In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry there is an exercise (Exercise 1.7 and 1.8) to prove the statement that the derived category $D^b(X)$ of a Calabi-Yau variety $X$ has no non-trivial semi-orthogonal decompositions. Yet also, it is later stated that it is known that every toric variety admits a full exceptional collection. Since every full exceptional collection gives rise to a semi-orthogonal decomposition (Example 1.60) and there exist toric Calabi-Yau varieties (though not smooth projective ones), these statements seem to contradict each other. Is someone able to explain the proof of this exercise and why it requires smoothness or projectivity?

  • $\begingroup$ the exercise, it seems, says "Calabi--Yau manifold", not "Calabi--Yau variety" (this does not substantially address your question but shows that there is no typo) $\endgroup$ – Aknazar Kazhymurat Mar 9 at 19:18
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    $\begingroup$ Directly above the exercise he writes: "Calabi-Yau manifolds, i.e. varieties with trivial canonical bundle", which shows that he is not assuming smooth or projective (this is the first mention of Calabi-Yau in this book) $\endgroup$ – Jeff Mar 9 at 19:48
  • $\begingroup$ The standard proof of this exercise very much relies on smoothness + properness, in that the key point is Serre duality. $\endgroup$ – dhy Mar 9 at 19:52
  • $\begingroup$ @dhy is it possible get away with gorensteinnes+properness? $\endgroup$ – Aknazar Kazhymurat Mar 9 at 20:01

For both statements properness is essential. The proof of the exercise (a theorem of Bridgeland) is based on Serre duality, and the construction of an exceptional collection (by Kawamata) is inductive with the base of induction given by the case of a weighted projective space. And in projective case no toric variety is Calabi-Yau, so there is no contradiction.


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