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, ··· , 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 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
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