Spherical objects on Kummer surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:16:51Z http://mathoverflow.net/feeds/question/38542 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38542/spherical-objects-on-kummer-surfaces Spherical objects on Kummer surfaces Andre Bloch 2010-09-13T06:40:36Z 2010-09-13T06:55:10Z <p>Spherical objects E in the derived category of coherent sheaves over a K3 surface satisfy:</p> <p>1) Hom(E,E)=C 2) Ext^2(E,E)=C 3 Ext^i=0 otherwise.</p> <p>Are the structure sheaf O_X and the sheaves associated with the exceptional divisors the only spherical objects on a Kummer surface?</p> http://mathoverflow.net/questions/38542/spherical-objects-on-kummer-surfaces/38543#38543 Answer by Sasha for Spherical objects on Kummer surfaces Sasha 2010-09-13T06:55:10Z 2010-09-13T06:55:10Z <p>Certainly not. The group of autoequivalences acts on the set of all spherical objects, so a twist of a spherical object is spherical, hence any line bundle is spherical. Also, you can apply the reflection in one spherical object to another. E.g., applying the reflection in $O_X$ to $O_E(t)$ (where $E$ is a rational $(-2)$ curve and $t \ge 0$) one concludes that $Cone(O_X^{t+1} \to O_E(t))$ is spherical. You can continue by acting with another reflections to get a huge number of spherical objects.</p>