Affine braid group actions on derived categories ofalgebraic surfaces
Hokuto Uehara

1Autoequivalences of derived categories

Let X be a smooth projective variety over C. The derived category D(X)of X is a triangulated category whose objects are bounded complexes ofcoherent sheaves on X. By an equivalence D(Y ) ' D(X), we always mean
a C-linear equivalence of triangulated categories.
We denote the group of isomorphism classes of autoequivalences of D(X)by Auteq D(X).
©
ªAuteq D(X) := Φ : D(X) ' D(X) autoequivalence / ∼=It seems natural to consider the following problem.
Problem 1.1. Describe Auteq D(X) for smooth projective varieties X.
We note that Auteq D(X) always contains the group
A(X) := (Aut X n Pic X) × Z,
generated by functors of tensoring with invertible sheaves, automorphismsof X and the shift functor. When KX or −KX is ample, it is shown thatAuteq D(X) ∼= A(X) in [BO01]. When X is an abelian variety, Orlov solves
Problem 1.1 in [Or02]. In this case, Auteq D(X) is strictly larger than A(X).

1.1Spherical objects and twist functors

The twist functors along spherical objects are autoequivalences of anotherkind that are not in A(X). Let us recall the definition of them.
For an object P ∈ D(X × Y ), an integral functor
ΦP
X→Y : D(X) → D(Y )is defined byL∗ΦP
X→Y (−) = RπY ∗ (P ⊗ LπX (−)),

where πX : X × Y → X and πY : X × Y → Y are the projections.
Definition 1.2 ([ST01]). (i) We say that an object α ∈ D(X) is spherical if we have α ⊗ ωX ∼= α and(0 k 6= 0, dim X
HomkD(X) (α, α) ∼=C k = 0, dim X.
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(ii) Let α ∈ D(X) be a spherical object. We consider the mapping cone
LC = Cone(π1∗ α∨ ⊗ π2∗ α → O∆ )Lof the natural evaluation π1∗ α∨ ⊗ π2∗ α → O∆ , where ∆ ⊂ X × X is thediagonal, and πi is the projection of X × X to the i-th factor. Thenthe integral functor Tα := ΦCX→X defines an autoequivalence of D(X),called the twist functor along the spherical object α.
By the definition of twist functors Tα , for any object β ∈ D(X), we have atriangle
LR HomD(X) (α, β) ⊗C α−→β −→ Tα (β)
so that, at the level of K-theory,[Tα (β)] = [β] − χ(α, β)[α].
Example 1.3. (i) Let X be a K3 surface and L a line bundle on Z. Then
L is a spherical object of D(X).
(ii) Let Z be the fundamental cycle of −2-curves in ADE configurations
on a smooth surface X and L a line bundle on Z. Then L is a sphericalobject of DZ (X). Here DZ (X) denotes the full subcategory of D(X)consisting of objects supported on Z.

1.2

Results in [IU04].

In our paper [IU04], we consider a chain Z of −2-curves on a smooth surface
X and study the autoequivalences of the derived category DZ (X).
Note that the twist functor Tα can be defined as long as the support of
α is projective, even if X is not projective. Moreover, the category DZ (X)depends only on the formal neighborhood of Z in X. Thus we can assume
as follows:
Y = Spec C[[x, y, z]]/(x2 + y 2 + z n+1 )is the An -singularity,f :X→Y
its minimal resolution andZ = f −1 (P ) = C1 ∪ · · · ∪ Cn
where P ∈ Y is the closed point.
For an autoequivalence Φ ∈ Auteq DZ (X), we don’t know if it is always
isomorphic to an integral functor. Here, an integral functor from DZ (X) to
DZ (X) is defined by an object P ∈ D(X × X) whose support is projective
over X with respect to each projection. If an autoequivalence is given as anintegral functor, we call it a Fourier-Mukai transform (FM transform). LetAuteqFM DZ (X) ⊂ Auteq DZ (X)be the subgroup consisting of FM transforms.
We denote the dualizing sheaf on Z by ωZ and putDE¯
B := TOC (−1) , TωZ ¯ 1 ≤ l ≤ n ⊂ AuteqFM DZ (X).
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Theorem 1.4 ([IU04]). We have
AuteqFM DZ (X) = (hB, Pic Xi o Aut X) × Z.
Here Z is the group generated by the shift [1].
Remark 1.5. We know more about subgroups of AuteqFM DZ (X), that is,
we have the following:• B ∩ Pic X = h⊗OX (C1 ), . . . , ⊗OX (Cn )i.
• hB, Pic Xi ∼= B o Z/(n + 1)Z.
®
­ ¯• B = Tα ¯ α ∈ DZ (X), spherical .
According to Orlov’s theorem [Or97], any autoequivalence Φ ∈ Auteq D(S)for a smooth projective variety S is isomorphic to an integral functor ΦP
S→S
for some P ∈ D(S × S). Using the Orlov’s theorem, we obtain Theorem 1.6
below. The proof is similar to the one of Theorem 1.4.
Theorem 1.6 ([IU04]). Let S be a smooth projective surface of general
type whose canonical model has An -singularities at worst. Then we have¯­®
Auteq D(S) = TOC (a) , A(S) ¯ C : −2-curve, a ∈ Z .

2Affine braid group actions and Bridgeland’s stability conditions

2.1

Affine braid group actions on derived categories

Let us consider the situation in Theorem 1.4. Put αi := OCi (−1) (1 ≤ i ≤ n)and α0 := αn+1 := ωZ . Then B is generated by all Tαi ’s by the definition.
The result in [ST01] implies that Tαi ’s satisfy the following relations.
(Tαi Tαi+1 Tαi ∼= Tαi+1 Tαi Tαi+1 if 0 ≤ i ≤ n,∼Tαi Tαj = Tαj Tαiif i − j 6= ±1.
In other words, the generators of B satisfy the defining relations of the affine
braid group of type Ãn , which is denoted by A(Ãn ). Therefore it follows
from Theorem 1.4 that the group A(Ãn ) acts on DZ (X).
Conjecture 2.1 ([IU04]). The action of the affine braid group A(Ãn ) onDZ (X) is faithful. Consequently, we have an isomorphism B ∼= A(Ãn ).
Actually we can show this conjecture is true when n = 1, namely when
X is the minimal resolution of the A1 -singularity.
Theorem 2.2 ([IU05]). Conjecture 2.1 is true when n = 1.

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2.2

Bridgeland’s stability conditions

Conjecture 2.1 is deeply related to the following conjecture proposed byBridgeland.
Motivated by some physical ideas, Bridgeland considers the set of stability conditions Stab D on a given triangulated category D ([Br02]). He also
shows that Stab D has a natural topology and that it is a complex manifold.
We call Stab D the space of stability conditions on D.
Let us consider below the space of stability conditions on DZ (X), whereX and Z are as in Theorem 1.4.
Conjecture 2.3 (cf. [Br03]). Stab DZ (X) is simply connected and connected.
Our interesting observation is as follows.
Proposition 2.4 ([IU05]). Assume that Stab DZ (X) is connected. Then
Conjecture 2.1 holds precisely when Conjecture 2.3 holds.
Combining Theorem 2.2 with Proposition 2.4, we conclude that Stab DZ (X)is simply connected when Stab DZ (X) is connected and n = 1.

References[BO01] A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math.
125 (2001), 327–344.
[Br02] T. Bridgeland, Stability conditions on triangulated categories,(math.AG/0212237).
[Br03] T.
Bridgeland,Stability(math.AG/0307164).

conditionson

K3surfaces,[IU04] A. Ishii, H. Uehara, Autoequivalences of derived categories
on the minimal resolutions of An -singularities on surfaces,(math.AG/0409151), to appear in J. Differential Geom.
[IU05] A. Ishii, H. Uehara, Affine braid group actions on derived categories
and Bridgeland’s stability conditions, (in preparation).
[Or97] D. Orlov, Equivalences of derived categories and K3 surfaces. Algebraic geometry, 7. J. Math. Sci. (New York) 84 (1997), 1361–1381.
[Or02] D. Orlov, Derived categories of coherent sheaves on abelian varieties
and equivalences between them, Izv. Math. 66 (2002), 569–594[ST01] P. Seidel, R. Thomas, Braid group actions on derived categories ofcoherent sheaves, Duke Math. J. 108, (2001), 37–108.

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