Rational or elliptic curves on Calabi-Yau threefolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:43:11Z http://mathoverflow.net/feeds/question/69716 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69716/rational-or-elliptic-curves-on-calabi-yau-threefolds Rational or elliptic curves on Calabi-Yau threefolds diverietti 2011-07-07T12:50:44Z 2011-07-07T17:48:08Z <p>Let $X$ be a Calabi-Yau threefold. From a complex analytic point of view, it is widely believed that it should not be Kobayashi hyperbolic, that is it should always admit some non-constant entire map from the complex plane $f\colon\mathbb C\to X$. </p> <p>One could be even more ambitious and ask whether a Calabi-Yau threefold always contains a rational or an elliptic curve (or, more generally a non-constant image of a complex torus).</p> <p>Mostly string theorists have produced lots of examples of such manifolds, mainly by adjunction or crepant resolution of singularities. So my question is:</p> <p><strong>Is it true that in all known examples of Calabi-Yau threefold one can always find a rational or an elliptic curve (or, more generally a non-constant image of a complex torus)?</strong></p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/69716/rational-or-elliptic-curves-on-calabi-yau-threefolds/69735#69735 Answer by Francesco Polizzi for Rational or elliptic curves on Calabi-Yau threefolds Francesco Polizzi 2011-07-07T17:30:03Z 2011-07-07T17:30:03Z <p>Let me give a partial answer.</p> <p>Most of the known examples of Calabi-Yau threefolds contain rational curves. However, there exist examples of Calabi-Yau threefolds <em>without</em> rational curves.</p> <p>You can find some of them in the paper by Oguiso and Sakurai <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>, Asian Journal of Mathematics 5 (2001).</p> <p>These threefolds, that the authors call "of Type A", are constructed as the quotient af an Abelian threefold $A$ by a suitable fixed-point free finite group of automorphisms. </p> <p>Moreover, a Calabi-Yau threefold $X$ is of type A if and only if $c_2(X)=0$, and in this case the Picard number $\rho(X)$ is either $2$ or $3$. </p> <p>In fact, the authors ask as an open question whether every Calabi-Yau threefold of Picard number $\rho \neq 2,3$ contains rational curves.</p> <p>I do not know whether Calabi-Yau threefolds of type A contain <em>elliptic</em> curves, but one can probably check this directly, since the construction is very explicit.</p>