18 TOM BRIDGELAND

As explained by Takahashi [46], the unfolding space T of an isolated hyper-

surface singularity X0 of dimension n should be related to the space of stability

conditions on the Fukaya category of the Milnor ﬁbre Xt of the singularity. Note

that

µ = dimC Hn(Xt, C) = dimC T.

Given a basis L1, ··· , Lµ of Hn(Xt, C), K. Saito’s theory of primitive forms shows

that for a suitable family of holomorphic n-forms Ωt on the ﬁbres Xt the periods

Z(Li) =

Li

Ωt,

form a system of flat co-ordinates on the unfolding space T . Since these periods

are the analogues of central charges this is exactly what one would expect from

Theorem 3.5.

On the other hand, the big quantum cohomology of a Fano variety Z seems to

be related to the space of stability conditions on the derived categories of Z and

of the corresponding local Calabi-Yau variety ωZ . In the case when the quantum

cohomology of Z is generically semisimple Dubrovin showed how to analytically

continue the Gromov-Witten prepotential from an open subset of H∗(X, C) to give

a Frobenius structure on a dense open subset of the conﬁguration space

M ⊂ Confn(C) = {(u1,··· , un) ∈

Cn

: i = j =⇒ ui = uj }.

According to a conjecture of Dubrovin [25] the quantum cohomology of Z is generi-

cally semisimple (so that one can deﬁne the above extended moduli space M) iﬀ the

derived category

Db

Coh(Z) has a full, strong exceptional collection (E0, ··· , En−1)

(so that one can understand the space of stability conditions by tilting as in Section

5). Moreover, in suitable co-ordinates, the Stokes matrix of the quantum cohomol-

ogy Sij (which controls the analytic continuation of the Frobenius structure on

M) is equal to the Gram matrix χ(Ei, Ej ) (which controls the tilting or mutation

process). For more on this see [16].

Finally, Barannikov and Kontsevich [5] showed that if X is a smooth complex

projective variety then the formal germ to deformations of X, whose tangent space

has dimension

H1(X,

TX ), is contained in a larger formal germ whose tangent space

has dimension

p,q

Hp(X,

q

TX ),

and which describes A∞ deformations of the category

Db

Coh(X). Suppose X1

and X2 are a mirror pair of Calabi-Yau threefolds. Complex deformations of X1

correspond to K¨ ahler deformations of X2. Passing to extended moduli spaces one

might imagine that some global form of Barannikov and Kontsevich’s space pa-

rameterising deformations of

Db

Coh(X1) should be mirror to the space of stability

conditions on X2. Very schematically we might write

Def(Db

Coh(X1))

∼

=

Stab(Db(Coh(X2)),

although to make the dimensions add up one should extend Stab(X2) so that its

tangent space is the whole cohomology of X2 as in Remark 3.7. Note that such

an isomorphism would be a mirror symmetry statement staying entirely within the

realm of algebraic geometry.

18