8 TOM BRIDGELAND

Proof. First note that the group

˜

GL +(2, R) can be thought of as the set of

pairs (T, f) where f : R → R is an increasing map with f(φ + 1) = f(φ) + 1, and

T :

R2

→

R2

is an orientation-preserving linear isomorphism, such that the induced

maps on

S1

= R/2Z =

(R2

\ {0})/R

0

are the same.

Given a stability condition σ = (Z, P) ∈ Stab(D), and a pair (T, f) ∈

˜

GL +(2, R),

deﬁne a new stability condition σ = (Z , P ) by setting Z = T

−1

◦ Z and P (φ) =

P(f(φ)). Note that the semistable objects of the stability conditions σ and σ are

the same, but the phases have been relabeled.

For the second action, note that an element Φ ∈ Aut(D) induces an automor-

phism φ of K(D). If σ = (Z, P) is a stability condition on D deﬁne Φ(σ) to be the

stability condition (Z ◦

φ−1,

P ), where P (t)=Φ(P(t)).

Neither of the two group actions of Lemma 3.8 will be free in general. In

particular, if σ = (Z, P) is a stability condition in which the image of the central

charge Z : K(D) → C lies on a real line in C then σ will be ﬁxed by some subgroup

of

˜

GL +(2, R). However there is a subgroup C ⊂

˜

GL +(2, R) which does act freely.

If λ ∈ C then λ sends a stability condition σ = (Z, P) to the stability condition

λ(σ) = (Z , P ) where Z (E) =

e−iπλZ(E)

and P (φ) = P(φ + Re(λ)). Note that

for any integer n the action of the shift functor [n] on Stab(D) coincides with the

action of n ∈ C.

Remark 3.9. Return for a moment to the discussion of Section 2.4 in which

D =

Db

Fuk(X, β + iω).

The action of C on Stab(D) clearly corresponds to rotating the holomorphic three-

form Ω. It also seems reasonable to guess that the action of Aut(D) on Stab(D)

corresponds to the discrete group quotient N → M. Thus we might expect an

embedding of the complex moduli space MC(X) in the double quotient

Aut(D)\ Stab(D)/C.

The mirror statement is that if X is a Calabi-Yau with a given complex structure

and D =

Db

Coh(X) then the above quotient contains the stringy K¨ ahler moduli

space MK(X). In the next section we will examine this suggestion in some simple

examples.

4. Compact examples

In this section I review some examples of stability conditions on smooth projec-

tive varieties. The only Calabi-Yau examples are elliptic curves and K3 and abelian

surfaces.

4.1. Elliptic curves. Let X be a complex projective curve of genus one. It

was shown in [12] that the action of

˜

GL +(2, R) on Stab(X) is free and transitive.

Thus

Stab(X)

∼

=

˜

GL +(2, R)

∼

=

C × H

where H ⊂ C is the upper half-plane. Quotienting by the group of autoequivalences

of D =

Db

Coh(X) gives

Stab(X)

Aut(D)

∼

=

GL+(2,

R)

SL(2, Z)

8