derived categories of twisted sheaves on calabi-yau manifolds

Theorem 5.1.10. Under the hypotheses <strong>of</strong> Theorem 5.1.6, the map ϕ is a Hodge
isometry between ˜ H(X, Q) and ˜ H(M, Q). It maps v ∈ ˜ H(X, Q) to the vector
(0, 0, ω) ∈ ˜ H(M, Q), and it therefore induces a Hodge isometry
v ⊥ /v ∼ = H 2 (M, Q),
the latter being computed inside ˜ H(X, Q). Restricted to v ⊥ /v, this isometry is
independent <strong>of</strong> the choice <strong>of</strong> quasi-universal bundle and is integral, i.e. it takes
integral vectors to integral vectors. It therefore induces a Hodge isometry
the latter now computed in ˜ H(X, Z).
H 2 (M, Z) ∼ = v ⊥ /v,
Remark 5.1.11. Using the Torelli theorem, this gives a complete description <strong>of</strong> the
moduli space M.
Pro<strong>of</strong>. When the moduli problem is fine, one first proves that U (the universal
sheaf) induces an equivalence <strong>of</strong> <strong>derived</strong> <strong>categories</strong> Db coh (X) ∼ = Db coh (M). We’ll
do this in a more general context, in Theorem 5.5.1. Then the result follows
immediately from the results in Section 3.1. The general case (when the moduli
problem is not fine) is done by a deformation argument in [25, Chapter 6].
Theorem 5.1.12. Let n be the greatest common divisor <strong>of</strong> the numbers (u, v),
where u runs over all ˜ H 1,1 (X) ∩ ˜ H(X, Z). Then the following statements hold:
1. There exist α-<strong>twisted</strong> locally free <strong>sheaves</strong> on M <strong>of</strong> ranks r1, r2, . . . , rk, with
gcd(r1, r2, . . . , rk) = n, and therefore α is n-torsion. (α is the obstruction
described in Definition 5.1.8.)
2. Denote by ϕ any cohomological Fourier-Mukai transform defined above (for
some choice <strong>of</strong> quasi-universal sheaf). Then ϕ maps TX into TM, (viewing
TX as a sublattice <strong>of</strong> ˜ H(X, Z) via the inclusion λ ↦→ (0, λ, 0)), and ϕ|TX is
independent <strong>of</strong> the choice <strong>of</strong> quasi-universal bundle used to define it.
3. There exists λ ∈ TX such that v + λ is divisible by n (in ˜ H(X, Z)); for such
a λ we have ϕ(λ) divisible by n (in TM).
4. ϕ is injective, and its cokernel is a finite, cyclic group <strong>of</strong> order n, generated
by ϕ(λ)/n for any λ satisfying the condition in (3).
Example 5.1.13. In Example 5.1.7 the number n = 2. Therefore the obstruction
α (which lives in the Brauer group <strong>of</strong> the moduli space, which is the double cover
<strong>of</strong> P 2 branched over a sextic) is <strong>of</strong> order 2.
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