derived categories of twisted sheaves on calabi-yau manifolds

which by definition associates to an S-scheme T <strong>of</strong> finite type the set <strong>of</strong> isomorphism
classes <strong>of</strong> T -flat families <strong>of</strong> semistable <strong>sheaves</strong> on the fibers <strong>of</strong> the morphism
XT := T ×S X → T with Hilbert polynomial P . In particular, for any closed
point s ∈ S one has MX/S(P )s ∼ = MXs(P ). Moreover there is an open subscheme
M s X/S (P ) ⊆ MX/S(P ) that universally corepresents the subfunctor M s X/S ⊆ MX/S
<strong>of</strong> families <strong>of</strong> stable <strong>sheaves</strong>.
Pro<strong>of</strong>. See [25, 4.3.7].
Proposition 3.3.2. Let X/S be a flat, projective morphism, and let O(1) be a
relatively ample sheaf on X/S. For a polynomial P , consider the relative moduli
space M s /S <strong>of</strong> stable <strong>sheaves</strong> with Hilbert polynomial P on the fibers <strong>of</strong> X/S. Then
there exists a covering {Ui} <strong>of</strong> M s (by analytic open sets in the analytic setting, and
by étale open sets in the algebraic setting) such that on each X ×S Ui there exists
a local universal sheaf Ui. Furthermore, there exists an α ∈ ˇ H2 (M s , O∗ M s) (that
only depends on X/S, O(1) and P ) and isomorphisms ϕij : Uj|Ui∩Uj → Ui|Ui∩Uj
that make ({Ui}, {ϕij}) into an α-sheaf (which we will call a universal α-sheaf).
Definition 3.3.3. The element α ∈ ˇ H2 (M s , O∗ M s) described above is called the
obstruction to the existence <strong>of</strong> a universal sheaf on X ×S M, and is denoted by
Obs(X/S, P ), with O(1) being understood.
Pro<strong>of</strong>. For simplicity, we prove the statement in the absolute setting as the relative
case is entirely similar; also, we work in the analytic category. We use the notations
<strong>of</strong> [25, Section 4.6]. Note that R s → M s is a principal PGL(V )-bundle ([25, 4.3.5])
and hence R s is isomorphic, locally on M s , to the product <strong>of</strong> M s and PGL(V ). If U
is any open set in M s over which R s is trivial, say isomorphic to PGL(V )×U → U,
we can find over it GL(V )-linearized line bundles <strong>of</strong> Z-weight 1: for example,
GL(V ) × U → PGL(V ) × U is such a line bundle. Now apply a local version <strong>of</strong> [25,
4.6.2] to conclude that for every open set U <strong>of</strong> M s such that Rs ×M s U is trivial
(as a PGL(V )-bundle), there exists a local universal sheaf U on X × U.
The existence <strong>of</strong> α ∈ ˇ H2 (M s , O∗ M s) and <strong>of</strong> the isomorphisms ϕij that make
({Ui}, {ϕij}) into a universal α-sheaf now follows in exactly the same way as in
the pro<strong>of</strong> <strong>of</strong> [32, A.6], and the uniqueness <strong>of</strong> α is routine checking.
Proposition 3.3.4 (Mukai, [32, A.6]). Let X, M, S be proper schemes or analytic
spaces over C, with M integral, and assume given morphisms X → S and
M → S, such that X → S is projective. Let p1 and p2 be the projections to X
and M from X ×S M. For α ∈ ˇ H2 (M, O∗ M ) assume that there exists a coherent
p∗ 2α-<strong>twisted</strong> sheaf F on X ×S M that is flat over M. Then α is in fact in Br(M).
Pro<strong>of</strong>. Using Lemma 1.2.6, represent F as ({Fi}, {ϕij}) on a cover {p −1
2 (Ui)},
where {Ui} is an open cover <strong>of</strong> M. Pull back a relatively ample sheaf on X/S
via p1 to get a relatively ample sheaf O(1) on X ×S M → M, flat over M. Now
using semicontinuity, for each i we can find n0 such that for n ≥ n0 we have
p2∗(Fi ⊗ O(n)) locally free on Ui. Since M is proper over C, we can choose n0
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