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 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
H2(X,
C) be the
class of a nonzero holomorphic two-form on X; we consider 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
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