derived categories of twisted sheaves on calabi-yau manifolds

But this is impossible, because then F would be the trivial extension OP ⊕ IQ,
contradiction. We conclude that F has no zero-dimensional sub<strong>sheaves</strong>, hence it
is pure.
Let H be a subsheaf <strong>of</strong> IQ. Viewing H as a subsheaf <strong>of</strong> OC via the natural
inclusion IQ → OC, we can consider the closed subscheme H that is determined
by H . Note that H is a subscheme <strong>of</strong> C and hence it is reduced at the generic
point <strong>of</strong> each component <strong>of</strong> C.
H could be one <strong>of</strong> the following:
1. Supp H = C; in this case H is a nilpotent ideal sheaf, hence 0 because C is
reduced. We have P (H ; t) = 0.
2. H is supported on one component, say l1, and possibly at some other isolated
points on l2; then P (OH; t) = t + 1 + k where k ≥ 0 takes into account the
extra isolated points on l2, as well as the possible nonreducedness <strong>of</strong> H at
the singular points <strong>of</strong> C. (Recall that we have P (O P 1; t) = t+1 by Riemann-
Roch.) Therefore P (H ; t) = t − k − 1. Since Q ∈ H, if Q is on l2 − l1 then
necessarily k ≥ 1.
3. H is supported at a number <strong>of</strong> points; then P (H , t) = 2t−k for some k ≥ 1,
and if H = IQ then k ≥ 2.
Let G be a nonzero proper subsheaf <strong>of</strong> F , and consider the composite map
G → F → OP . There are two cases to consider, when this map is zero or when it
is surjective. In the first case, G is a subsheaf <strong>of</strong> IQ so by the previous analysis
p(G ; t) = t − k/n where n is 1 or 2 and k ≥ 1 and hence p(G ; t) < t = p(F ; t), so
that G cannot be a destabilizing sheaf for F .
Assume we’re in the second case, and consider H to be the kernel <strong>of</strong> the
composite map G → F → OP , which is a subsheaf <strong>of</strong> IQ. Since we have the
exact sequence
0 → H → G → OP → 0
we see that we have P (G ; t) = P (H ; t) + 1. Checking each possibility for H in
turn, we see that indeed p(G ) ≤ t = p(F ).
Since we proved that p(G ; t) ≤ p(F ; t) for all proper sub<strong>sheaves</strong> G <strong>of</strong> F , we
conclude that F is semistable.
Now assume P and Q are lying in the same component <strong>of</strong> C, and none is a
singular point <strong>of</strong> C. Without loss <strong>of</strong> generality, assume P, Q ∈ l1. In order to have
p(G ; t) = t for some proper subsheaf G <strong>of</strong> F , the map G → OP must be surjective,
and P (H ; t) = t−1 (H is, as before, the kernel <strong>of</strong> G → OP ). (If P (H ; t) = 2t−1,
then H = IQ, and hence G = F , contradicting our assumption that G is a proper
subsheaf <strong>of</strong> F .) Therefore H = Il1 or H = Il2. Since H ⊆ IQ, and Q ∈ l2,
H must be Il1. But then H is zero around P , hence G is the trivial extension
between H and OP . This is a contradiction, because if this were the case OP
would be a subsheaf <strong>of</strong> F , contradicting the fact that F is pure. We conclude
that p(G ; t) < t for all proper sub<strong>sheaves</strong> G <strong>of</strong> F .
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