

E extracta mathematicae Vol. 31, Núm. 1, 89 – 107 (2016)

On the Moduli Space of Donaldson-Thomas Instantons

Yuuji Tanaka

Graduate School of Mathematics, Nagoya University

Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

yu2tanaka@gmail.com

Presented by Oscar Garćıa Prada Received May 19, 2015

Abstract: In alignment with a programme by Donaldson and Thomas, Thomas [48]
constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is
now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves
by using algebraic geometry techniques. In the same paper [48], Thomas noted that certain
perturbed Hermitian-Einstein equations might possibly produce an analytic theory of the
invariant. This article sets up the equations on symplectic 6-manifolds, and gives the local
model and structures of the moduli space coming from the equations. We then describe
a Hitchin-Kobayashi style correspondence for the equations on compact Kähler threefolds,

which turns out to be a special case of results by Álvarez-Cónsul and Garćıa-Prada [1].

Key words: gauge theory; the Donaldson-Thomas theory.

AMS Subject Class. (2010): 53C07.

1. Introduction

In [21], Donaldson and Thomas suggested higher-dimensional analogues
of gauge theories, and proposed the following two directions: gauge theories
on Spin(7) and G2-manifolds; and gauge theories in complex 3 and 4 dimen-
sions. The first ones could be related to “Topological M-theory” proposed
by Nekrasov and others [39], [15]. The second ones are a “complexification”
of the lower-dimensional gauge theories. In this direction, Thomas [48] con-
structed a deformation invariant of smooth projective Calabi-Yau threefolds
from the moduli space of (semi-)stable sheaves, which he called the holomor-
phic Casson invariant because it can be viewed as a complex analogue of the
Taubes-Casson invariant [47]. It is now called the Donaldson-Thomas invari-
ant (D-T invariant for short), and further developed by Joyce-Song [28] and
Kontsevich-Soibelman [32], [33], [34]. Later, Donaldson and Segal [20] further
promoted the programme, taking into account the progress made after the
proposal. Recently, more breakthroughs concerning the “categorification” of
the D-T invariant by using perverse sheaves were made by a group led by
Joyce [7], [27], [8], [9], [4], also by Kiem-Li [29].

89


90 y. tanaka

Let us mention here a conjecture (called the MNOP conjecture) posed
by Maulik-Nekrasov-Okounkov-Pandharipande [37], [38], which insists that
the rank one D-T invariants (“counting” of ideal sheaves on a Calabi-Yau
threefold) can be determined by only the Betti numbers and the Gromov-
Witten invariants. Assuming the conjecture is true, one can observe that
the rank one D-T invariants are symplectic invariants, as the Gromov-Witten
invariants are symplectic invariants. One might further speculate that the full
D-T invariants defined by Joyce and Song could be also symplectic invariants.
One of our goals is to work toward proving this by using a gauge-theoretic
equation (we call it the Donaldson-Thomas equation) on a compact symplectic
6-manifold, which ought to be an analytic counterpart of the notion of stable
holomorphic vector bundles, as the problem is analytic in nature.

Perhaps, one might think of that a gauge-theoretic equation which would
describe the D-T invariant could be the Hermitian-Einstein equations, as the
Hitchin-Kobayashi correspondence [17], [18], [50], [51] (see also [31], [36])
insists that there is a one-to-one correspondence between the existence of
the Hermitian-Einstein connection and the Mumford-Takemoto stability of
an irreducible vector bundle over a compact Kähler manifold. However, the
Hermitian-Einstein equations do not form an elliptic system even with a gauge
fixing equation in complex dimension three and more (see Section 2.1), so this
might cause a little problem.

In order to work out this issue, Donaldson and Thomas [48] suggested
a perturbation of the Hermitian-Einstein equations described below. This
perturbation was also brought in by Baulieu-Kanno-Singer [3] and Iqbal-
Nekrasov-Okounkov-Vafa [26] in String Theory context.

Let Z be a compact symplectic 6-manifold with symplectic form ω, P
a principal U(r)-bundle on Z, and E the associated unitary vector bundle
on Z. The equations we consider are ones for a connection A of P and an
Ad(P)-valued (0,3)-form u on Z of the following form:

F
0,2
A + ∂̄

∗
Au = 0 , F

1,1
A ∧ ω

2 + [u,ū] + 2πiµ(E)IdEω
3 = 0 ,

where F
0,2
A and F

1,1
A are the (0,2) and (1,1) components of the curvature FA

of A, and

µ(E) :=
1

r

∫
Z
c1(E) ∧ ω2.

Here we picked up an almost complex structure compatible with ω to get the
splitting of the space of the complexified two forms. We call the equations
the Donaldson-Thomas equations (D-T equations for short) and a solution


moduli space of donaldson-thomas instantons 91

to the equations a Donaldson-Thomas instanton (D-T instanton for short).
These equations with a gauge fixing equation form an elliptic system. We aim
at developing an analytic theory concerning the D-T invariant by using the
moduli space coming from these equations.

In [44], [45], we studied some analytic properties of solutions to the equa-
tions on compact Kähler threefolds. In [44], we proved that a sequence of
solutions to the D-T equation has a subsequence which smoothly converges
to a solution to the D-T equation outside a closed subset of the Hausdorff
dimension two. In [45], we proved some of singularities which appeared in the
above weak limit can be removed.

In this article, we describe the infinitesimal deformation and the Kuran-
ishi model of the moduli space of D-T instantons by using familiar techniques
in gauge theory, for example, the corresponding results for the anti-self-dual
instantons in real four dimensions were studied by Atiyah-Hitchin-Singer [2]
(see also [22], [19]), and for the Hermitian-Einstein connections by Kim [30]
(see also [31], [36]). We then describe a Hitchin-Kobayashi style correspon-
dence for the D-T instanton on compact Kähler threefolds, which turns out
to be a special case of results by Álvarez-Cónsul and Garćıa-Prada [1].

The organisation of this article is as follows. In Section 2, we briefly
recall the Hermitian-Einstein connections, subsequently, we introduce the D-
T equations on symplectic 6-manifolds. We also mention a relation between
the D-T equations and the complex anti-self-dual equations by dimensional
reduction argument. In Section 3, we give the Kuranishi model of the space
of the D-T instantons. In Section 4, we describe a Hitchin-Kobayashi style
correspondence for the D-T instanton on compact Kähler threefolds.

2. The Donaldson-Thomas instantons

2.1. The Hermitian-Einstein connections on compact Kähler
manifolds. We first recall the notion of the Hermitian-Einstein connections
on compact Kähler manifolds. General references for the Hermitian-Einstein
connections are [31] and [36].

Let X be a compact Kähler manifold of complex dimension n with Kähler
form ω, E a hermitian vector bundle over X with hermitian metric h. A metric
preserving connection A of E is said to be a Hermitian-Einstein connection
if A satisfies the following equations:

F
0,2
A = 0 , iΛF

1,1
A = 2nπµ(E)IdE , (2.1)

where F
0,2
A and F

1,1
A are the (0,2) and (1,1) components of the curvature FA


92 y. tanaka

of A, Λ := (ω)∗, and

µ(E) :=
1

r

∫
X
c1(E) ∧ ωn−1.

The existence of a solution to the equations (2.1) is related to the notion of
stability for holomorphic vector bundles. In fact, Donaldson [17], [18] and
Uhlenbeck-Yau [50], [51] proved that there is a one-to-one correspondence
between the existence of the Hermitian-Einstein connection and the Mumford-
Takemoto stability of an irreducible vector bundle over a compact Kähler
manifold (see also [31], [36]).

The infinitesimal deformation of a Hermitian-Einstein connection A was
studied by Kim [30] (see also [31], [41]), and it is described by the following:

0 −→ Ω0(X,u(E)) dA−−→ Ω1(X,u(E))
d+
A−−→ Ω+(X,u(E))

D̄′
A−−→ A0,3(X,u(E)) D̄A−−→ A0,4(X,u(E))

D̄A−−→ · · · D̄A−−→ A0,n(X,u(E)) −→ 0,

(2.2)

where A0,q(X,u(E)) := C∞(u(E) ⊗ A0,q), u(E) = End(E,h) is the bundle
of skew-Hermitian endomorphisms of E, A0,p is the space of real (0,p)-forms
(see [42, pp. 32-33]) over X, defined by A0,p ⊗R C = Λ0,p ⊕ Λp,0,

Ω+(X,u(E)) := A0,2(X,u(E)) ⊕ Ω0(X,u(E))ω

=
{
ϕ + ϕ̄ + fω : ϕ ∈ Ω0,2(X,u(E)), f ∈ Ω0(X,u(E))

}
,

D̄A : A
0,p(X,u(E)) → A0,p+1(X,u(E)) is defined by

D̄Aα = ∂̄Aα
0,p + ∂Aα0,p

for α = α0,p + α0,p, where α0,p ∈ Ω0,p(X,u(E)), and

d+A := π
+ ◦ dA , D̄′A := D̄A ◦ π

0,2,

where π+ and π0,2 are respectively the orthogonal projections from Ω2 to Ω+

and A0,2.
Kim proved that (2.2) is an elliptic complex if A is a Hermitian-Einstein

connection. However, it is obviously not the Atiyah-Hitchin-Singer type com-
plex [2] if n ≥ 3, since there are additional terms such as A0,3(X,u(E)) and
so on. Hence, the Hermitian-Einstein connections would not work for an an-
alytic construction of the Donaldson-Thomas invariant just as it is. But, in


moduli space of donaldson-thomas instantons 93

[48], Thomas noted a perturbed Hermitian-Einstein equation, which basically
corresponds to a “holding” of the extra term A0,3(X,u(E)) in (2.2) (we shall
see it in Section 3.1), could possibly work for an analytic definition of the
Donaldson-Thomas invariant. We introduce that perturbed equation in the
next subsection.

2.2. The Donaldson-Thomas instantons on compact symplectic
6-manifolds. Let Z be a compact symplectic 6-manifold with symplectic
form ω, and E a unitary vector bundle of rank r over Z. We take an almost
complex structure on Z compatible with the symplectic form ω. Then the
almost complex structure induces the splitting of the complexified two forms
as Λ2 ⊗ C = Λ2,0 ⊕ Λ0,2 ⊕ Λ1,1. We consider the following equations for
a connection A of E, which preserves the hermitian structure of E, and a
u(E)-valued (0,3)-form u on Z.

F
0,2
A + ∂̄

∗
Au = 0 , (2.3)

F
1,1
A ∧ ω

2 + [u,ū] + 2πiµ(E)IdE ω
3 = 0 , (2.4)

where F
0,2
A and F

1,1
A are the (0,2) and (1,1) components of the curvature FA

of A, and

µ(E) :=
1

r

∫
Z
c1(E) ∧ ω2.

We call these equations (2.3), (2.4) the Donaldson-Thomas equations, and
a solution (A,u) to these equations a Donaldson-Thomas instanton (D-T
instanton for short).

One may think of these equations as the Hermitian-Einstein equations with
a perturbation u. However, we think of u as a Higgs field, namely, a new vari-
able. One of advantages of bringing in the new field u is that the Donaldson-
Thomas equations form an elliptic system after fixing a gauge transformation,
despite the fact that the Hermitian-Einstein equations on compact Kähler
threefolds do not form it in the same way.

These equations (2.3), (2.4) were also studied in physics such as in [3]. In
that context, these equations are interpreted as a bosonic part of dimensional
reduction equations of the N = 1 super Yang-Mills equation in 10 dimensions
to 6 dimensions (see also [26], [40]).

The equations in the Kähler case. If the almost complex struc-
ture is integrable, then we have ∂̄AF

0,2
A = 0 by the Bianchi identity. Hence


94 y. tanaka

∂̄A∂̄
∗
Au = 0 by (2.3), thus we have ∂̄

∗
Au = 0 on compact Kähler threefolds.

Therefore, the Donaldson-Thomas equations (2.3), (2.4) becomes

∂̄∗Au = 0, F
0,2
A = 0 ,

F
1,1
A ∧ ω

2 + [u,ū] + 2πiµ(E)IdE ω
3 = 0 .

The above equations could be thought of as a generalization of the Hitchin
equation on Riemann surfaces [24] to Kähler threefolds in the same way as the
Vafa-Witten equations on Kähler surfaces as mentioned in [46]. In Section 4 to
this article, we describe the corresponding Hitchin-Kobayashi correspondence
in this setting, which turns out to be a special case of results by Álvarez-Cónsul
and Garćıa-Prada [1].

2.3. The complex ASD and the Donaldson-Thomas instantons.
In this section, we see that the Donaldson-Thomas equations on Calabi-Yau
threefolds can be thought of as the dimensional reduction of the complex ASD
equations on Calabi-Yau fourfolds, this was pointed out by Tian [49], and it
is analogous to the Hitchin pair [24].

Complex ASD equations on Calabi-Yau fourfolds. Let X be a
compact Calabi-Yau fourfold with Kähler form ω and holomorphic (4,0)-form
θ. We assume the normalization condition θ ∧ θ̄ = 16

4!
ω4 on ω and θ. Let E

be a hermitian vector bundle over X. By using the holomorphic (4,0)-form
θ, we define the complex Hodge operator ∗θ : Λ0,2 → Λ0,2 by

tr(ϕ ∧ ∗θψ) = ⟨ϕ,ψ⟩θ̄ , ϕ,ψ ∈ Λ0,2.

Then ∗2θ = 1, and the space of (0,2)-forms further decomposes into Λ
0,2 =

Λ
0,2
+ ⊕ Λ

0,2
− , where

Λ
0,2
+ = {ϕ ∈ Λ

0,2 : ∗θϕ = ϕ} , Λ
0,2
− = {ϕ ∈ Λ

0,2 : ∗θϕ = −ϕ}.

Note that the operator ∗θ is an anti-holomorphic map, hence Λ
0,2
+ and Λ

0,2
−

are real subspaces of Λ0,2.
We consider the following equations for connections of E:

(1 + ∗θ)F
0,2
A = 0 , iΛF

1,1
A = 8πµ(E)IdE, (2.5)

where

µ(E) :=
1

r

∫
X
c1(E) ∧ ω3.


moduli space of donaldson-thomas instantons 95

We call these equations complex ASD equations, and a solution to these equa-
tions a complex ASD instanton. These were brought in by Donaldson and
Thomas in [21]. These equations with a gauge fixing equation form an elliptic
system. Analytic properties of the complex ASD instantons were studied by
Tian [49].

Note that the complex ASD instantons are special cases of Spin(7)-instan-
tons on Spin(7)-manifolds (see [43, §3.1]).

More recently, Donaldson-Thomas style invariants for Calabi-Yau four-
folds, which concerns the moduli space of the solutions to the above complex
ASD equations, were defined by Borisov-Joyce [5], Cao [10] and Cao-Leung
[11] (see also [12], [13], [14]).

Dimensional reduction. We describe a relation between the Donald-
son-Thomas equations (2.3), (2.4) and the complex ASD equations (2.5) by
dimensional reduction argument. This was pointed out by Tian [49].

Let Z be a compact Calabi-Yau threefold with Kähler form ω0 and holo-
morphic (3,0)-form θ0, and T

2 a torus of complex dimension one. We consider
the direct product of Z and T2, and denote it by X, namely, X := Z × T2.
We define a Kähler form ω and a holomorphic (4,0)-form on X by

ω := ω0 + dz ∧ dz̄ , θ := θ0 ∧ dz ,

where dz is the standard flat (1,0) form on T2.
Let E be a hermitian vector bundle with structure group SU(r) over Z,

and p : X = Z × T2 → Z. We then consider T2-invariant solutions to the
complex ASD equations (2.5) on p∗E → X. Then these solutions satisfy the
Donaldson-Thomas equations on Z. In fact, if we write a connection A on
X = Z × T2 as AX = A + ϕdz + ϕ̄dz̄, where A is the Z-component of the
connection AX and ϕ ∈ Γ(Z,su(E)), then the curvature becomes

FAX = FA + dAϕ ∧ dz + dAϕ̄ ∧ dz̄ + [ϕ,ϕ̄]dz ∧ dz̄ .

Hence, if we put u := ϕθ̄0 ∈ Ω0,3(Z,su(E)), then A and u satisfy the
Donaldson-Thomas equations, provided that this AX is a T

2-invariant so-
lution to the complex ASD equations.

3. Local model for the moduli space of
Donaldson-Thomas instantons

Let Z be a compact symplectic 6-manifold with symplectic form ω, (E,h)
a hermitian vector bundle over Z with hermitian metric h.


96 y. tanaka

We denote by A(E) = A(E,h) the set of all connections of E which pre-
serve the hermitian structure of E, and put C(E) := A(E) × Ω0,3(Z,u(E)).
We denote by G(E) = G(E,h) the gauge group, the group of unitary auto-
morphism of (E,h), where the action of the gauge group on C(E) is defined
by g(A,u) = (A − (dAg)g−1,g−1ug). These spaces C(E), G(E) can be seen as
Fréchet spaces with C∞-norms, but we shall use Sobolev completions of them
in Section 3.2.

We denote by Γ(A,u) the stabilizer at (A,u) ∈ C(E) of the gauge group
G(E), namely,

Γ(A,u) :=
{
g ∈ G(E) : g(A,u) = (A,u)

}
.

We call (A,u) ∈ C(E) irreducible if Γ(A,u) coincides with the centre of the
structure group of E, and reducible otherwise. We denote by C∗(E) the set
of all irreducible pair (A,u) ∈ C(E). Note that the action of G(E) is not free
on C∗(E), but the action of Ĝ(E) = G(E)/U(1) is free on C∗(E).

We denote by D(E) the set of all D-T instantons of E, and by D∗(E) the
set of all irreducible D-T instantons of E. We call M(E) = D(E)/G(E) the
moduli space of the Donaldson-Thomas instantons.

3.1. Linearization. The infinitesimal deformation of a D-T instanton
(A,u) is described by the following sequence:

0 −→ Ω0(Z,u(E))
D(A,u)

−−−−−−→ Ω1(Z,u(E)) ⊕ A0,3(Z,u(E))
D+

(A,u)
−−−−−−→ Ω+(Z,u(E)) −→ 0 ,

(3.1)

where

D(A,u)(s) = (dAs, [ũ,s]) , ũ = u + ū ,

D+
(A,u)

(α,υ) = d+Aα + Λ
2([u,ῡ] + [υ,ū]) + D̄∗Aυ

for s ∈ Ω0(Z,u(E)) and (α,υ) ∈ Ω1(Z,u(E)) ⊕ A0,3(Z,u(E)). If (A,u) is a
D-T instanton, then (3.1) is a complex. In fact, D+

(A,u)
D(A,u) = 0 follows

directly from the equations (2.3), (2.4). The complex (3.1) can be seen as
“holding” of the A0,3(Z,u(E))-term in (2.2), namely, it is equivalent to con-
sider the following complex instead of (3.1):

0 −→ Ω0(X,u(E)) dA−−→ Ω1(X,u(E))
d+
A−−→ Ω+(X,u(E))

D̄′
A−−→ A0,3(X,u(E)) −→ 0 .

(3.2)


moduli space of donaldson-thomas instantons 97

This is the same as that of the Hermitian-Einstein connections in Section 2.2,
but it still makes sense in the almost complex setting. Hence the following
just reduces to the case in (3.2), and it was proved by Reyes Carrión [41].

Proposition 3.1. If (A,u) ∈ D(E), then the complex (3.1) is elliptic.

We denote by Hi
(A,u)

= Hi
(A,u)

(Z,u(E)) the i-th cohomology of the complex

(3.1) for i = 0,1,2.

The complex (3.2) has the associated Dolbeault complex as Kim [30] de-
scribed it in the Kähler case (see also [31, Chapter VII, §2]):

0 −−−−→ Ω0 dA−−−−→ Ω1
d+
A−−−−→ Ω+

D̄′
A−−−−→ A0,3 D̄A−−−−→ 0yj0 yj1 yj2 yj3

0 −−−−→ Ω0,0 ∂̄A−−−−→ Ω0,1 ∂̄A−−−−→ Ω0,2 ∂̄A−−−−→ Ω0,3 ∂̄A−−−−→ 0 ,

(3.3)

where j0 is injective, j1 is bijective, j2 is surjective with the kernel {βω : β ∈
Ω0}, and j3 is bijective. Hence the index of the complex (3.2), thus that of
the complex (3.1), can be expressed by that of the Dolbeault complex above,

which is given by
∫
Z
Â(Z)∧ch

(
K

1/2
Z

)
∧ch(u(E)) (see [23, §3.5]). In the Kähler

case, the index can be computed as

∫
Z
c1(Z) ∧

(
r − 1
2

c1(E)
2 − rc2(E)

)
+ r2

3∑
i=0

(−1)i dimH0,i(Z) .

Note that the index is zero if Z is a Calabi-Yau threefold.

3.2. Kuranishi model and the local description of the moduli
space. We denote by Ck(E), C∗k(E), Dk(E), D

∗
k(E) the L

2
k-completions of

C(E), C∗(E), D(E), D∗(E) respectively, and by Gk+1(E) the L2k+1-completion
of G(E). We take k sufficiently large so that Gk+1 becomes a Hilbert Lie group
acting smoothly on Ck(E), the quotient topology Ck(E)/Gk+1(E) becomes
Hausdorff (see e.g. [22, §3]), and to use implicit function theorems for the
Sobolev spaces. A general reference for the Sobolev spaces and the implicit
function theorems on them for our purpose is, for example, [52].


98 y. tanaka

Slice. We define slice S(A,u),ε at (A,u) in Ck(E) by

S(A,u),ε :=
{
(α,υ) ∈ L2k

(
u(E) ⊗

(
Λ1 ⊕ A0,3

))
: D∗(A,u)(α,υ) = 0 , ||(α,υ)||L2k ≤ ε

}
.

This set S(A,u),ε is transverse to the Gk+1-orbit through (A,u) as kerD∗(A,u)
is orthogonal, with respect to the L2-norm in L2k

(
u(E) ⊗ (Λ1 ⊕ A0,3)

)
, to

ImD(A,u). There is a natural map

P(A,u),ε : S(A,u),ε −→ Ck(E)/Gk+1(E) , (α,υ) 7−→ [(A + α,u + υ
′)] ,

where υ′ = j3(υ), and j3 : A
0,3 → Ω0,3 is the map in (3.3).

In the following, we take (A,u) ∈ C∗k(E) for simplicity.

Proposition 3.2. Let (A,u) ∈ C∗k(E). Then there exists ε > 0 such that
S(A,u),ε is diffeomorphic to P(A,u),ε

(
S(A,u),ε

)
in C∗k(E)/Ĝk+1(E).

Proof. This is a familiar claim in gauge theory, the proof is a modification
of known results for the ASD and the Hermitian-Einstein connections (cf.
[16, Theorem 6], [22, Theorem 3.2 and Theorem 4.4], [31, Chapter VII, §4,
Theorem 4.16], and [36, Proposition 4.2.1]). We divide the proof into two
steps:

Step 1. We consider a map f(A,u) : S(A,u),ε × Ĝk+1(E) → C∗k(E) defined by
f(A,u)((α,υ),g) = g(A+α,u+υ

′). Then the differential of f(A,u) at ((0,0), id)
is given by

Df(A,u)|((0,0),id)((β,φ),s) = (β,φ) + D(A,u)(s) .

As ImD(A,u) and kerD
∗
(A,u)

are L2-orthogonal in L2k
(
u(E) ⊗ (Λ1 ⊕ A0,3)

)
,

Df(A,u)|((0,0),id) is injective if (A,u) is irreducible.
On the other hand, associated to the operator

D∗(A,u)D(A,u) : L
2
k+1(u(E) ⊗ Λ

0)/u(1) −→ L2k−1(u(E) ⊗ Λ
0)/u(1) ,

where

L2k+1(u(E) ⊗ Λ
0)/u(1) =

{
s ∈ L2k+1(u(E) ⊗ Λ

0) :
∫
Z
tr (s) volg = 0

}
,

there exist the Green operator G0 : L2k(u(E)⊗Λ
0)/u(1) → L2k(u(E)⊗Λ

0)/u(1)
and the harmonic projection H0 : L2k(u(E) ⊗ Λ

0)/u(1) → L2k(u(E) ⊗ Λ
0)/u(1)

with the identity:
Id = H0 + D∗(A,u)D(A,u) ◦ G

0


moduli space of donaldson-thomas instantons 99

(see e.g. [52, Chapter IV, §5]). From the identity, we obtain

D∗(A,u)((γ,χ) − D(A,u)G
0D∗(A,u)(γ,χ)) = 0

for any (γ,χ) ∈ L2k(u(E) ⊗ (Λ
1 ⊕ A0,3)). Thus, for a given (γ,χ) ∈ L2k(u(E) ⊗

(Λ1⊕A0,3)), we take (β,φ) = (γ,χ)−D(A,u)G0D∗(A,u)(γ,χ), s = G
0D∗

(A,u)
(γ,χ)

to get (γ,χ) = (β,φ) + D(A,u)(s). Therefore Df(A,u)|((0,0),id) is surjective.
We then use an inverse mapping theorem for the Hilbert spaces (see e.g.

[35, Chapter 6]) to deduce that around (A,u), C∗k(E) is locally diffeomorphic
to a neighbourhood of ((A,u), id) in S(A,u),ε × Ĝk+1(E).

Step 2. We then prove that if for (α1,υ1),(α2,υ2) ∈ S(A,u),ε there exists
g ∈ Gk+1(E) such that

(A + α1, ũ + υ1) = g(A + α2, ũ + υ2) , (3.4)

then cg is close to idE in L
2
k+1 for some c ∈ U(1).

Since we assume that (A,u) is irreducible, we can take c ∈ U(1) so that
g′ = cg − idE ∈ ker

(
D(A,u)

)⊥
. From (3.4), we get

dAg
′ = α1g

′ − g′α2 + (α1 − α2) , [ũ,g′] = g′υ1 − υ2g′ + υ1 − υ2 .

Hence,
D(A,u)g

′ =
(
α1g

′ − g′α2 + α12,g′υ1 − υ2g′ + υ12
)
, (3.5)

where α12 = α1 − α2,υ12 = υ1 − υ2.
Since g′ lies in

(
kerD(A,u)

)⊥
, there exists a constant C > 0 independent

of (A,u) and g′ such that ||g′||L2
k+1

≤ C||D(A,u)g′||L2
k
. Thus, using (3.5), we

obtain

||g′||L2
k+1

≤ C
(
||g′||L2

k

(
||α1||L2

k
+ ||α2||L2

k
+ ||υ2||L2

k

)
+ ||α12||L2

k
+ ||υ12||L2

k

)
.

Hence,

||g′||L2
k+1

≤
C

1 − 3εC

(
||α12||L2

k
+ ||υ12||L2

k

)
for ε < 1/3C. Thus, we get ||cg − idE||L2

k+1
< C′ε for ε small, where C′ is a

positive constant.
From this, the assertion of the lemma is reduced to Step 1.

Remark 3.3. By modifying the proof of Lemma 3.2, one can prove that
for (A,u) ∈ Ck(E), there exists ε > 0 such that S(A,u),ε/Γ̂(A,u) is diffeomorphic
to P(A,u)

(
S(A,u),ε/Γ̂(A,u)

)
in Ck(E)/Ĝk+1(E), where Γ̂(A,u) = Γ(A,u)/U(1), fol-

lowing, for example, [22, Theorem 4.4].


100 y. tanaka

Kuranishi model. This is also a familiar picture in gauge theory. We
describe it for the Donaldson-Thomas instanton case, modifying known results
in the ASD and Hermitian-Einstein connections (cf. [16, Proposition 8], [31,
Chapter VII, §4, Theorem 4.20], and [36, Proposition 4.5.3]). We take (A,u) ∈
Dk(E), and consider a deformation (A + α,u + υ′) ∈ Dk(E), where

(α,υ) ∈ L2k
(
u(E) ⊗ (Λ1 ⊕ A0,3)

)
.

Then, (α,υ) satisfies the following:

d+Aα + π
+(α ∧ α) + Bu(υ) + Λ2[υ,ῡ] + D̄∗Aυ + ∗̄α∗̄υ = 0 , (3.6)

where Bu(υ) := Λ
2([u,ῡ] + [υ,ū]).

Associated to the operator

D+
(A,u)

(D+
(A,u)

)∗ : L2k(u(E) ⊗ Λ
+) −→ L2k(u(E) ⊗ Λ

+) ,

there exist the Green operator G2 : L2k(u(E) ⊗ Λ
+) → L2k(u(E) ⊗ Λ

+) and the
harmonic projection H : L2k(u(E) ⊗ Λ

+) → L2k(u(E) ⊗ Λ
+) with the identity:

Id = H + D+
(A,u)

(D+
(A,u)

)∗ ◦ G2

(see e.g. [52, Chapter IV, §5]). Using these, we define a map

K(A,u) : L
2
k(u(E) ⊗ (Λ

1 ⊕ A0,3)) −→ L2k(u(E) ⊗ (Λ
1 ⊕ A0,3))

by

K(A,u)(α,υ) :=
(
α +

(
d+A

)∗ ◦ G2 ◦ (π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ) ,
υ +

(
D̄′A + (B

∗
u)

′) ◦ G2 ◦ (π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ)) ,
where (B∗u)

′ = B∗u ◦ πω, B∗u : Ω0ω → A0,3 is the adjoint of Bu, and πω is the
orthogonal projection from Ω2 to Ω0ω.

Lemma 3.4. A pair (α,υ) ∈ L2k(u(E) ⊗ (Λ
1 ⊕ A0,3)) satisfies (3.6) if and

only if it satisfies D+
(A,u)

K(A,u)(α,υ) = 0 and

H
(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
= 0 .


moduli space of donaldson-thomas instantons 101

Proof. Using the identity Id = H + D+
(A,u)

(
D+

(A,u)

)∗ ◦ G2, we rewrite the
left-hand side of (3.6) as

d+A
(
α + (d+A)

∗ ◦ G2 ◦
(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

))
+ Bu(υ)

+ D̄∗A
(
υ +

(
D̄′A + (B

∗
u)

′) ◦ G2 ◦ (π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ))
+ H ◦

(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
= D+

(A,u)
K(A,u) + H ◦

(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
.

(3.7)

Hence, if D+
(A,u)

K(A,u) = 0 and H ◦
(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
= 0, then

(3.6) holds.
Conversely, if (3.6) holds, then from (3.7) we get

D+
(A,u)

K(A,u) + H ◦
(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
= 0 .

Thus,
(
D+

(A,u)

)∗
D+

(A,u)
K(A,u) = 0. This implies

∥∥D+
(A,u)

K(A,u)
∥∥
L2
k−1(u(E)⊗Λ

+)
=

0, hence, D+
(A,u)

K(A,u) = 0 and H ◦
(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
= 0.

We put Sd
(A,u),ε

:=
{
(α,υ) ∈ S(A,u),ε : (α,υ) satisfies (3.6)

}
, and denote

by Hi
(A,u)

(Z,u(E)) (i = 0,1,2) the harmonic spaces of the complex (3.1).

Lemma 3.5. K(A,u)
(
Sd
(A,u),ε

)
⊂ H1

(A,u)
(Z,u(E)) .

Proof. From the definition of the map K(A,u), we have

D∗(A,,u)K(A,,u)(α,υ) = D
∗
(A,u)(α,υ)

+ D∗(A,u)(D
+
(A,u)

)∗
(
G2 ◦

(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

))
for (α,υ) ∈ Sd

(A,u),ε
. This is equal to 0, because D∗

(A,u)
(α,υ) = 0 for (α,υ) ∈

Sd
(A,u),ε

, and D∗
(A,u)

(
D+

(A,u)

)∗
= 0 as D+

(A,u)
D(A,u) = 0. From Lemma 3.4, we

also have D+
(A,u)

K(A,u) = 0. Thus Lemma 3.5 holds.

From Lemmas 3.4 and 3.5, we deduce the following.

Lemma 3.6. A pair (α,υ) ∈ L2k
(
u(E) ⊗ (Λ1 ⊕ A0,3)

)
lies in Sd

(A,u),ε
if and

only if K(A,u)(α,υ) ∈ H1(A,u)(Z,u(E)) and

H
(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
= 0 .


102 y. tanaka

We now prove the following.

Theorem 3.7. Let (A,u) ∈ D∗(E). Then there exists a neighbourhood
U of 0 in H1

(A,u)
(Z,u(E)) such that around [(A,u)] the moduli space

M∗(E) = D∗(E)/Ĝ(E) is locally modeled on the zero set of a real analytic
map κ(A,u) : U → H2(A,u)(Z,u(E)) with κ(A,u)(0) = 0, and the first derivative
of κ(A,u) at 0 also vanishes.

Proof. From the definition of the map K(A,u), we have K(A,u)(0) = 0.
Since the differential of K(A,u) at 0 is identity, we can deduce, from the inverse
mapping theorem on the Hilbert spaces (see e.g. [35, Chapter 6]), that there
exist a neighbourhood U of 0 in H1

(A,u)
(Z,u(E)) and a map

K−1
(A,u)

: U −→ L2k
(
u(E) ⊗

(
Λ1 ⊕ A0,3

))
such that K−1

(A,u)
is a diffeomorphism between U and K−1

(A,u)
(U). We then define

a map κ(A,u) : U → H2(A,u) by κ(A,u) = ψ ◦ K
−1
(A,u)

, where ψ : H1
(A,u)

→ H2
(A,u)

is defined by

ψ(α,υ) = H
(
π+(α ∧ α) + Λ2[υ,ῡ] + ∗̄α∗̄υ

)
.

We now take ε sufficiently small so that all the following hold. Firstly,
from Lemma 3.6, the zero set of κ(A,u) is mapped by K

−1
(A,u)

diffeomorphically

to an open subset in Sd
(A,u),ε

. Next, from Proposition 3.2, Sd
(A,u),ε

is diffeo-

morphic to p(A,u),ε(S
d
(A,u),ε

) in D∗k(E)/Ĝk+1(E). Hence, the zero set of κ(A,u)
is diffeomorphic to a neighbourhood of [(A,u)] in D∗k(E)/Ĝk+1(E). More-
over, from the elliptic regularity, the harmonic elements are actually smooth,
therefore the neighbourhood of [(A,u)] in D∗k(E)/Ĝk+1(E) is isomorphic to a
neighbourhood of [(A,u)] in M∗(E).

The assertions that κ(A,u) = 0 and the derivative of κ(A,u) at 0 is zero just

follow from the definition κ(A,u) = ψ ◦K
−1
(A,u)

and the fact that the differential
of K(A,u) at 0 is the identity.

From Theorem 3.7, one can deduce that M∗(E) is smooth around [(A,u)] if
H2

(A,u)
(Z,u(E)) = 0. But, as in the case of the Hermitian-Einstein connections

(cf. [30], [31, Chapter VII, §4], [25, Chapter 2, §2.1], [36, Chapter 4, §4.5]),
it can be improved in the following way. Firstly, we note that, corresponding
to the decomposition of u(r) = iR ⊕ su(r), the bundle u(E) naturally decom-
poses into R and u(E)0 over Z, where u(E)0 is the bundle of trace-free skew-
Hermitian endmorphisms of E, and there is a subcomplex of the complex (3.1),


moduli space of donaldson-thomas instantons 103

which is defined by using the bundle u(E)0 instead of u(E). The decomposition
is preserved by the operators of the complex, hence it induces a corresponding
splitting of Hi

(A,u)
(Z,u(E)) (i = 0,1,2). For (αc,υc) ∈ Λ1(Z) ⊕ A0,3(Z), it is

always H
(
π+(αc ∧αc) + Λ2[υc, ῡc] + ∗̄αc∗̄υc

)
= 0, hence the map κ(A,u) values

in H2(Z,u(E)0). In particular, we obtain the following.

Corollary 3.8. Around [(A,u)] ∈ M∗(E) with H2
(A,u)

(Z,u(E)0) = 0,

the moduli space M∗(E) is smooth.

Remark 3.9. Around (A,u) ∈ D(E), which is not irreducible, one can
prove that H1

(A,u)
(Z,u(E)) and H2

(A,u)
(Z,u(E)) are Γ(A,u)-invariant, and the

map κ(A,u) is Γ(A,u)-equivariant. Hence, combining the claim in Remark 3.3,
one can deduce that around [(A,u)] the moduli space M(E) is locally modeled
on κ−1

(A,u)
(0)/Γ(A,u).

4. The Hitchin-Kobayashi correspondence for the
Donaldson-Thomas instantons on compact Kähler threefolds

Perhaps one might ask what kind of a Hitchin-Kobayashi style correspon-
dence would hold for the Donaldson-Thomas instanton on compact Kähler
threefolds. In this section, we describe this, which actually follows from a
result by Álvarez-Cónsul and Garćıa-Prada [1].

Let Z be a compact Kähler threefold, and E = (E,h) a Hermitian vector
bundle over Z with Hermitian metric h. If (A,u) is a D-T instanton on E,
then the connection A defines a holomorphic structure ∂̄A on E as F

0,2
A = 0,

thus, we can think of E as a locally free sheaf O(E,∂̄A). In addition, the
End(E)-valued (0,3)-form u is naturally identified with a section of the bundle
End(E) ⊗ K−1Z , so ∗̄u is a section of the bundle End(E) ⊗ KZ. The equation
∂̄∗Au = 0 implies ∂̄A∗̄u = 0, hence, φ := ∗̄u is a holomorphic section of
End(E) ⊗ KZ.

We then consider a pair (E,φ) consisting of a torsion-free sheaf E and a
holomorphic section φ of End(E) ⊗ KZ. A subsheaf F of E is said to be a
φ-invariant if φ(F) ⊂ F ⊗KZ. We define a slope µ(F) of a coherent subsheaf
F of E by

µ(F) :=
1

rank(F)

∫
Z
c1(det F) ∧ ω2 .

Definition 4.1. A pair (E,φ) consisting of a torsion-free sheaf E and a
holomorphic section φ of End(E)⊗KZ is called semi-stable if µ(F) ≤ µ(E) for


104 y. tanaka

any φ-invariant coherent subsheaf F with rank(F) < rank(E). A pair (E,φ)
is called stable if µ(F) < µ(E) for any φ-invariant coherent subsheaf F with
rank(F) < rank(E).

Definition 4.2. A pair (E,φ) consisting of a torsion-free sheaf E and a
holomorphic section φ of End(E)⊗KZ is said to be poly-stable if it is a direct
sum of stable sheaves with the same slopes in the sense of Definition 4.1.

Then the correspondence can be stated as a one-to-one correspondence
between a pair (E,φ), where E is a locally-free sheaf on a Kähler threefold
Z and a holomorphic section φ of End(E) ⊗ KZ, which is stable in the sense
of Definition 4.1; and the existence of a solution to the Donaldson-Thomas
equations on E. This fits into a setting studied by Álvarez-Cónsul and Garćıa-
Prada [1] (see also [6]), and it is stated as a special case of their results as
the case of a twisted quiver bundle with one vertex and one arrow, whose
head and tail conincide, and with twisting sheaf the anti-canonical bundle.
We state it in our setting as follows.

Theorem 4.3. ([1]) Let Z be a compact Kähler threefold with Kähler
form ω. Let (E,φ) be a pair consisting of a locally-free sheaf E on Z and a
holomorphic section φ of End (E)⊗KZ. Then, (E,φ) is poly-stable if and only
if E admits a unique Hermitian metric h satisfying

ΛFh + Λ
3[φ,φ̄h] + 6πiµ(E)IdE = 0 ,

where Fh is the curvature form of h, and Λ := (∧ω)∗.

Note that the equation ∂̄∗Au = 0 in the Donaldson-Thomas equations on
a compact Kähler threefold is implicitly addressed in Theorem 4.3 by saying
that φ = ∗̄u is a holomorphic section of End(E) ⊗ KZ. One more remark
is that a proof of the Hitchin-Kobayashi correspondence using the Mehta-
Ramanathan argument for the Vafa-Witten equations in [46] could also apply
to the Donaldson-Thomas instanton on smooth projective threefold as men-
tined in [46].

Acknowledgements

I would like to thank Mikio Furuta, Ryushi Goto, Ryoichi Kobayashi,
Hiroshi Ohta for valuable comments, and referees for many useful advice.
I am also grateful to Katrin Wehrheim for wonderful encouragement. A
part of this article was written when I had visited Beijing International
Center for Mathematical Research, Peking University in 2008 – 2009,


moduli space of donaldson-thomas instantons 105

I am very grateful to Gang Tian and the institute for their support and
hospitality. A part of revision was made during my visit to Institut des
Hautes Études Scientifiques in February to March of 2012. I would like
to thank the institute for the support and giving me an excellent research
environment. Last but not least, I would like to thank Dominic Joyce for
enlightening me on these subjects over the years. This work was partially
supported by JSPS Grant-in-Aid for Scientific Research No. 15H02054.

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