derived categories of twisted sheaves on calabi-yau manifolds

Applying (ρ −1
i )∗ we get
and thus
(ρ −1
i )∗ O X k ij (ti − tj) = OXij (ρi(ti) − ρi(tj))
= OXij (si − τij(sj))
= OXij (si)
−1
⊗ OXij (τij(sj))
= OXij (si) ⊗ τ ∗ jiOXij (−sj)
= OXij (si) ⊗ OXij (−sj) ⊗ L −1
ji
= OXij (si) ⊗ OXij (−sj) ⊗ OXij ((k − 1)(si − sj))
= OXij (k(si − sj)),
ϕ ∗ (µ k i ) ∗ O X k ij (ti − tj) = µ ∗ i (ρ −1
i )∗ O X k ij (ti − tj)
= µ ∗ i OXij (k(si − sj)),
which shows that if X is represented by α ∈ XS(JX), then X k is represented by
(ϕ −1 ) ∗ α k in XS(J X k).
The result about the existence <strong>of</strong> a universal sheaf is known (see, for example,
[6]). But here is a quick pro<strong>of</strong> <strong>of</strong> the existence <strong>of</strong> a universal sheaf when
(k, n) = 1: we only need to show that we can twist the universal <strong>sheaves</strong> described
in the beginning <strong>of</strong> the pro<strong>of</strong> in a way that would make them glue. A computation
similar to the one done in Section 4.3 shows that
Mij = Uj|Pij
⊗ U −1
i |Pij
is equal to the pull-back <strong>of</strong> a line bundle Fij from X k ij, and Fij can be taken to
be <strong>of</strong> the form
ρ ∗ i L −1
ij = ρ∗ i OXij ((k − 1)(sj − si)).
Now let S be a line bundle on X, <strong>of</strong> fiber degree n (such a line bundle exists
globally on X, by the comments at the end <strong>of</strong> the previous section), and consider
). We have
the line bundles S ′
i = ρ ∗ i Si (where Si = S |Xi
(ρ −1
j )∗ (S ′
j ⊗ (S ′
i ) −1 ) = Sj ⊗ τ ∗ ijS −1
i
= Sj ⊗ S −1
i
⊗−n
⊗ L
= OXij (−n(k − 1)(si − sj)).
Therefore, considering the collections <strong>of</strong> line bundles {O X k i (ti)} and {S ′
i }, we
see that their coboundaries are {O(k(si −sj))} and {O(−n(k −1)(si −sj))} (where
we have omitted the subscript Xij and the pull-backs by isomorphisms to X k
because these do not matter, since we are dealing with line bundles <strong>of</strong> degree 0).
Since k and −n(k − 1) are coprime, we can obtain any multiple <strong>of</strong> {O(si − sj)} out
<strong>of</strong> them, in particular we can construct {O((k − 1)(sj − si))}, which we have seen
is the obstruction to gluing the Ui’s together. Therefore this obstruction is trivial,
and hence the Ui’s can be glued to form a global universal sheaf, thus finishing
the pro<strong>of</strong>.
ij
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