balanced curves in Calabi-Yau 3-folds - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:16:59Z http://mathoverflow.net/feeds/question/27481 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27481/balanced-curves-in-calabi-yau-3-folds balanced curves in Calabi-Yau 3-folds Mohammad F.Tehrani 2010-06-08T15:34:40Z 2010-06-08T20:31:15Z <p>A balanced smooth rational curve in a calabi-Yau X is a smooth rational curve whose normal bundle is \$O(-1)\oplus O(-1)\$.</p> <p>We usually like these curves because of their rigidity. </p> <p>But, Is there any theorem that guaranty the existence of at least one such curve. For example for Quintic? Or for any other example?</p> http://mathoverflow.net/questions/27481/balanced-curves-in-calabi-yau-3-folds/27513#27513 Answer by mdeland for balanced curves in Calabi-Yau 3-folds mdeland 2010-06-08T20:31:15Z 2010-06-08T20:31:15Z <p>Perhaps you already know this: but we don't even know how to show that there are finitely many rational curves of a given degree \$d\$ on the general quintic threefold. This was originally conjectured by Clemens. However, for low degrees (up to \$d = 11\$ or something close to that), the conjecture is verified in the "strong form": any smooth rational curve of low degree (again, at most \$11\$) on the general quintic has normal bundle \$O(-1) \oplus O(-1)\$. As far as I know, that's the current state of affairs. It can often be difficult to produce rational curves with the "expected" normal bundle in any given situation! </p>