SPACES OF STABILITY CONDITIONS 9

which is thus a

C∗−bundle

over the modular curve H/ PSL(2, Z). In fact this is

the

C∗−bundle

parameterising equivalence classes of data consisting of a complex

structure on X together with a non-zero holomorphic one-form.

According to Remark 3.9 we expect an inclusion of the stringy K¨ ahler moduli

space in the quotient

Aut(D)\ Stab(X)/C

∼

=

H/ PSL(2, Z).

We also know that

MK(X) = MC(

ˇ

X ),

and since tori are self-mirror the latter is just the moduli of complex structures on

a two-torus. Thus we obtain perfect agreement in this case.

Remark 4.1. The calculation of the space of stability conditions on an elliptic

curve has been generalised by Burban and Kreussler to include irreducible singular

curves of arithmetic genus one [19]. The resulting space of stability conditions and

the quotient by the group of autoequivalences is the same as in the smooth case.

Remark 4.2. It is possible that spaces of stability conditions can be (partially)

compactiﬁed by adding non-locally-ﬁnite stability conditions. In the example of

the elliptic curve the non-locally-ﬁnite stability conditions up to the action of C

are parameterised by R \ Q. It might be interesting to think this point through in

some other examples.

4.2. K3 surfaces. Let X be an algebraic K3 surface and set D =

Db

Coh(X).

I use the notation introduced in Section 2.3; in particular Λ denotes the lattice

H∗(X,

Z) equipped with the Mukai symmetric form. Let Ω ∈

H2(X,

C) be the

class of a nonzero holomorphic two-form on X; we consider Ω as an element of

Λ ⊗ C. The sublattice

N (X) = Λ ∩

Ω⊥

⊂

H∗(X,

Z)

can be identiﬁed with Z ⊕ Pic(X) ⊕ Z and has signature (2, ρ), where 1 ρ 20 is

the Picard number of X. Write

O+(Λ,

Ω) for the subgroup of

O+(Λ)

consisting of

isometries which preserve the class [Ω] ∈ P(Λ ⊗ C). Any such isometry restricts to

give an isometry of N (X). Set ∆(Λ, Ω) = ∆(Λ) ∩

Ω⊥

and for each δ ∈ ∆(Λ, Ω) let

δ⊥

= { ∈ N (X) ⊗ C : ( , δ) = 0} ⊂ N (X) ⊗ C

be the corresponding complex hyperplane.

Deﬁne an open subset P(X) ⊂ N (X) ⊗ C consisting of vectors ∈ N (X) ⊗ C

whose real and imaginary parts span a positive deﬁnite two-plane in N (X) ⊗ R.

Taking orthogonal bases in the two-plane shows that P(X) is a GL(2, R)-bundle

over the set

Q(X) = { ∈ P(N (X) ⊗ C) : ( , ) = 0 and ( ,

¯)

0}.

These spaces P(X) and Q(X) have two connected components that are exchanged

by complex conjugation. Let

P+(X)

and Q+(X) denote the components containing

vectors of the form (1, iω, −ω2/2) with ω ∈ NS(X) the class of an ample line bundle.

The Mukai vector of an object E ∈ D(X) is deﬁned to be

v(E) = ch(E) td(X) ∈ N (X),

9