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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?

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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.

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