabi-yau-varieties/65210#65210 Answer by Sándor Kovács for Fundamental groups of Calabi-Yau varieties Sándor Kovács 2011-05-17T08:15:55Z 2011-05-17T08:15:55Z <p>This is definitely not an easy question. For \$n=3\$ checkout the paper <a href="http://www.intlpress.com/AJM/p/2001/5_1/AJM-5-1-043-078.pdf" rel="nofollow">Calabi-Yau Threefolds of Quotient Type</a> by Oguiso and Sakurai. They are ("only") concerned with whether the fundamental group is finite. However, that should perhaps be the first step towards answering your question.</p> http://mathoverflow.net/questions/65208/fundamental-groups-of-calabi-yau-varieties/65266#65266 Answer by Jason Starr for Fundamental groups of Calabi-Yau varieties Jason Starr 2011-05-17T19:01:20Z 2011-05-17T19:01:20Z <p>If n is even, then I believe you can use the Atiyah-Bott fixed point formula to rule out many cases. For instance, let G be a simple, non-cyclic group. Consider the action of G on the Hodge group \$H^{0,n}(X)\$. Since G has only the trivial character, this action must be trivial. Then for every element g in G, the holomorphic Lefschetz number is 2 (if n is odd, the number is 0, which doesn't help). Therefore g has a fixed point. </p>