10 TOM BRIDGELAND
where ch(E) is the Chern character of E and td(X) is the Todd class of X. The fact
that the Mukai form is non-degenerate means that for any σ = (Z, P) Stab(X)
we can write the central charge Z in the form
Z(E) = (π(σ), v(E))
for some vector = π(σ) N (X). This defines a map π : Stab(X) N (X) C.
It was proved in [13] that there is a connected component
Stab†(X)
Stab(X)
that is mapped by π onto the open subset
P0
+
(X) =
P+(X)
\
δ∈∆(Λ,Ω)
δ⊥
N (X) C.
Moreover, the induced map
Stab†(X)
P0
+(X)
is a covering map, and if
Aut†(D)
is the subgroup of Aut(D) preserving the connected component
Stab†(X),
then
Stab†(X)
Aut†(D)

=
P0
+(X)
O+(Λ, Ω)
.
Comparing this result with the discussion in Section 2.3 we see that the space
Aut†(D)\ Stab†(X)/C
agrees closely with the leaf MK(X) MK3 corresponding to the fixed holomorphic
two-form Ω, but that there are two differences.
Firstly, the space MK(X) consists of points in the projective space of
Ω⊥

¯


=
2
p=0
Hp,p(X)

H∗(X,
C),
whereas the stability conditions space only sees the algebraic part N (X) C. This
problem could presumably be fixed by changing the definition of the map Z as
suggested in Remark 3.7.
The second point is that the vectors in MK(X) satisfy ( , ) = 0 whereas the
space of stability conditions has no such normalisation. This means that Stab(X)
is one complex dimension larger than one would otherwise expect. This may seem a
minor point but is in fact very important, being the first glimpse of Hodge theoretic
restrictions on the central charge. We discuss such restrictions further in Section 7.
4.3. Other examples. The only other compact varieties for which the space
of stability conditions is well-understood are abelian surfaces and varieties of di-
mension one. The case of abelian surfaces is very similar to the K3 case but easier;
it is covered in [13]. It might also be possible to calculate the space of stability con-
ditions on higher-dimensional abelian varieties, although I have not thought about
this in any detail. Turning to non-Calabi-Yau examples S. Okada [41] proved that
Stab(P1)

=
C2
and E. Macri [40] proved that for any curve X of genus g 2 one has
Stab(X)

=
˜
GL +(2, R)

= C × H.
It is also possible that the definition of stability condition needs to be changed for
non-Calabi-Yau categories to take account of the non-trivial Serre functor in some
way.
It would be extremely interesting to calculate the space of stability conditions
on a compact Calabi-Yau threefold such as the quintic. Unfortunately we don’t
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