STABILITY CONDITIONS ON An-SINGULARITIES Akira Ishii, Kazushi Ueda & Hokuto Uehara ...

40 Pages·2010·0.33 MB·English

j. differentialgeometry
84(2010)87-126
STABILITY CONDITIONS ON An-SINGULARITIES
Akira Ishii, Kazushi Ueda & Hokuto Uehara
Abstract
We study the spaces of locally finite stability conditions on the
derived categories of coherent sheaves on the minimal resolutions
of An-singularities supported at the exceptional sets. Our main
theorem is that they are connected and simply-connected. The
proof is based on the study of spherical objects in [30] and the
homological mirror symmetry for An-singularities.
1. Introduction
The theory of stability conditions on triangulated categories is intro-
duced by Bridgeland [9] based on the work of Douglas et al. [2, 15,
16, 17, 18] on the stability of BPS D-branes. It is a fine mixture of
the theory of t-structures [3] and the slope stability [36], which allows
us to represent any object in a triangulated category as a successive
mapping cone of semistable objects in a unique way. He proved that
the set Stab of stability conditions on a triangulated category sat-
T T
isfying an additional assumption called local-finiteness has a natural
structure of a complex manifold, and proposed to study this manifold
as an invariant of . Since the definition of stability conditions uses
T
only the triangulated structure of , the group Auteq of triangle au-
T T
toequivalences of naturally acts on the manifold Stab , suggesting
T T
a geometric approach to study the structure of Auteq .
T
Such an approach has been pursued by Bridgeland himself [11] when
is the bounded derived category DbcohX of coherent sheaves on a
T
complex algebraic K3 surface X, leading him to the following remark-
able result and conjecture: There is a distinguished connected com-
ponent Σ(X) of the space StabX of locally finite, numerical stability
conditions on DbcohX. His conjecture is:
(i) Σ(X) is preserved by AuteqDbcohX, and
(ii) Σ(X) is simply-connected.
Assuming this conjecture, he could prove that AuteqDbcohX is an
extension of the index two subgroup Aut+H (X,Z) of the group of
∗
Received 02/12/2008.
87
88 A. ISHII, K. UEDA & H. UEHARA
Hodge isometries of the Mukai lattice of X by the fundamental group
π +(X) of the period domain of X.
1P0
For a positive integer n, let
f :X Y = SpecC[x,y,z]/(xy +zn+1)
→
be the minimal resolution of the A -singularity. Let further be the
n
D
bounded derived category Dbcoh X of coherent sheaves on X sup-
Z
ported at the exceptional set Z, and be its full triangulated subcate-
C
gory consisting of objects E satisfying Rf E = 0. The categories and
∗ C
serve as toy models of the derived categories of coherent sheaves on
D
K3 surfaces. The main result in this paper is the following:
Theorem 1. Stab and Stab are connected, and Stab is simply-
C D D
connected.
Thisresult,togetherwiththesimply-connectednessofadistinguished
connected component of Stab proved by Thomas [42], shows that the
C
above conjecture of Bridgeland holds in these cases. When n = 1, the
connectedness of Stab has also been proved by Okada [37].
D
The basic strategy of our proof for the connectedness of Stab is to
D
find a stability condition such that structure sheaves of all the closed
points are stable in any given connected component of Stab . Since
D
the set of such stability conditions form a distinguished connected open
subset of Stab , the connectedness of Stab follows.
D D
Theorem7duetoBridgeland shows thatthesimply-connectedness of
Stab followsfromthefaithfulnessofanaffinebraidgroupactionon .
D D
We prove this using homological mirror symmetry for A -singularities
n
and ideas from Khovanov and Seidel [33]. Unfortunately, we have to
work over a field of characteristic two in order to apply Theorem 27 by
Khovanov and Seidel, and we lift this faithfulness result to any charac-
teristic using the deformation theory of complexes by Inaba [29].
In contrast to the case of Stab , we cannot use algebro-geometric
D
argument in the proof of the connectedness of Stab , since does not
C C
contain any skyscraper sheaves. Instead, we use a result in [30] and
ideas from [33] to reduce the problem of the connectedness of Stab to
C
that of configurations of curves on a disk.
The organization of this paper is as follows: In 2, we collect basic
§
definitions and known results used in this paper. In 3, we recall the
§
McKay correspondence for A -singularities in such a way that is valid
n
in any characteristic. In 4, we give the proof of the connectedness of
§
Stab . We prove the faithfulness in characteristic two in 5 and lift it
D §
to any characteristic in 6. The connectedness of Stab is proved in 7.
§ C §
In the appendix, we prove that every autoequivalence of is given by
D
an integral functor.
Acknowledgment: We thank Jun-ichi Matsuzawa and Yukinobu
Toda for valuable discussions and suggestions. A. I. is supported by
STABILITY CONDITIONS ON An-SINGULARITIES 89
the Grants-in-Aid for Scientific Research (No.18540034). K. U. is sup-
ported by the 21st Century COE Program of Osaka University. H. U. is
supported by the Grants-in-Aid for Scientific Research (No.17740012).
2. Generalities
We collect basic definitions and known results in this section. All
the categories appearing in this paper will be essentially small. For a
triangulated category , K( ) denotes its Grothendieck group, and for
T T
an object E , [E] will denote its class in K( ). For two objects
∈ T T
E,F and i Z, Hom∗(E,F), Hom≤i(E,F), and Hom≥i(E,F) will
∈ T ∈ j T jT T j
denote Hom (E,F), Hom (E,F), and Hom (E,F),
j Z j i j i
respectively∈. T ≤ T ≥ T
L L L
2.1. Stability conditions on triangulated categories.The follow-
ing definition is introduced by Bridgeland [9] based on the work of
Douglas et al. [2, 15, 16, 17, 18] on the stability of BPS D-branes:
Definition 2. A stability condition σ = (Z, ) on a triangulated
P
category consists of
T
a group homomorphism Z : K( ) C, and
• T →
full additive subcategories (φ) for φ R
• P ∈
satisfying the following conditions:
(i) If 0 = E (φ), then Z(E) = m(E)exp(iπφ) for some m(E)
6 ∈ P ∈
R ,
>0
(ii) for all φ R, (φ+1) = (φ)[1],
∈ P P
(iii) for A (φ ) (j = 1,2) with φ > φ , we have Hom (A ,A ) =
j j 1 2 1 2
∈ P T
0,
(iv) for every nonzero object E , there is a finite sequence of real
∈ T
numbers
φ > φ > > φ
1 2 n
···
and a collection of triangles
0 = E E E E E =E
0 1 2 n 1 n
··· −
(1)
A A A
1 2 n
with A (φ ) for all j.
j j
∈ P
Z is called the central charge, and the collection of triangles in (1) is
called the Harder-Narasimhan filtration. It follows from the definition
that (φ) is an abelian category, and its non-zero object (φ)
P E ∈ P
is said to be semistable of phase φ. is said to be stable if it is a
E
simple object of (φ), i.e., there are no proper subobjects of in P(φ).
P E
By [9, proposition 5.3], to give a stability condition on a triangulated
category is equivalent to giving a bounded t-structure on and a
T T
90 A. ISHII, K. UEDA & H. UEHARA
stability function (previously called a centered slope-function) on its
heart with the Harder-Narasimhan property. For the definitions of a
stability function and the Harder-Narasimhan property, see [9, 2].
§
The set of stability conditions satisfying a certain technical condition
called local-finiteness [9, definition 5.7] is denoted by Stab . This
T
condition ensures that each (φ) is a finite length category so that each
P
semi-stable object has a Jordan-H¨older filtration. By combining it with
the Harder-Narasimhan filtration, any non-zero object E admits
∈ T
a decomposition as in (1) such that A (φ ) is stable for all j and
j j
∈ P
φ φ φ . Bridgeland introduces a natural topology on
1 2 n
≥ ≥ ··· ≥
Stab such that the forgetful map
T
: Stab Hom(K( ),C)
Z T → T
∈ ∈
(Z, ) Z
P 7→
satisfies the following:
Theorem 3 ([9, theorem 1.2]). For each connected component Σ
of Stab , there is a linear subspace V(Σ) Hom(K( ),C) with a
T ⊂ T
well-defined linear topology such that the restriction gives a local
Σ
Z|
homeomorphism.
Hence Stab forms a (possibly infinite-dimensional) complex man-
T
ifold modeled on the topological vector space V(Σ). When = or
T C
, K( ) is finite-dimensional, and we prove in Lemma 15 that V(Σ)
D T
always coincides with Hom(K( ),C).
T
Since the definition of Stab uses only the triangulated structure of
T
, the group Auteq of triangle autoequivalences of acts naturally
T T T
on Stab from the left; for σ = (Z, ) Stab and Φ Auteq ,
T P ∈ T ∈ T
Φ(σ) = (Φ Z,Φ( ))
∗
P
where Φ is the pull-back by the inverse of the automorphism Φ :
∗
∗
K( ) K( ) induced by Φ. This action commutes with the right
T → T
action of the universal cover G]L+(2,R) of the general linear group
GL+(2,R)withpositivedeterminant,which“rotates”thecentralcharge
[9, lemma 8.2].
2.2. Minimal resolutions of A -singularities. We consider an ar-
n
bitrary field k. The case char(k) = 2 will be important later. For a
positive integer n, let
f :X Speck[x,y,z]/(xy +zn+1)
→
be the minimal resolution of the A -singularity. The exceptional set of
n
f will be denoted by
Z = f 1(0) = C C ,
− 1 n
∪···∪
STABILITY CONDITIONS ON An-SINGULARITIES 91
where C ’s are irreducible ( 2)-curves such that C C = if i j >
i i j
− ∩ ∅ | − |
1. Let be the bounded derived category of coherent sheaves on X
k
D
supported at Z and be its full triangulated subcategory consisting
k
C
of objects E satisfying Rf E = 0. Put E = ω and E = ( 1) for
∗ 0 Z i OCi −
i = 1,...,n. Here, ω is the dualizing sheaf of Z. Then we have
Z
= E ,...,E
k 1 n
C h i
and
= E ,...,E ,
k 0 n
D h i
where denotes the smallest full triangulated subcategory of con-
k
h•i D
taining them. We simply write and instead of and , respec-
C C
C D C D
tively.
For E,F , define the Euler form by
k
∈ D
(2) χ(E,F) = ( 1)idimHomi (E,F),
− Dk
i
X
which descends to a bilinear form on K( ). By the Riemann-Roch
k
D
formula, we have
χ(E,F) = c (E) c (F).
1 1
− ·
The Euler form χ endows K( ) with the structure of the affine root
k
D
(1)
latticeoftypeA . Anon-zeroelementα K( )isarootifχ(α,α) 2
n
∈ D ≤
and it is a real root if χ(α,α) = 2. An imaginary root is a root that is
not a real root. Let δ K( ) be the class of the structure sheaf of a
k
∈ D
closed point with residue field k. Then an imaginary root is a non-zero
element of Zδ K( ).
k
⊂ D
Lemma 4. If E is stable with respect to some stability condition,
∈ D
then [E] K( ) is a root.
∈ D
Proof. ThestabilityofEimpliesHom≤−1(E,E) = 0andHom (E,E) ∼=
C. The Serre duality shows Hom≥3(E,DE) = 0 and Hom2(E,ED) ∼= C.
Hence χ(E,E) 2 and [E] is a roDot. D q.e.d.
≤
Definition 5. (i) An object E is spherical if
k
∈ D
k if i = 0,2,
Homi (E,E) =
Dk ∼ (0 otherwise.
(ii) An ordered set (E ,...,E ) of spherical objects in is an A -
1 n k n
D
configuration if
1 if i j = 1,
dimHom∗ (Ei,Ej) = | − |
Dk (0 if i j 2.
| − | ≥
The proof of Lemma 4 shows the following:
Lemma 6. If the class [E] K( ) of a stable object E is a real
∈ D ∈ D
root, then E is spherical.
92 A. ISHII, K. UEDA & H. UEHARA
A spherical object E gives rise to an autoequivalence of
k k
∈ D D
through the twist functor T , defined as the Fourier-Mukai transform
E
with
E ⊠E DbcohX X
∨ ∆
{ → O } ∈ ×
as the kernel [41]. Define
Br( ) = T ,...,T Auteq
Dk h E0 Eni ⊂ Dk
and
Br( ) = T ,...,T Auteq .
Ck h E1 Eni ⊂ Ck
Define the braid group B as the group generated by σ ,...,σ sub-
n 1 n
ject to relations
σ σ σ = σ σ σ , i= 1,...,n 1,
i i+1 i i+1 i i+1
−
σ σ = σ σ , i j > 2.
i j j i
| − |
It has the following topological description: Let
h = (a ,...,a ) Cn+1 a + +a = 0
1 n+1 1 n+1
{ ∈ | ··· }
be a Cartan subalgebra of the complex simple Lie algebra of type A ,
n
and hreg be the complement of its root hyperplanes,
hreg = (a ,...,a ) h a = a for i = j .
1 n+1 i j
{ ∈ | 6 6 }
The Weyl group W = S acts on h by permutations, and hreg is
∼ n+1
the set of regular orbits of W. Then B is isomorphic to the funda-
n
mental group of the quotient hreg/W [13, 14]. It follows that B has
n
another topological description: Let ∆ = 1,ζ,ζ2,...,ζn be the set of
{ }
(n+1)th roots of unity and Diff (C) be the group of diffeomorphisms
0
of C which are the identity map outside compact sets. Then there is a
map Diff (C) hreg/W which sends φ Diff (C) to [ φ(1) c,φ(ζ)
0 0
c,...,φ(ζn) →c ] with c = n φ(ζn)∈/(n + 1). This{map−is a Serr−e
− } i=0
fibration whose fiber over [∆] is the subgroup Diff (C;∆) Diff (C)
0 0
P ⊂
which fixes ∆ as a set. From the long exact sequence of homotopy
groups associated to this fibration, we can see that
B = π (Diff (C;∆)).
n ∼ 0 0
The assignment σ T for i = 1,...,n defines a homomorphism
i 7→ Ei
from B to Br( ), which is injective by Khovanov, Seidel, and Thomas
n
C
[33, 41]. This result is the key to the proof of the simply-connectedness
of a distinguished connected component of Stab by Thomas [42].
C
(1)
Now define the affine braid group B to be the group generated by
n
σ ,...,σ subject to relations
0 n
σ σ σ = σ σ σ , i = 0,...,n,
i i+1 i i+1 i i+1
σ σ = σ σ , i j > 2.
i j j i
| − |
STABILITY CONDITIONS ON An-SINGULARITIES 93
Here, we put σ = σ by notation. Let hreg be the complement of the
n+1 0
affine root hyperplanes in h = h C:
⊕
b
hreg = (a ,...,a ,b) h C a a +bd = 0 for i = j and d Z .
1 n+1 b i j
{ ∈ ⊕ | − 6 6 ∈ }
TheaffineWeylgroupW acts freelyonhreg,andthefundamentalgroup
b
of the orbit space hreg/W is given by B(1) Z by [19].
n
×
(1) c b
The group B also admits the following topological interpretation:
n
Let Diff (C ) bebthe gcroup of diffeomorphisms of C which are the
0 × ×
identity maps outside compact sets and Diff (C ;∆) be its subgroup
0 ×
fixing ∆ as a set. We can define a homomorphism
B(1) π (Diff (C ;∆))
n → 0 0 ×
(1)
from B to the group of connected components of Diff (C ;∆) by
n 0 ×
sending σ to the class of a diffeomorphism of C which permutes two
i ×
neighboring points ζi and ζi+1 for i = 0,...,n. This homomorphism is
known to be injective (cf. [32]).
The assignment σ T for i = 0,...,n defines a homomorphism
i 7→ Ei
ρ:B(1) Br( ),
n → Dk
which can be extended to a surjective homomorphism
ρ˜: B(1) Z Br( ) Z.
n × → Dk ×
Here, since Br( ) does not contain any power of the shift functor, the
k
D
right-hand side is considered as a subgroup of Auteq so that the
k
D
second factor Z corresponds to the group generated by the shift functor
[2]. In [11], Bridgeland shows the following theorem for any Kleinian
singularities, that is, rational double points over C.
Theorem 7 ([11, theorem 1.3]). There is a connected component of
Stab whichisacoveringspace ofhreg/W suchthatthegroupBr( ) Z
D D ×
acts as the group of deck transformations.
b c
Hence we have the canonical group homomorphism
π (hreg/W)= B(1) Z Br( ) Z.
1 n × → D ×
Bridgelandalsoshowsthatthishomomorphismcoincideswithρ˜. There-
b c
fore if ρ is injective, we conclude that the connected component in The-
orem 7 is simply connected.
We prove theconnectedness of Stab in 4 andtheinjectivity of ρin
D §
5and 6. Theseresults,togetherwiththeabovetheoremofBridgeland,
§ §
gives the following explicit description of Stab :
D
Theorem 8. Stab is the universal cover of hreg/W.
D
b c
94 A. ISHII, K. UEDA & H. UEHARA
As for Stab , a result of Thomas [42] shows that there is a distin-
C
guished connected component of Stab which is the universal cover of
C
hreg/W. We will prove the connectedness of Stab in 7, so that this
C §
connected component is the whole of Stab .
C
3. The McKay correspondence
We collect basic facts on the McKay correspondence in this sec-
tion. We expect that the result in this section is well-known to ex-
perts, although we have been unable to locate an appropriate reference.
Throughout this section, k will denote a field of any characteristic. We
restrict our discussion to the case of A -singularities since it is the only
n
case in need in this paper. For a noetherian k-algebra A, the abelian
categoryoffinitelygeneratedrightA-moduleswillbedenotedbymodA.
3.1. Path algebra and the endomorphism algebra of a reflexive
(1)
module.Let be the preprojective algebra for the affine Dynkin
n
A
(1)
quiver of type A , described explicitly as follows: As a k-vector space,
n
(1)
isgenerated by thesymbols(i i ... i )for l 1, i Z/(n+1)Z
n 1 2 l m
A | | | ≥ ∈
and i = i 1. The multiplication is defined by
m+1 m
±
(i ... i j ... j ) if i = j ,
1 l 2 m l 1
(i ... i )(j ... j )= | | | | |
1 l 1 m
| | | | (0 otherwise,
and the relations are generated by
(ii+1i) = (ii 1i)
| | | − |
for i Z/(n+1)Z.
∈
Let = k[x,y,z]/(xy + zn+1) be the affine coordinate ring of the
O
rational double point of type A . For an integer a = (n+1)q+r with
n
0 r n, consider the fractional ideal I = (yq+1,yqzr) of . I ’s
a a
≤ ≤ O O
are reflexive -modules such that I = I if and only if a b is divisible
O a ∼ b −
by n+1. For i Z/(n+1)Z, we lift i to a Z with 0 a n and
∈ ∈ ≤ ≤
put E = I . For an integer b = q(n+1)+r with 0 r n, we fix
i a
≤ ≤
the isomorphism I = E = I given by the multiplication by
b ∼ (b modn+1) r
y q. Consider the reflexive -module
−
O
E = E .
i
i Z/(n+1)Z
∈ M
(1)
E is an -module in the following way. The idempotent (i)
n k
A ⊗ O
acts as the projection of E to I . The path (ii + 1) corresponds to
i
z |
the homomorphism Ia · Ia+1 given by the multiplication by z, where
−→
a Z is a lift of i. The path (ii 1) goes to the inclusion I ֒ I .
a a 1
∈ | − (1) → −
Then it is easy to see that we obtain an -action on E. Thus
n k
A ⊗ O
STABILITY CONDITIONS ON An-SINGULARITIES 95
we have a k-algebra homomorphism
η : (1) End (E).
An → O
Proposition 9. η is an isomorphism.
(1)
Proof. We firstnote that an element of is a k-linear combination
n
A
of the elements P(i,l,m) defined as follows for i Z/(n + 1)Z and
∈
l,m Z with m 0:
∈ ≥
(ii+1i)m(ii+1 ... i+l 1i+l) if l 0,
P(i,l,m) = | | | | | − | ≥
((ii+1i)m(ii 1 ... i+l+1i+l) otherwise.
| | | − | | |
To show that η is an isomorphism, it is sufficient to show that the
restricted map
η :(i) (1)(j) Hom (E ,E )
ij An → O i j
is bijective for i,j Z/(n+1)Z. Lift i,j to a,a+r Z with 0 r n.
∈ ∈ ≤ ≤
Then we have isomorphisms
Hom (E ,E )= Hom (I ,I ) = I = (y,zr) .
i j ∼ a a+r ∼ r
O O O
Ifl = q(n+1)+rwithq 0,thenP(i,l,m)ismappedtoxqzm+r I ,
r
≥ ∈
and if l = q(n+1)+r < 0 with q > 0, then P(i,l,m) is mapped to
−
yqzm I . Moreover, the monomials xqzm+r (q 0,m 0) and yqzm
r
∈ ≥ ≥
(q > 0,m 0) form a k-linear basis of I . Therefore, η must be
r ij
≥
bijective. q.e.d.
3.2. A full sheaf as a projective generator.Let f : X Y =
→
Spec be the minimal resolution and 1Per(X/Y) be the abelian cat-
−
O
egory of perverse sheaves introduced by Bridgeland [8]; an object E
∈
1Per(X/Y) is a bounded complex of coherent sheaves on X such that
−
its cohomology sheaves satisfy
f ( 1(E)) = 0, R1f ( 0(E)) = 0, i(E) =0 for i = 1,0,
−
∗ H ∗ H H 6 −
and
Hom ( 0(E),F) = 0
X
H
for any coherent sheaf F on X satisfying Rf F = 0.
∗
For a reflexive -module F, put
O
F := f (F)/torsion,
∗
which is a locally free sheaf on X(see [1]). A locally free sheaf of this
e
form is called a full sheaf. It is proved in [20] that a locally free sheaf
on X is a full sheaf if and only if the following two conditions are
F
satisfied:
(i) is generated by its global sections.
F
(ii) R1f ( )= 0.
∨
∗ F
The following result is due to Van den Bergh:
96 A. ISHII, K. UEDA & H. UEHARA
Proposition 10 ([43, proposition 3.2.7, corollary 3.2.8]). (i) Afull
sheaf is a projective generator of 1Per(X/Y) if and only if its
−
M
first Chern class c ( ) is ample and is a direct summand of
1 X
M O
a for some positive integer a.
⊕
M
(ii) Assumethatafullsheaf isaprojectivegeneratorof 1Per(X/Y)
−
M
and put A = End ( ). Then the functor RHom ( , ) gives
X X
M M •
an equivalence between DbcohX and DbmodA, whose inverse is
L
given by the functor .
A
• ⊗ M
The above proposition yields the following:
Theorem 11. The bounded derived category DbcohX of coherent
sheaves on the crepant resolution X of the A -singularity is equivalent
n
to the bounded derived category Dbmod (1) of finitely generated right
n
A
(1)
-modules.
n
A
Proof. Put E = E be as in the previous subsection. Since
i Z/nZ i
⊕∈
c (E ) C =δ ,
1 i j ij
·
the corresponding full sheaf E has an ample first Chern class and is
f
hence a projective generator of 1Per(X/Y) satisfying End(E) = (1).
− ∼ n
A
e q.e.d.
e
(1)
Let mod be the abelian category of finitely generated nilpotent
0 n
A
(1)
right -modules. Under the above equivalence, corresponds to
n k
A D
the bounded derived category Dbmod (1) of mod (1).
0 n 0 n
A A
Whenk contains aprimitive(n+1)throotζ ofunity, Y isisomorphic
to the quotient A2/G of the affine plane by the natural action of the
subgroup G of SL (k) generated by the diagonal matrix diag(ζ,ζn).
2
(1)
Since is isomorphic to the crossed product k[x,y] ⋊ k[G] of the
n
A
polynomial ring with the group ring, the category of finitely gener-
ated nilpotent (1)-modules is equivalent to the category cohGA2 of
An 0
G-equivariant coherent sheaves on A2 supported at the origin:
mod (1) = cohGA2.
0An ∼ 0
Hence Theorem 11 in this case gives the equivalence
Dbcoh X = DbcohGA2
Z ∼ 0
of triangulated categories, first proved by Kapranov and Vasserot [31]
(see also Bridgeland, King, and Reid [12]).
4. Connectedness of Stab
D
We prove the connectedness of Stab in this section. Our strategy
D
is the following: