SPACES OF STABILITY CONDITIONS 13
to S. One can either view S as the torsion part of a torsion pair on A, in which
case the torsion-free part is
F = {E A : HomA(S, E) = 0},
or as the torsion-free part, in which case the torsion part is
T = {E A : HomA(E, S) = 0}.
The corresponding tilted subcategories are defined to be
LS A = {E D :
Hi(E)
= 0 for i / {0, 1},
H0(E)
F and
H1(E)
S}
RS A = {E D :
Hi(E)
= 0 for i / {−1, 0},
H−1(E)
S and
H0(E)
T }.
We can now return to stability conditions. Suppose we are in the situation of
Lemma 5.2 and σ = (Z, P) is a stability condition in the boundary of the region
U(A). Then there is some i such that Z(Si) lies on the real axis. Forgetting about
higher codimension phenomena for now let us assume that Im Z(Sj ) 0 for every
j = i. Since each object Si is stable for all stability conditions in U(A), by Remark
3.3, each Si is at least semistable in σ, and hence Z(Si) is nonzero. The following
result is easily checked.
Lemma 5.5. In the situation of Lemma 5.2 suppose σ = (Z, P) Stab(D) lies
on a unique codimension one boundary of the region U(A) so that Im Z(Si) = 0
for a unique simple Si. Assume the categories LSi (A) and RSi (A) are of finite
length. Then either Z(Si) R
0
and a neighbourhood of σ is contained in U(A)
U(LSi (A)), or Z(Si) R
0
and a neighbourhood of σ is contained in U(A)
U(RSi (A)).
In general the tilted subcategories LSi (A) and RSi (A) need not be of finite
length and so we cannot necessarily repeat this process indefinitely. But in many
examples we can. Then we obtain a subset of Stab(D) covered by regions isomorphic
to Hn,
each one corresponding to a given heart A D, and with different regions
glued together along boundaries corresponding to pairs of hearts related by tilts at
simple objects. Thus understanding the algebra of the tilting process can lead to
a combinatorial description for certain spaces of stability conditions. In the next
section we shall see some examples of this.
6. Non-compact examples
Given the primitive state of knowledge concerning coherent sheaves on pro-
jective varieties of dimension at least three it is natural to study quasi-projective
varieties instead. A particularly amenable class of examples consists of varieties for
which there exists a derived equivalence
D Qcoh(X)

=
D Mod(B),
where Mod(B) is the category of modules over some non-commutative algebra B. In
practice, the non-compact variety X is often the total space of a holomorphic vector
bundle on a lower-dimensional variety Z; such examples are called local varieties in
the physics literature. The derived equivalence is then obtained using the theory
of exceptional collections [44] and the relevant algebras B can be described via a
quiver with relations.
13
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