SPACES OF STABILITY CONDITIONS 7

It is clear that there is a forgetful map

Z : Stab(D) −→ HomZ(K(D), C)

sending a stability condition to its central charge. The following result was proved

in [12]. Its slogan is that deformations of the central charge lift uniquely to de-

formations of the stability condition. One has to be a little bit careful because in

general (in fact even for D =

Db

Coh(X) with X an elliptic curve) the group K(D)

has inﬁnite rank.

Theorem 3.5. Let D be a triangulated category. For each connected com-

ponent

Stab∗(D)

⊂ Stab(D) there is a linear subspace V ⊂ HomZ(K(D), C),

with a well-deﬁned linear topology, such that the restricted map Z :

Stab∗(D)

→

HomZ(K(D), C) is a local homeomorphism onto an open subset of V .

In the cases of most interest for us we can get round the problem of K(D) being

of inﬁnite rank as follows.

Definition 3.6. If X is a smooth complex projective variety we write Stab(X)

for the set of locally-ﬁnite stability conditions on D =

Db

Coh(X) for which the

central charge Z factors via the Chern character map

K(D)

ch

Z

C

H∗(X, Q)

In physics language we are insisting that the central charge of a brane depends only

on its topological charge.

Theorem 3.5 immediately implies that Stab(X) is a ﬁnite-dimensional complex

manifold.

Remark 3.7. There are other more natural ways to get round the problem

of K(D) being inﬁnite rank which will almost certainly play a role in the further

development of the theory. One possibility would be to take some homology theory

H(D) associated to D for which there is a natural Chern character map K(D) →

H(D), and then insist that the central charge Z factors via H(D). For example

one could take periodic cyclic homology. In the case when D =

Db

Coh(X), with X

a smooth projective variety, it is known [53] that this homology theory coincides

as a vector space with the de Rham cohomology of X, and in particular is ﬁnite-

dimensional. But in the absence of any real understanding of why this would be

particularly a sensible thing to do we shall stick with the ad hoc deﬁnition above.

3.3. Group actions. The space of stability conditions of any triangulated

category has some natural group actions which will be important in what follows.

Lemma 3.8. The space Stab(D) carries a right action of the group

˜

GL +(2, R),

the universal covering space of

GL+(2,

R), and a left action of the group Aut(D) of

exact autoequivalences of D. These two actions commute.

7