6 TOM BRIDGELAND
(c) if φ1 φ2 and Aj ∈ P(φj) then HomD(A1, A2) = 0,
(d) for each nonzero object E ∈ D there is a ﬁnite sequence of real numbers
φ1 φ2 ··· φn
and a collection of triangles
0 E0 E1 E2
. . .
En−1 En E
A1 A2 An
with Aj ∈ P(φj) for all j.
Remark 3.2. The central charge part of the deﬁnition of a stability condition
is mathematically quite bizarre. For example it means that even if one takes the
derived category of a variety deﬁned over a ﬁeld of positive characteristic one still
obtains a complex manifold as the space of stability conditions. Omitting the central
charge in Deﬁnition 3.1 gives the more natural notion of a slicing. Note however
that the universal cover of the group of orientation-preserving homeomorphisms of
the circle acts on the set of slicings of D by relabelling the phases φ, so that spaces
of slicings will always be hopelessly inﬁnite-dimensional.
Given a stability condition σ = (Z, P) as in the deﬁnition, each subcategory
P(φ) is abelian. The nonzero objects of P(φ) are said to be semistable of phase
φ in σ, and the simple objects of P(φ) are said to be stable. It follows from the
other axioms that the decomposition of an object 0 = E ∈ D given by axiom (d)
is uniquely deﬁned up to isomorphism. Write φσ +(E) = φ1 and φσ −(E) = φn. The
mass of E is deﬁned to be the positive real number mσ(E) =
For any interval I ⊂ R, deﬁne P(I) to be the extension-closed subcategory
of D generated by the subcategories P(φ) for φ ∈ I. Thus, for example, the full
subcategory P((a, b)) consists of the zero objects of D together with those objects
0 = E ∈ D which satisfy a φσ −(E) φσ +(E) b.
In order for Theorem 3.5 below to hold it is necessary to restrict attention to
stability conditions satisfying an extra technical condition. A stability condition
is called locally ﬁnite if there is some 0 such that each quasi-abelian category
P((φ − , φ + )) is of ﬁnite length. For details see .
3.2. Spaces of stability conditions. We always assume that our triangu-
lated categories are essentially small, i.e. equivalent to a category with a set of
objects. Write Stab(D) for the set of locally-ﬁnite stability conditions on a ﬁxed
triangulated category D. It has a natural topology induced by the metric
d(σ1, σ2) = sup
(E) − φσ1
(E)|, | log
| ∈ [0, ∞].
Remark 3.3. Note that the set of stability conditions σ ∈ Stab(D) for which
a given object E ∈ D is semistable is closed. Indeed, a nonzero object E ∈ D is
semistable in a stability condition σ precisely if φσ
Remark 3.4. Is it possible to use properties of the above metric to show that
spaces of stability conditions are always contractible? Such a result would have non-
trivial implications for computing groups of autoequivalences of derived categories.
There are no known counterexamples.