derived categories of twisted sheaves on calabi-yau manifolds

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2. If one has Db coh (X, α) ∼ = Db coh (Y ) for smooth spaces X and Y , and for α ∈
Br ′ (X), does it imply that α ∈ Br(X)? This would have implications for the
conjecture that Br(X) = Br ′ (X) for a smooth space X.
3. What are the conditions for being able to find a smooth space Y with
Db coh (X, α) ∼ = Db coh (Y ), given X and α?
4. What is the most general setup in which one has
D b coh(X, α) ∼ = D b coh(X, α k )
for smooth X, α ∈ Br(X), k coprime to the order <strong>of</strong> α?
5. Is there more geometry to the “virtual” K3 surfaces from Remark 5.5.4?
6. In Example 6.2.3, is X 2 deformation equivalent to X? This would give a
complete counterexample to the Torelli problem (and it would also prove
that X and Y are not birational to each other). Note that in order to prove
this, it would be enough to solve the following problem: let C be a smooth
curve <strong>of</strong> genus 1, let r and d be coprime positive integers, and let V be an
irreducible vector bundle <strong>of</strong> rank r and degree d. Then one would need to
find a natural isomorphism
H 0 (C, V ) ∼ = H 0 (C, det V ).
7. What kind <strong>of</strong> singularities can arise through processes similar to the one
that generate the singularities <strong>of</strong> the relative Jacobian? In all the “generic”
examples I know <strong>of</strong> these singularities are ordinary double points. Are there
other cases, more degenerate?
8. How should one understand the Vafa-Witten, Aspinwall-Morrison example?
What is the relationship <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> with physics?
The main cause we cannot solve more <strong>of</strong> these issues is our lack <strong>of</strong> examples.
These would likely provide us with a better understanding <strong>of</strong> the theory <strong>of</strong> <strong>twisted</strong>
<strong>sheaves</strong>, which would bring forth answers to some <strong>of</strong> these questions.