derived categories of twisted sheaves on calabi-yau manifolds

- if P = Q, Ext i
X(F 0 (P ), F 0 (Q)) = 0, for all i;
- Ext i
X(F 0 (P ), F 0 (P )) = 0, for all P , and for i > 3;
- HomX(F 0 (P ), F 0 (P )) = C, for all P ;
- F 0 (P ) ⊗ ωX ∼ = F 0 (P ), for all P .
104
The last property follows easily from the adjunction formula and the fact that
the fibers <strong>of</strong> X have trivial canonical bundles.
If P and Q lie in different fibers <strong>of</strong> ¯ J → S, there is nothing to prove as
Supp F 0 (P ) and Supp F 0 (Q) are disjoint. So assume that P and Q lie in the
same fiber ¯ Js (over some s ∈ S). Let i : Xs → X be the inclusion, so that
F 0 (P ) = i∗F (P ) and F 0 (Q) = i∗F (Q).
Using Proposition 3.3.5 we reduce to checking that
- if P = Q, Ext i Xs (F (P ), F (Q)) = 0 for all i;
- Ext i
Xs (F (P ), F (P )) = 0 for all P , and for i > 1;
- HomXs(F (P ), F (P )) = C for all P .
For irreducible fibers (smooth, nodal or cuspidal), the <strong>sheaves</strong> F (P ) are stable,
so the first and third statements follow from [25, 1.2.7 and 1.2.8], while for I2
fibers they follow from Lemma 6.3.6. Since Xs is Cohen-Macaulay and has trivial
dualizing sheaf, we get by Serre duality
Ext i
Xs
(F (P ), F (Q)) = Ext1−i(F
(Q), F (P )).
Xs
which proves the second statement, thus finishing the pro<strong>of</strong>.
Since the actual result is quite central to the theory, we restate it in a slightly
different form:
Theorem 6.5.4. Let X → S be a generic elliptic Calabi-Yau threefold, let J → S
be its relative Jacobian, and let ¯ J be an analytic small resolution <strong>of</strong> the singularities
<strong>of</strong> J. Let U = S \∆, where ∆ is the discriminant locus <strong>of</strong> X → S. If α ∈ Br(JU) is
the obstruction to the existence <strong>of</strong> a universal sheaf on XU ×U JU (or, alternatively,
the element <strong>of</strong> the Tate-Shafarevich group representing X → S), then there is a
unique extension <strong>of</strong> α to an element ¯α <strong>of</strong> Br( ¯ J). For this ¯α, there is an equivalence
<strong>of</strong> <strong>derived</strong> <strong>categories</strong>
D b coh(X) ∼ = D b coh( ¯ J, ¯α −1 ).
Remark 6.5.5. The fact that X is a Calabi-Yau threefold is irrelevant for the actual
pro<strong>of</strong>. We only needed that KX is trivial along the fibers <strong>of</strong> X → S, which is an
immediate consequence <strong>of</strong> the adjunction formula, if X → S has fibers which are
plane curves <strong>of</strong> arithmetic genus 1.
As an easy corollary <strong>of</strong> the above theorem, we obtain the following result: