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