SPACES OF STABILITY CONDITIONS 15

It is usually convenient to restrict attention the full subcategory D ⊂ D Qcoh(X)

consisting of bounded complexes with coherent cohomology sheaves supported on

the zero-section Z ⊂ X. In a sense this is the interesting part of the category, the

rest being rather flabby. The equivalence above induces an equivalence between

D and the full subcategory of D Mod(B) consisting of bounded complexes with

ﬁnite-dimensional and nilpotent cohomology modules. The standard t-structure

on D Mod(B) then induces a bounded t-structure on D whose heart A ⊂ D is

equivalent to the ﬁnite length category Mod0(B) of ﬁnite-dimensional and nilpo-

tent B-modules. We call the hearts A ⊂ D obtained from geometric exceptional

collections in this way exceptional.

6.2. Tilting, mutations and Seiberg duality. In order to apply the theory

of Section 5 to the situation described above we need to understand the tilting

process for exceptional subcategories. This problem is closely related to the theory

of mutations of exceptional collections, as studied in the Rudakov seminar [44].

The connection is worked out in detail in [18], see also [3, 8, 31].

In the case when Z satisﬁes

n = dim K(Z) ⊗ C = 1 + dim Z

(for example when Z is a projective space) the relationship between tilting and

mutations is particularly straightforward and is described precisely in [14]. In par-

ticular any tilt of an exceptional subcategory A ⊂ D is the image of an exceptional

subcategory by some autoequivalence of D. This means that the tilting process

can be continued indeﬁnitely. The combinatorics of the process is controlled by the

aﬃne braid group

Bn = τ0, ··· , τn−1 | τiτj τi = τj τiτj if j − i ≡ ±1(n) and τiτj = τj τi otherwise .

For details on this result we refer the reader to [14]; the main input is Bondal

and Polishchuk’s work [10]. In the case Z =

P2

these results allow one to give a

combinatorial description of a connected component of Stab(D).

Theorem 6.3 ([16]). Set Z =

P2

and deﬁne D ⊂

Db

Coh(ωZ ) as above. Then

there is a subset of Stab(D) that can be written as a disjoint union of regions

g∈G

U(g),

where G = B3 is the aﬃne braid group on three strings. Each region U(g) is

isomorphic to H3 and consists of stability conditions with a given heart A(g) ⊂ D.

The closures of two regions U(g1) and U(g2) intersect along a codimension one

boundary precisely if g1g2

−1

=

τi±1

for some i. Each of the categories A(g) is

equivalent to a category of nilpotent representations of a quiver with relations of the

form

•

a

•

b

•

c

where the positive integers a, b, c counting the numbers of arrows connecting the

vertices always satisfy the Markov equation

a2

+

b2

+

c2

= abc.

15