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 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
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