derived categories of twisted sheaves on calabi-yau manifolds

and therefore
Now consider the map
given by
Br(M) = (H 2 (M, Z)/ NS(M)) ⊗ Q/Z.
p : H 2 (M, Z) → T ∨
M
p(v)(w) = (v, w)
for w ∈ TM. It is obviously a linear map, it is surjective (because H 2 (M, Z) is
unimodular) and its kernel consists <strong>of</strong> all v ∈ H 2 (M, Z) such that (v, w) = 0 for
all w ∈ TM, which is T ⊥ M . Since NS(M) is a primitive sublattice in H2 (M, Z), we
have
T ⊥ M = (NS(M) ⊥ ) ⊥ = NS(M),
and therefore we conclude that
thus obtaining the result <strong>of</strong> the lemma.
T ∨
M = H 2 (M, Z)/ NS(M),
Lemma 5.4.2. Let H be a unimodular lattice, let N be a primitive sublattice in
H, and let n be a fixed integer. Define Nn to be the sublattice <strong>of</strong> N consisting <strong>of</strong> all
x ∈ N such that (x, y) is divisible by n for all y ∈ N. Also, let T = N ⊥ , and let Tn
be defined just as Nn. Then there is a natural group isomorphism Nn/nN ∼ = Tn/nT
defined by sending an element v ∈ Nn/nN to the unique element λ <strong>of</strong> Tn/nT such
that v − λ is divisible by n inside H.
Pro<strong>of</strong>. Begin by proving that for every v ∈ Nn there exists a λ ∈ Tn such that
v − λ is divisible by n. The idea is from [32]: consider the functional
1
n v∨ = 1
∨
(v, ·) ∈ N
n
(it takes integer values because v ∈ Nn). Since N is a primitive sublattice in
the unimodular lattice H, there exists w ∈ H such that w∨ |N = 1
nv∨ . Thus
(v − nw, x) = 0 for all x ∈ N, and thus λ = v − nw ∈ T . Obviously, we have
v − λ divisible by n inside H (equaling nw). The fact that λ is in fact in Tn is
immediate:
(λ, y) = (λ − v, y) = −n(w, y)
for any y ∈ T .
It is obvious that the map is well-defined, bijective, and a group homomorphism,
so we’re set.
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