derived categories of twisted sheaves on calabi-yau manifolds

In order to identify the kernel, note that the map ϕ becomes invertible when
tensored with Q, and the same holds also for ϕ∨ . If ψ ∈ T ∨
X is such that ψ(λ) =
1 mod n, then ψ ′ = ψ◦(ϕ⊗Q) −1 is an element <strong>of</strong> T ∨
∨
M ⊗Q whose image in TM ⊗Q/Z
generates the kernel. Indeed, its image in T ∨
X ⊗Q is ψ, which is integral (and hence
becomes zero in T ∨
X ⊗ Q/Z), so it belongs to the kernel. Since ψ generates the
cokernel <strong>of</strong> the map T ∨
∨
M → TX , ψ′ generates the kernel <strong>of</strong> T ∨
∨
M ⊗ Q/Z → TX ⊗ Q/Z.
It is easy to see that all these constructions are natural, and that having chosen
a particular generator µ for the cokernel <strong>of</strong> TX → TM, we have singled out a
generator <strong>of</strong> the kernel <strong>of</strong> Br(M) → Br(X).
Now consider the result <strong>of</strong> Theorem 5.3.1: it states that in order to obtain α,
we have to find u ∈ ˜ H(X, Z) such that (u, v) = 1 mod n, and consider the image
α = ϕ(u) ∈ Br(M).
Tracing through the identifications <strong>of</strong> Br(M) with T ∨
M ⊗ Q/Z, we find that ϕ(u) ∈
Br(M) corresponds to the functional ψ ′ ∈ T ∨
M ⊗ Q/Z given by
ψ ′ ( · ) = (ϕ(u), · ),
where ϕ(u) is now viewed as an element <strong>of</strong> ˜ H(M, Q). But it is easy to see that
ψ ′ (ϕ(λ)) = (ϕ(u), ϕ(λ)) = (u, λ) = 1 mod n,
the last equality following from the fact that v − λ is divisible by n by the construction
<strong>of</strong> λ, and (u, v) = 1 mod n.
Therefore
ψ( · ) = ψ ′ ◦ ϕ( · ) = (ϕ(u), ϕ( · )) = (u, · )
is an integral functional on TX, satisfying ψ(λ) = 1 mod n, and thus ψ ′ is the
distinguished generator <strong>of</strong> the kernel <strong>of</strong> the map
T ∨
M ⊗ Q/Z → T ∨
X ⊗ Q/Z.
We conclude that α, the obstruction to the existence <strong>of</strong> a universal sheaf on X ×
M, is precisely the distinguished generator <strong>of</strong> the kernel <strong>of</strong> the map Br(M) →
Br(X).
Remark 5.4.5. This result should be contrasted with [11, 5.3.5], where the authors
observe that in the case <strong>of</strong> an elliptically fibered K3 X, Br(X) = Br(J), where J is
the relative Jacobian. Of course, J can be viewed as a moduli space <strong>of</strong> semistable
<strong>sheaves</strong> on X (Chapter 4), so that one should in fact get
0 → Z/nZ → Br(J) → Br(X) → 0,
with the kernel generated by α ∈ Br(J) which is the obstruction to the existence
<strong>of</strong> a universal sheaf on X × J, which coincides with the element α ∈ XS(J) that
represents X. The explanation for this apparent mismatch is the following: Br(X)
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