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) =

∑

i

|Z(Ai)|.

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 [12].

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

0=E∈D

|φσ2

−

(E) − φσ1

−

(E)|, |φσ2(E)

+

− φσ1

+

(E)|, | log

mσ2 (E)

mσ1 (E)

| ∈ [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

+(E)

= φσ

−(E).

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.

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