derived categories of twisted sheaves on calabi-yau manifolds

for all F , G stable <strong>sheaves</strong> on X with Mukai vector v.
Since X is smooth <strong>of</strong> dimension 2, Ext i (F , G ) = 0 for i < 0 or i > 2 anyway.
Since F and G are stable, Hom(F , G ) = 0 if F = G . Therefore Ext 2 (F , G ) = 0
by Serre duality (use the fact that ωX is trivial). Since
χ(F , G ) = (v(F ), v(G )) = (v, v) = 0
by Lemma 3.1.7, we conclude that Ext 1 (F , G ) = 0 as well. Hom(F , F ) = C
because a stable sheaf is simple. Finally, the last condition holds automatically
because ωX is trivial.
As an application <strong>of</strong> this we prove the following:
Theorem 5.5.2. Let M be a K3 surface, let α ∈ Br(M), and let k ∈ Z be such
that gcd(k, ord(α)) = 1. Assume that there exist K3 surfaces X and X ′ , and Mukai
vectors v ∈ ˜ H 1,1 (X, Z), v ′ ∈ ˜ H 1,1 (X ′ , Z) such that we have isomorphisms
M ∼ = MX(v) ∼ = MX ′(v′ ),
and such that under these isomorphisms, Obs(X, v) −1 = α and Obs(X ′ , v ′ ) −1 = α k .
Then we have
D b coh(M, α) ∼ = D b coh(M, α k ).
Pro<strong>of</strong>. We have seen (Remark 5.4.6) that under these assumptions TX is Hodge
isometric to TX ′. Orlov’s theorem (Theorem 5.1.14) then tells us that
D b coh(X) ∼ = D b coh(X ′ ).
But Theorem 5.5.1 shows that D b coh (M, α) ∼ = D b coh (X) and Db coh (M, αk ) ∼ =
D b coh (X′ ), and thus
D b coh(M, α) ∼ = D b coh(M, α k ).
It is tempting (especially in view <strong>of</strong> the results <strong>of</strong> Chapter 6) to conjecture that
the conclusion <strong>of</strong> the previous theorem holds more generally:
Conjecture 5.5.3. For any K3 surface M, and any twisting α ∈ Br(M), we have
for any k with gcd(k, ord(α)) = 1.
D b coh(M, α) ∼ = D b coh(M, α k )
The difficulty in proving such a theorem would lie on one hand in the fact that
if one takes TX = Ker(α), then TX may not be embeddable in the K3 lattice LK3.
Even if one assumes this embeddability condition, there are problems with finding
the Mukai vector v.
83