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⊥

∩

¯

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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