derived categories of twisted sheaves on calabi-yau manifolds

Pro<strong>of</strong>. Trivial chase through the definitions. The last statement follows from the
commutativity <strong>of</strong> the diagram
H 1 (X, P GL(n))
H 1 (X, P GL(n))
t
✲ H 2 (X, Z/nZ)
a
❄
✲ ′
Br (X),
which is deduced from the map <strong>of</strong> short exact sequences
0 ✲ Z/nZ ✲ SL(n) ✲ PGL(n) ✲ 0
❄
❄
0 ✲ ∗
O ✲
X GL(n) ✲ PGL(n) ✲ 0.
Pro<strong>of</strong>. (<strong>of</strong> Theorem 5.2.1) Let n = rk(E ), and consider the projective bundle
associated to E , Y = Proj(E ) → X. As such, it has an associated topological
twisting class, t = t(Y/X) ∈ H 2 (X, Z/nZ), for which we have ϕ(t) = α, where
ϕ is the map from the previous proposition. Since the topological twisting class
is natural and is a topological invariant, we must have t|Xs = t(Ys/Xs) for every
s ∈ S. But on X0, we must have t|X0 = −c1(E0) mod n by Proposition 5.2.4. Since
the map ϕ is compatible with restriction, we have
α = ϕ(t) = ϕ(t|X0) = ϕ(−c1(E0) mod n) = − 1
n c1(E0).
Remark 5.2.6. In particular, this shows that on Xs we must have
as well.
α|Xs = − 1
n c1(E0)
5.3 Identifying the Obstruction
In this section we compute, for a moduli problem on a K3 surface, the obstruction α
to the existence <strong>of</strong> a universal sheaf (Definition 5.1.8), in terms <strong>of</strong> the cohomological
Fourier-Mukai transform described in Definition 5.1.9.
The setup is, as before, an algebraic K3 surface X, with a fixed polarization P,
and v a primitive, isotropic vector in ˜ H 1,1 (X)∩ ˜ H(X, Z), such that the moduli space
<strong>of</strong> semistable <strong>sheaves</strong> <strong>of</strong> Mukai vector v, M(v), is non-empty and does not contain
any properly semistable points. Then, according to Theorem 5.1.6, M = M(v) is
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