derived categories of twisted sheaves on calabi-yau manifolds

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where now F |l2 = (F |l2) free ⊕ OQ. Composing the surjection F → F |l2 with the
surjection F |l2 → (F |l2) free , we get the desired surjection in this case.) Denote by
Gi the kernel <strong>of</strong> this surjection.
Following through the analysis <strong>of</strong> Proposition 6.3.5, we find that the only possible
sub<strong>sheaves</strong> <strong>of</strong> F that could destabilize it are <strong>of</strong> the form Gi. The degree <strong>of</strong>
Gi on li is si − 2 − δiq, where δiq is 1 if Q is in li and a smooth point <strong>of</strong> C, and 0
otherwise. Thus the Hilbert polynomial <strong>of</strong> Gi is
P (t; Gi) = pit + si − 1 − δiq.
The reduced Hilbert polynomial <strong>of</strong> Gi is then
p(t; Gi) = t + si − 1 − δiq
.
On the other hand, the Hilbert polynomial <strong>of</strong> F is
and its reduced Hilbert polynomial is
pi
P (t; F ) = (p1 + p2)t + s1 + s2 − 1
p(t; F ) = t + s1 + s2 − 1
.
p1 + p2
In order for Gi to be destabilizing, we should have
But this is equivalent to
si − 1 − δiq
pi
≥
k
p1 + p2
si − 1 − δiq ≥ kpi
which is impossible since δiq ≥ 0, and
kpi
si =
p1 + p2
p1 + p2
(Note that if kpi is divisible by p1 + p2, we can have properly semistable <strong>sheaves</strong>,
as we have seen before.)
Therefore we conclude that all the <strong>sheaves</strong> UQ for Q ∈ C are stable, and a
computation similar to Lemma 6.3.6 shows that they are all distinct.
6.7 Applications to the Torelli Problem
Proposition 6.7.1. Let X and Y be Calabi-Yau threefolds, (complex projective
manifolds, simply connected and with trivial canonical class) such that Db coh (X) ∼ =
Db coh (Y ). Then there exists a Hodge isometry
.
H 3 (X, Z[1/2])free ∼ = H 3 (Y, Z[1/2])free,
.
,