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arXiv:0809.2994v3 [math.AG] 4 Dec 2008

Derived categories of small toric Calabi-Yau

3-folds and curve counting invariants

Kentaro Nagao

December 4, 2008

Abstract

We first construct a derived equivalence between a small crepant res-

olution of an affine toric Calabi-Yau 3-fold and a certain quiver with

a superpotential. Under this derived equivalence we establish a wall-

crossing formula for the generating function of the counting invariants of

perverse coherent sheaves. As an application we provide some equations

on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi’s invari-

ants. Finally, we show that moduli spaces associated with a quiver given

by successive mutations are realized as the moduli spaces associated the

original quiver by changing the stability conditions.

Introduction

This is a subsequent paper of [NN]. We study variants of Donaldson-Thomas

(DT in short) invariants on small crepant resolutions of affine toric Calabi-Yau

varieties.

The original Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are

defined by virtual counting of moduli spaces of ideal sheaves IZof 1-dimensional

closed subschemes Z ⊂ Y ([Tho00], [Beh]). These are conjecturally equivalent to

Gromov-Witten invariants after normalizing the contribution of 0-dimensional

sheaves ([MNOP06]).

A variant has been introduced Pandharipande and Thomas (PT in short)

as virtual counting of moduli spaces of stable coherent systems ([PTa]). They

conjectured these invariants also coincide with DT invariants after suitable nor-

malization and mentioned that the coincidence should be recognized as a wall-

crossing phenomenon. Here, a coherent system is a pair of a coherent sheaf and

a morphism to it from the structure sheaf, which is first introduced by Le Potier

in his study on moduli problems ([LP93]). Note that an ideal sheaf IZ is the

kernel of the canonical surjections from the structure sheaf OY to the structure

sheaf OZ. So in this sense DT invariants also count coherent systems.

On the other hand, a variety sometimes has a derived equivalence with a non-

commutative algebra. A typical example is a noncommutative crepant resolution

of a Calabi-Yau 3-fold introduced by Michel Van den Bergh ([VdB04], [VdB]).

In the case of [VdB04], the Abelian category of modules of the noncommuta-

tive crepant resolution corresponds to the Abelian category of perverse coherent

sheaves in the sense of Tom Bridgeland ([Bri02]). Recently, Balazs Szendroi pro-

posed to study counting invariants of ideals of such noncommutative algebras

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([Sze]). He call these invariants noncommutative Donaldson-Thomas (NCDT in

short) invariants. He originally studied on the conifold, but his definition works

in more general settings ([Youb], [MR]).

Inspired by his work, Hiraku Nakajima and the author introduced perverse

coherent systems (pairs of a perverse coherent sheaf and a morphism to it from

the structure sheaf) and study their moduli spaces and counting invariants

([NN]). This attempt seems successful since

• we can describe explicitly a space of stability parameters with a chamber

structure, and

• at certain chambers, the moduli spaces in DT, PT and NCDT theory are

recovered.

Moreover, in the conifold case, we established the wall-crossing formula for

the generating functions of counting invariants of perverse coherent systems

and provide some equations on DT, PT and NCDT invariants. The chamber

structure and the wall-crossing formula formally look very similar to the counter

parts for moduli spaces of perverse coherent sheaves on the blow-up of a complex

surface studied earlier by Nakajima and Yoshioka [NYa, NYb].

The purpose of this paper is to show the wall-crossing formula (Theorem

2.18) for general small crepant resolutions of toric Calabi-Yau 3-folds. Here

we say a crepant resolutions of affine toric Calabi-Yau 3-fold is small when the

dimensions of the fibers are less than 2. In such cases, the lattice polygon in

R2corresponding to the affine toric Calabi-Yau 3-fold does not have any lattice

points in its interior. Such lattice polygons are classified up to equivalence into

the following two cases:

• trapezoids with heights 1, or

• the right isosceles triangle with with length 2 isosceles edges.

In this paper we study the first case. Our argument works for the second case

as well.

In §1, we construct derived equivalences between small crepant resolutions

of affine toric Calabi-Yau 3-folds and certain quivers with superpotentials. In

§1.1, using toric geometry, we construct tilting vector bundles given by Van den

Bergh ([VdB04]) explicitly. Then, we review Ishii and Ueda’s construction of

crepant resolutions as moduli spaces of representations of certain quivers with

superpotentials ([IU]) in §1.2. In §1.3 we show the tautological vector bundles

on the moduli spaces coincide with the tilting bundles given in §1.1. Using such

moduli theoretic description, we calculate the endomorphism algebras of the

tilting bundles in §1.4.

The argument in §2 is basically parallel to [NN]. In our case, the fiber on

the origin of the affine toric variety is the type A configuration of (−1,−1)-

or (0,−2)-curves. A wall in the space of stability parameters is a hypersurface

which is perpendicular to a root vector of the root system of typeˆA. Stability

parameters in chambers adjacent to the wall correspondingto the imaginary root

realize DT theory and PT theory ([NN, §2]). Note that the story is completely

parallel to that of typeˆA quiver varieties (of rank 1), which are the moduli spaces

of framed representations of typeˆA preprojective algebras ([Nak94], [Nak98],

[Nak01]). Quiver varieties associated with a stability parameter in a chamber

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adjacent to the imaginary wall realize Hilbert schemes of points on the minimal

resolution of the Kleinian singularitie of type A, whose exceptional fiber is the

type A configurations of (−2)-curves ([Nak99], [Kuz]).

Our main result is the wall-crossing formula for the generating functions of

the Euler characteristics of the moduli spaces (Theorem 2.18). The contribution

of a wall depends on the information of self-extensions of stable objects on the

wall. Note that in the conifold case ([NN]) every wall has a single stable object

on it and every stable object has a trivial self-extension. Computations of self-

extensions are done in §2.6.

Note that the sets of torus fixed points on the moduli spaces in DT, PT and

NCDT theory are isolated, and we can show that DT, PT and NCDT invariants

coincide with the Euler characteristics of the moduli spaces. In particular, the

wall-crossing formula provides a product expansion formula of the generating

functions of PT invariants. The indices in this formula are nothing but the BPS

state counts ng,β([GV], [HST01], [Tod]) in the sense of Pandeharipande-Thomas

([PTa, §3]). Although an algorithm to extract Gopakumar-Vafa invariants of

our toric Calabi-Yau 3-folds from the topological vertex expression is known

([IKP]1), the explicit formula in this paper is new as far as the author knows.

In §3, we provide alternative descriptions of the moduli spaces. Given a

quiver with a superpotential A = (Q,ω), we can mutate it at a vertex k to

provide a new quiver with a superpotential µk(A) = (µk(Q),µk(ω)). For a

generic stability parameter ζ, we can associate a sequence k1,...krof vertices

and the moduli space of ζ-stable A-modules is isomorphic to the moduli space

of cyclic modules over the quiver with the potetial µkr◦ ··· ◦ µk1(A). As an

application, we show that for a stability parameter ”between DT and NCDT”

the set of torus fixed points on the moduli space is isolated.

As in [NN], our formula does not cover the wall corresponding to the DT-

PT conjecture. We can provide the wall-crossing formula for this wall applying

Joyce’s formula ([Joy])2. In §4, we make some observations on how the virtual

counting version of the wall-crossing formula would be induced from the per-

spective of the recent work of Kontsevich and Soibelman ([KS]). In fact, the

wall-crossing formula (and hence the invariants) coincides with the Euler char-

acteristic version up to sign. The author learned from Tom Bridgeland that he

and Balazs Szendroi reproved the Young’s product formula ([Youa]) for NCDT

invariants of the conifold. Their idea looks quite similar to our observation.

Acknowledgement

The author is grateful to Hiraku Nakajima for collaborating in the paper [NN]

and for many valuable discussion.

He thanks Kazushi Ueda for patiently teaching his work on brane tilings,

Tom Bridgeland for explaining his work with Balazs Szendroi, Yan Soibelman

for helpful comments on §4.

He also thanks Yoshiyuki Kimura and Michael Wemyss for useful discus-

sions, Akira Ishii, Yukari Ito, Osamu Iyama, Yukiko Konishi, Sergey Mozgovoy

1The author was informed on this reference by Yukiko Konishi.

2Yukinobu Toda informed me that it is possible to prove the (Euler characteristic version

of) DT-PT correspondence conjecture for arbitrary projective Calabi-Yau 3-folds using Joyce’s

formula.

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and Yukinobu Toda for helpful comments. The author is supported by JSPS

Fellowships for Young Scientists No.19-2672.

1 Derived equivalences

1.1tilting generators

Let N0> 0 and N1≥ 0 be integers such that N0≥ N1and set N = N0+ N1.

We set

I = {1,...,N − 1},

ˆI = {0,1,...,N − 1},

˜I =

?1

?

2,3

n +1

2,...,N −1

2

?

,

˜Z =

2

???n ∈ Z

?

.

For l ∈ Z and j ∈˜Z, let l ∈ˆI and j ∈˜I be the elements such that l−l ≡ j−j ≡ 0

modulo N.

We denote by Γ the quadrilateral (or the triangle in the cases N1= 0) in

R2= {(x,y)} with vertices (0,0), (0,1), (N0,0) and (N1,1). Let M∨≃ Z3be

the lattice with basis {x∨,y∨,z∨}, and we identify the plane

{(x,y,1)} ⊂ M∨

with the one where the quadrilateral Γ is. Let M be the dual lattice of M∨.

We denote the cone of Γ in M∨

Rby ∆ and consider the semigroup

R:= M∨⊗ R

S∆= ∆∨∩ M := {u ∈ M | ?u,v? ≥ 0 (∀v ∈ ∆)}.

Let R = RΓ := C[S∆] be the group algebra and X = XΓ := Spce(RΓ) the

3-dimensional affine toric Calabi-Yau variety corresponding to ∆.

Let {x,y,z} ⊂ M be the dual basis. The semigroup is generated by

X := x,

Y := −x − (N0− N1)y + N0z,

Z := y,

W := −y + z,

and they have a unique relation X + Y = N0Z + N1W. So we have

(1)

(2)

(3)

(4)

R ≃ C[X,Y,Z,W]/(XY − ZN0WN1).

A partition σ of Γ is a pair of functions σx:˜I →˜Z and σy:˜I → {0,1} such

that

• σ(i) := (σx(i),σy(i)) gives a permutation of the set

??1

2,0

?

,

?3

2,0

?

,...,

?

N0−1

2,0

?

,

?1

2,1

?

,

?3

2,1

?

,...,

?

N1−1

2,1

??

,

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• if i < j and σy(i) = σy(j) then σx(i) > σx(j).

Giving a partition σ of Γ is equivalent dividing Γ into N-tuples of triangles

{Ti}i∈˜Iwith area 1/2 so that Tihas (σx(i) ± 1/2,σy(i)) as its vertices. Let Γσ

be the corresponding diagram, ∆σ be the fun and fσ: Yσ→ X be the crepant

resolution of X.

We denote by Dε,x(ε = 0,1 and 0 ≤ k ≤ Nε) the divisor of Yσcorresponding

to the lattice point (x,ε) in the diagram Γσ. Note that any torus equivariant

divisor is described as a linear combination of Dε,x’s. For a torus equivariant

divisor D let D(ε,x) denote its coefficient of Dε,x. The support function ψDof

D is the piecewise linear function on |∆σ| such that ψD((x,ε,1)) = −D(ε,x) and

such that ψDis linear on each cone of ∆σ. We sometimes denote the restriction

of ψDon the plane {z = 1} by ψDas well.

Definition 1.1. For i ∈˜I and k ∈ I we define effective divisors E±

by

i and F±

k

E+

i=

Nσy(i)

?

σx(i)−1

?

j=σx(i)+1

2

Dσy(i),j,F+

k=

k−1

?

N−1

?

2

i=1

2

E+

i,

E−

i=

2

j=0

Dσy(i),j,F−

k=

2

i=k+1

2

E−

i.

Example 1.2. Let us consider as an example the case N0= 4, N1= 2 and

(σ(i))i∈˜I=

??7

2,0

?

,

?3

2,1

?

,

?5

2,0

?

,

?3

2,0

?

,

?1

2,1

?

,

?1

2,0

??

.

We show the corresponding diagram Γσ in Figure 1. The divisors are given as

follows:

E+

1

2:=

?

?

?

?

?

?

0

0

0

0

0

001

?

?

?

?

?

?

,F+

1:=

?

?

?

?

?

?

0

0

0

0

0

001

?

?

?

?

?

?

,

E+

3

2:=

0

0

0

0

1

000

,F+

2:=

0

0

0

0

1

001

,

E+

5

2:=

0

0

0

0

0

011

,F+

3:=

0

0

0

0

1

012

,

E+

7

2:=

0

0

0

0

0

111

,F+

4:=

0

0

0

0

1

123

,

E+

9

2:=

0

0

1

0

1

000

,F+

5:=

0

0

1

0

2

123

,

E+

11

2

:=

0

0

0

1

0

111

,F+

6:=

0

0

1

1

2

234

.

Here the (ε,x)-th matrix element represent the coefficient of the divisor Dε,x.

Lemma 1.3.

(1) OYσ(E+

i+ E−

i) ≃ OYσ,

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