Morse theory on moduli spaces of instantons
on ALE scalar-flat Kähler surfaces

Hiraku Nakajima
Mathematical Institute, Tôhoku University

Abstract. LeBrun constructed a scalar-flat Kähler metric on the total space ofC

Chern class −n line bundle O(−n) → P1 . We study moduli spaces of ASD connections on it. It is known that the natural L2 -metrics on them are kählerian. We
study them when the metric is complete. We give an algorithm to compute their
Betti numbers. On the way of the proof, we also show that their homology groups
have no torsion and vanish in odd degrees. Our method is the Morse theory, can beapplied to a wider class of noncompact 4-manifolds.

1. Introduction
In the conference I talked about homology groups of moduli spaces of instantons
on ALE hyper-Kähler 4-manifold constructed by Kronheimer [Kr1]. (The paper[Na2] will appear elsewhere.) After the talk, Y.S. Poon suggested me to apply thesame technique to moduli spaces of instantons on other ALE Kähler surfaces (e.g.,
ALE scalar-flat Kähler metrics constructed by LeBrun [Le1, 2]). This paper gives
an affirmative answer.
Let (X, g) be an ALE scalar-flat Kähler surface (see §2 for the definition). Takea hermitian vector bundle E over X and consider the moduli space M of ASD
connections on E (see §3 for the precise definition). It is known that M has anatural Kähler structure induced from that on X [Na1]. We feel an interst in further
geometric properties of M, e.g., the topology, when the metric is complete. Though
the metric is incomplete in general, we shall give a criterion for the completeness ofthe metric (Proposition 3.4). Now suppose that the base space X has a U(1)-action
preserving the Kähler structure and satisfying the asymptotic condition (4.1). Thenit induces a U(1)-action on the moduli space. Following the approach due to Frankel[Fr] (see also [CS, At, Ki]), we use the corresponding moment map as a Morsefunction. The critical points set corresponds to the fixed points set of the action,and its components are submanifolds of M. Under the further assumption (4.3),
we can perturb the Morse function to have only isolated critical points of evenindices. Critical points and their indices have geometric meaning, thus we have analgorithm to compute homology groups of M (Theorem 4.7).
Our previous result [Na2] gives an improvement of the algorithm in the case of
ALE hyper-Kähler 4-manifolds, but relies on the ADHM description of the moduli
space [KN]. On the other hand, our result is purely geometric.
Supported in part by Grant-in-Aid for Scientific Reserch (No. 03740014), Ministry of Education, Science and Culture, Japan
Typeset by AMS-TEX
1

2. ALE scalar-flat Kähler surfaces
Let X be a complex surface. A Kähler metric g on X is called scalar-flat if its
scalar curvature is identically zero. Such metrics attract our interest since they haveanti-self-dual Weyl curvature [Ga]: hence twistor spaces with integrable complex
structures [AHS].
Though lots of compact scalar-flat Kähler surfaces are known and studied extensively, we here will study noncompact spaces with the ALE condition. ALEstands for asymptotically locally Euclidean and means that our 4-manifold (X, g) isassumed to be complete, and that there exists a compact set K such that X \ K
is diffeomorphic to (R4 \ BR )/Γ for some finite subgroup Γ of O(4) acting freely on
R4 \ BR , and the metric g approximates the Euclidean metric. Such a coordinate
system is called a coordinate system at infinity. Since we are considering Kähler
metrics, the group Γ is a subgroup of U(2), and the complex structure I on X
approximates that on the Euclidean space C2 = R4 .
Many examples of such spaces are known. A trivial example is, of course, theEuclidean space C2 . Kronheimer [Kr1] constructed such metrics on the minimal
resolution of C2 /Γ, where Γ is a finite subgroup of SU(2). His spaces have thefurther property: hyper-Kähler structures. Other examples were given by LeBrun[Le1, 2]. He constructed a scalar-flat Kähler metric on the total space of any
complex line bundle over CP1 with the first Chern class c1 < −2. The fundamentalgroup Γ of the end is the cyclic group of order #Γ = −c1 . He also constructed
such a metric on the blow-up of C2 at n-points situated along a straight complex
line. The projection onto C2 gives a coordinate system at infinity. In particular,Γ = {e}.
The above constructions seems closely related to the geometry of the moduli
spaces of ASD connections on them. However we do not review them here as
we only give a general algorithm for the calculation of Betti numbers. If one
wants to know about further properties of geometry, he/she certainly needs toknow the constructions. For example, we describe our algorithm in terms of theYoung tableaux when the space is Kronheimer’s one [Na2]. We needed the detailed
information about his construction.
3. Moduli spaces of ASD connections
Constructions of the moduli spaces of ASD connections on ALE spaces are discussed in [Na1] in detail. We must introduce weighted Sobolev norms in order toargue rigorously, but here we omit the analytic details for the sake of brevity. Theinterested reader should consult the above mentioned paper.
As in the previous section, (X, g) is assumed to be an ALE scalar-flat Kähler
surface with a coordinate system at infinity X \ K → (R4 \ BR )/Γ. Let us take arepresentation ρ: Γ → U(r). Take a hermitian vector bundle E over X and supposethat there exists a connection A0 on E whose restriction to the end X \ K is flatand corresponds to the representation ρ. Let A be the set of connections A on E
such thatl timesz}|{| ∇A0 · · · ∇A0 (A − A0 )| = O(r−3−l ),where r is the distance function from a point in X. Let G0 be the group of gauge
2transformations s withl timesz}|{| ∇A0 · · · ∇A0 (s − id)| = O(r−2−l ).
This group acts on A by pull-back. Then the moduli space of ASD connections isdefined bydef.

M = {A ∈ A | ∗RA = −RA }/G0 .
It is shown that the moduli space is a C ∞ -manifold near the gauge equivalenceclass [A], if
0 = L2 – Ker dA∗ : Ω+ (Endskew E) → Ω1 (Endskew E),
where Endskew E is the bundle of skew-adjoint endomorphisms of E. Thanks tothe anti-self-duality of the Weyl tensor and the vanishing of the scalar curvature,
we can prove that this condition is satisfied for any ASD connection A on X using
Bochner-Weitzenböck formula (see [Na1, 5.1]). Hence the moduli space M is asmooth manifold. Its dimension is given by the index formula (see [Na1, 2.7]). The
tangent space at [A] can be identified with
L2 – Ker(dA+ ⊕ dA∗ ): Ω1 (Endskew E) → Ω+ (Endskew E) ⊕ Ω0 (Endskew E).
The L2 -inner product gives a Riemannian metric on M. The almost complex
structure I on X induces an endomorphism on the cotange bundle T ∗ X, and theL2 – Ker(dA+ ⊕ dA∗ ) is invariant under it. Hence we have an almost complex structure IM on M. And it is known that IM is covariant constant with respect to theLevi-Civita connection of the L2 -metric [Na1, 2.6]. Summarizing the above results,
we have
Theorem 3.1. The moduli space M of ASD connections on the ALE scalar-flatKähler surface X is a Kähler manifold.
Before discussing the further properties of moduli spaces, we relate our moduli
space to the moduli space of ASD connections on the one-point compactificationb = X ∪{∞}. The ALE condition allows us to give X
b the structure of the orbifold.
Xb which is conformal to g on X. (See [Kr2,There exists an orbifold metric gb on X
b Theyp.686].) ASD connections with the above asymptotic condition extend to X.
b The fiber E
b over ∞ has a Γ-action which
all live a fixed orbifold vector bundle E.
is isomorphic to ρ. Then it is not hard to see
Proposition 3.2. The moduli space M is homeomorphic to the framed moduli
b that is the set of isomorphism classes of pairs:
space of ASD connections on E,b Γ-equivariant isomorphism ϕ: E
b∞ → Cr ).
(ASD connection A on E,
We return to study geometric properties of the moduli space M. The first is thenatural ‘symmetry’. The change of the framing induces an action on M:
3

Proposition 3.3. Let Gρ be the stabilizer of the representation ρ:
def.

Gρ = {s ∈ U(r) | sρs−1 = ρ}.
Then there exists an action of Gρ on the moduli space M which preserves the L2 metric and the complex structure.
Next we want to discuss the completeness of the metric. It relates to Uhlenbeck’sb Let [Ai ] be a sequence in M. Thencompactness theorem applied to the orbifold X.
we have a subsequence [Aj ] such thatb such that Aj converges to an ASD(1) there exists a finite set {x1 , . . . , xn } ⊂ X
connection A∞ outside it after gauge transformations,(2) there exist constants ak (k = 1, . . . , n) such that the curvature densities|RAj |2 dV converge to
X|RA∞ |2 dV +ak δxk .
k

The above constant ak relates to the curvature integral of ASD connection bubblingb (i.e., xk ∈ X), ak is an integer multiple
out around xk . If xk is a regular point of X2
of 8π . On the other hand, if xk = ∞, ak is an integer multiple of 8π 2 /#Γ, where#Γ is the order of Γ.
Proposition 3.4. The L2 -metric on the moduli space M is complete if we have
S = ∅ or S = {∞} for any sequence [Ai ] as above.
Proof. Let [At ] (t ∈ [0, t0 )) be an open curve of finite length. We want to showthat [At ] has a limit point. As is shown in [Na1, 5.4], it is enough to show that
|RAt | ≤ Cr−2
for some constant C independent of t. But this can be proved exactly as in [Ba,
Proposition 3]. We omit the detail. ¤There are many examples satisfying the above condition. For example, if
Z|RA |2 dV < 8π 2for [A] ∈ M,
X

then ak < 8π 2 for any k in the above statement (2). Thus the singular set S cannot
contain regular points. For Kronheimer’s ALE spaces, many examples are given
by [KN, 9.2 and Remark following 9.2]. However, when Γ is the trivial group, theabove condition is rarely satisfied. When X = R4 , the L2 -metric is never complete.
4. Morse theory on the moduli space
From now on we assume that
(4.1) the ALE space X has a U(1)-action which preserves both the Riemannian
metric and the complex structure, and approximates the following U(1)action on C2 /Γ under the coordinate system at infinity:for (z1 , z2 ) mod Γ ∈ C2 /Γ, t ∈ U(1),(z1 , z2 ) mod Γ 7→ (tz1 , tz2 ) mod Γ(4.2) H 1 (X; R) = 0,
(4.3) the group Γ is a cyclic group.
4The blow-up of C2 at n-points situated along a straight complex line has an
U(1)-action, but it is asymptotically given by(z1 , z2 ) 7→ (tz1 , z2 ).
It does not satisfy the condition (4.1). The total space of complex line bundle over
CP1 with the first Chern class c1 < 2 with LeBrun’s metric space satisfies all these
conditions. Kronheimer’s space satisfies them if the space are biholomorphic to theminimal resolution of C2 /Γ. (In general, his space is only diffeomorphic to it.)
By the condition (4.2), there exists a function µ: X → R such that Igrad µ is
the vector generating the U(1)-action in (4.1). This is essentially the moment mapof the U(1)-action.
As in §3, we take a hermitian vector bundle E over X admitting a flat connection
A0 on the end.
It is not clear that the U(1)-action in (4.1) can lift to an action on E at firstsight, but easy to see the infinitesimal action is always liftable: (In fact, we shallsee that the action is liftable. See Remark after Lemma 4.5 and Remark 4.9(1).)
Lemma 4.4 (see [GP, 4.3], [Ma, §4]). The U(1)-action in (4.1) induces an infinitesimal U(1)-action V on M given by the formuladef.

V ([A]) = (Igrad µ) y RA ∈ L2 – Ker(dA+ ⊕ dA∗ ) ∼= T[A] M,
where y denotes the interior product.
The vector field V is holomorphic and Killing. The corresponding moment map
is given by [Ma]Z
def.

µ|RA |2 dV.

F0 ([A]) =X

So we have
grad F0 = IV.
Lemma 4.5. The function F0 is proper if the L2 -metric is complete.
Proof. Since the U(1)-action approximates the standard action on C2 , the momentmap µ has the following asymptotic behaviour:√µ≈Hence if F is bounded,

−1 2
r .
2

Z
r2 |RA |2 dVX

is also bounded. Hence the curvature density cannot converges to |RA∞ |2 dV + aδ∞with a > 0. Proposition 3.4 ensures that [A] stays in a compact set. ¤Remark. The above implies that the vector field V is complete.
The function F is a non-degnereate Morse function on M in the sense of Bott.
The critical points are the fixed points of the U(1)-action. This is, in general, aunion of submanifolds of M, and the index along a critical submanifold is an eveninteger.
5

It seems that it is not so easy to determine the fixed point set explicitly. So we
use the Gρ action (see (3.3)) to perturb F0 as follows. Take a maximal torus T r ofGρ . Under the assumption (4.3), we have r = rank E. Consider the corresponding
moment map coupled with an element ε ∈ tr :
Z
lim

r→∞

i(
Sr

∂
) tr(εRA ) ∧ ω,∂rwhere Sr is the distance sphere of the radius r, i denotes the interior product, ω is
the Kähler form, and ε is considered as a section of End E near infinity. Let
(4.6)

Z

def.

F ([A]) = F0 ([A]) + lim

r→∞

i(
Sr

∂
) tr(εRA ) ∧ ω.
∂r

Theorem 4.7. Assume (4.1)–(4.3). Suppose that the L2 -metric is complete andε is sufficiently small and generic. Then the function F satisfies the following
properties.
(1) F is proper.
(2) The gauge equivalence class [A] is a critical point of F if and only if there
exists a T r -invariant decomposition E = L1 ⊕ · · · ⊕ Lr into sum of line bundles
such that the connection A decomposes accordingly.
(3) F is a Morse function (in the usual sense) and the index at each criticalpoint is an even number. In particular, the homology of M has no torsion andvanishes in odd degrees, and every component of M is simply-connected.
Proof. (1) If ε are sufficiently small, F ≤ c implies F0 is bounded. Hence F ≤ c iscompact.
(2) Since F is essentially the moment map of the torus U(1) × T r = T r+1 -action
coupling with ε, the critical points of F are precisely the fixed points if ε are generic.
Take a gauge equivalence class of A. It is fixed by T r if and only if for each h ∈ T r
there exists a gauge transformation γ such that γ ∗ A = A andlim γ(x) = h.

x→∞Then A decomposes as the bundle decomposes into eigenspaces of γ. If the eigenvalues of h are all distinct, the bundle is a direct sum of line bundles, that is
L1 ⊕ · · · ⊕ Lr .
Since H 1 (X; R) = 0, the gauge equivalence classes of connections on a line
bundles are classified by their curvature form. If the curvature form is ASD, it isuniquely determined by its cohomology class, that is the first Chern class of the line
bundle. In particular, the moduli space on La consists of one point, so the point
must be fixed by the U(1)-action. Therefore the direct sum is also a fixed point.
(3) This statement holds for a general function arising from a moment map (see[At, Ki]). But we give the proof for our situation.
Take a fixed point [A] in M. The complex structure I on X induces a complex
structure IM on the tangent space of M at [A]. This complex tangent space of M
at [A] is identified with the L2 -kernel of the operator∗∂A ⊕ ∂A : Ω0,1 (End E) → Ω0,0 (End E) ⊕ Ω0,2 (End E).
6Since [A] corresponds to the sum of line bundles L1 ⊕ · · · ⊕ Lr , the L2 kernel has a
C-vector space decomposition
M∗(L2 -kernel of ∂A ⊕ ∂A ) ∩ Ω0,1 (L∗a ⊗ Lb ).
a,b

Since [A] is a fixed point, there exists a lift t̃ to E of t: X → X which respects
the connection A, preserves the decomposition E = L1 ⊕ · · · ⊕ Lr and acts as theidentity on E∞ = ⊕(La )∞ . Hence T[A] Mζ becomes a U(1)-module and decomposes
into the sum
MM
m
Ha,b
a,b m∈Zmof complex subspaces where U(1) acts on Ha,bwith weight m. Then the hessian ofm
F0 acts on Ha,b as multiplication by m. Suppose ε, regarded as an endomorphism√
on E∞ , acts on (La )∞ as the multiplication by −1εa . Then the hessian of themsecond term in (4.6) acts on Ha,b
as multiplication by εb − εa . So the hessian of F
is non-degenerate, if all εa ’s are distinct, as we have been assuming. The index isgiven by

(4.8)

X
a,b

Ã

X

m<0

X

m
dimR Ha,b+!
mdimR Ha,b.

m=0,εa >εbm
Since Ha,b
is a complex vector space, the index is even. ¤Remarks 4.9. (1) Since F takes a minimum at a point, the moduli space M contains
at least one point which comes from the direct sum of line bundles. Since the moduli
space on a line bundle consists of one point, the U(1)-action lifts to the line bundle.
Therefore the U(1)-action lifts to the direct sum E.
(2) The index (4.8) can be calculated as follows:i) Determine the fixed points set of the U(1)-action on the base manifold X.
ii) Calculate weights for the normal bundles of components F of the fixed
points set.
iii) Calculate the weights for the fiber of L∗a ⊗ Lb over F.
iv) Calculate the eta invariants for the Dirac operators on S 3 /Γ twisted by theflat connection, to which the ASD connection on L∗a ⊗ Lb is asymptotic.
v) Substituting the above data to the Lefschetz fixed points formula, we get
the weights space decomposition of the L2 -kernel of the Dolbeault operator.
(For Kronheimer’s ALE spaces, see [Na2]).
Examples 4.10. (1) Examples for Kronheimer’s ALE spaces can be found in [Na2].
(2) Let X be the total space of the Chern class −n-bundle O(−n) → CP1(n = 2, 3, . . . ) with LeBrun’s metric. The zero section of O(−n), considered as adivisor of X, produces the line bundle L. Then L has a unique ASD connection
which is asymptotic to the trivial connection. Set E = C ⊕ L and consider themoduli space M of ASD connections asymptotic to the trivial connection. Thenthe dimension formula [Na1, 2.7] shows dimR M = 2n.
7

Claim. The L2 -metric is complete.
Proof. We use Proposition 3.4. Suppose we have a sequence [Ai ] in M with {∞} 6=
S. We may assume that ∞ ∈/ S. By Uhlenbeck’s removable singularities theorem
the limit [A∞ ] extends to a connection on a possibly different bundle E 0 over X.
Let M0 be the moduli space containing [A∞ ]. By Remark 4.9(1), M0 contains a
reducible connection. Thena) A∞ is also asymptotic to the trivial connection since ∞ ∈/ S,b) the first Chern class is preserved under the weak convergence, so we havec1 (E 0 ) = c1 (E).
Therefore the reducible connection must be in the form:L⊗m ⊕ L⊗1−m ,where L⊗−m = (L∗ )⊗m for m > 0. We haveZZ
0

c2 (E ) =
X

Z
c2 (L

⊗m

⊗1−m

⊕L

X

)≥

c2 (E)
X

with the equality if and only if m = 0, 1. However, by the lower semi-continuityof the action under the weak convergence, we have the inequality of the opposite
direction. Hence c2 (E 0 ) = c2 (E) and S = ∅. ¤We have two fixed points of the T 3 -action corresponding to C ⊕ L and L ⊕ C.
(Since we are discussing the framed moduli space, these two points are different !)As in [Na2] we can show that one has index 0 and the other has 2. Hence thePoincaré polynomial is 1 + t2 .
(3) Let X be as in (2). Take the line bundle L such that the first Chern classc1 (L) is a generator of H 2 (X; Z) ∼= Z. It has an ASD connection A0 asymptotic toa flat connection with the associated representation
µ
¶
2πik
2πik
ρ exp() = exp()
nn

for k = 0, 1, . . . , n − 1.

Set E = L ⊕ L∗ and consider the moduli space M of ASD connections asymptotic
to ρ ⊕ ρ∗ . SinceZZ
1c2 (E) = −c1 (L)2 = < 1,nXX
the L2 -metric is complete. The dimension formula [Na1, 2.7] shows dimR M = 2 if
n > 2, and dimR M = 4 if n = 2.
When n = 2, we have two fixed points of T 3 -action corresponding to L ⊕ L∗ ,∗L ⊕ L. (Note that ρ∗ ∼= ρ if n = 2.) One is of index 0 and the other of 2. Hencethe Poincaré polynomial is 1 + t2 . In fact, it can be shown that the moduli spaceis isomorphic to the cotangent bundle of CP1 (see [KN]).
When n > 2, we have only one fixed point L ⊕ L∗ . Hence M is diffeomorphic to
the 2-ball B 2 .
Acknowledgement. I would like to thank Y.S. Poon for suggesting me the problem.
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C

C

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Aramaki, Aoba-ku, Sendai 980, Japan

E-mail address: nakajima@math.tohoku.ac.jp9