Spherical objects $E$ in the derived category of coherent sheaves over a K3 surface satisfy:

  • $\operatorname{Hom}(E,E)=\mathbb{C}$,
  • $\operatorname{Ext}^2(E,E)=\mathbb{C}$,
  • $\operatorname{Ext}^i(E,E)=0$ otherwise.

Are the structure sheaf $O_X$ and the sheaves associated with the exceptional divisors the only spherical objects on a Kummer surface?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.