SPACES OF STABILITY CONDITIONS 11

know enough about coherent sheaves on threefolds to be able to do this. In par-

ticular it would be useful to know the set of Chern characters of Gieseker stable

bundles. In fact it is not known how to write down a single stability condition on a

Calabi-Yau threefold. Nonetheless, physicists have used mirror symmetry together

with a certain amount of guesswork to make some nontrivial computations [1, 2].

One way to proceed for the quintic would be to construct its derived category

via matrix factorisations as in [43] and then to somehow construct the stability con-

dition corresponding to the Gepner point in the stringy K¨ ahler moduli space. This

seems to be an interesting project. For more on stability conditions on categories

of matrix factorisations see [37, 46, 52].

5. T-structures and tilting

In this section I explain the connection between stability conditions and t-

structures. This is the way stability conditions are constructed in practice. I also

explain how the method of tilting can be used to give a combinatorial description

of certain spaces of stability conditions. This technique will be applied in the next

section to describe spaces of stability conditions on some non-compact Calabi-Yau

varieties.

5.1. Stability conditions and t-structures. A stability function on an

abelian category A is deﬁned to be a group homomorphism Z : K(A) → C such

that

0 = E ∈ A =⇒ Z(E) ∈ R

0

exp(iπφ(E)) with 0 φ(E) 1.

The real number φ(E) ∈ (0, 1] is called the phase of the object E.

A nonzero object E ∈ A is said to be semistable with respect to Z if every

subobject 0 = A ⊂ E satisﬁes φ(A) φ(E). The stability function Z is said to

have the Harder-Narasimhan property if every nonzero object E ∈ A has a ﬁnite

ﬁltration

0 = E0 ⊂ E1 ⊂ ··· ⊂ En−1 ⊂ En = E

whose factors Fj = Ej /Ej−1 are semistable objects of A with

φ(F1) φ(F2) ··· φ(Fn).

Given a stability condition σ = (Z, P) on a triangulated category D, the full

subcategory A = P((0, 1]) ⊂ D is the heart of a bounded t-structure on D. It

follows that A is an abelian category and we can identify its Grothendieck group

K(A) with K(D). We call A the heart of the stability condition σ. The central

charge Z deﬁnes a stability function on A, and the decompositions of axiom (d)

give Harder-Narasimhan ﬁltrations for objects of A.

Conversely, given a bounded t-structure on D together with a stability func-

tion Z on its heart A ⊂ D with the Harder-Narasimhan property, we can deﬁne

subcategories P(φ) ⊂ A ⊂ D to be the semistable objects in A of phase φ for each

φ ∈ (0, 1]. Axiom (b) then ﬁxes P(φ) for all φ ∈ R, and identifying K(A) with

K(D) as before, we obtain a stability condition σ = (Z, P) on D. The decompo-

sitions of axiom (d) arise by combining the truncation functors of the t-structure

with the Harder-Narasimhan ﬁltrations of objects of A. Thus we have

Proposition 5.1. To give a stability condition on a triangulated category D

is equivalent to giving a bounded t-structure on D together with a stability function

on its heart with the Harder-Narasimhan property.

11