A convergence theorem for Einstein metrics and the ALE spaces

Hiraku Nakajima
Mathematical Institute, Tôhoku University

Introduction
A Riemannian metric g on a manifold is called an Einstein metric, if its Ricci
curvature is a constant multiple of g, i.e., Ric = kg for some constant k. Someone
says that it is an equation for the space-time in the relativity, so quite important.
However it seems diﬃcult to understand the meaning of this definition itself at
first sight. We know its importance only by experiences of seeing that Einstein
metrics appeared in many fields of diﬀerential geometry. For example, if you readthe book by Besse [Be], you know how many fields relate to Einstein metrics. Oneof the most important results is the existence of Einstein-Kähler metric obtainedby Aubin [Au] and Yau [Ya] (shown by Yau when c1 = 0, and by Aubin and Yau
independently when c1 < 0). The result had many applications to the algebraic
geometry. Recently in works by Narashimhan-Seshadri [NaSe], S.Kobayashi [Kos],
Lübke [Lü], Donaldson [Do1, 2], Uhlenbeck-Yau [UY] and others, it was pointed outthat an Einstein-Hermitian metric, which is a counterpart of the Einstein-Kähler
metric in the gauge theory, has a close relationship with stable holomorphic vector
bundles on Kähler manifolds. By their results it becomes possible to study themoduli space of holomorphic vector bundle from the diﬀerential geometric view
point. (See the textbook by Siu [Si].)
It is believed that Einstein-Kähler metrics also relate to the moduli space ofcomplex structures. Then it is natural to consider its compactification. Anderson [An1] and Bando-Kasue-Nakajima [BaKaNa1 ] obtained a convergence theorem
for Einstein metrics when the manifold is 4-dimensional. We used the theory of
the Hausdorﬀ distance, which is introduced by Gromov, and developed by many
peoples. (It should be noticed that our result is motivated by the study of a degeneration of K3 surfaces by R.Kobayashi-Todorov [KorTo].) In particular, we obtaina compactification of the moduli space when the Ricci curvature is positive, and acompletion when nonpositive. As a limit we get a space called an orbifold, which
is not a manifold.
From the analytic point of view regarding the Einstein metric as a solution of
PDE, one can find many cousins of the convergence theorem in other nonlinear
PDE’s of elliptic type which have the “the scaling invariance”. In these problems,
the Palais-Smale condition (C) does not hold, i.e., there may not exist a convergent
subsequence in a given sequence of solutions with bounded energy. But one canshow that solutions converge outside a finite set thanks to the scaling invariance.
1 This is a joke by B&O. “Bakana” means foolish in Japanese !

Typeset by AMS-TEX

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One can find examples in Sacks-Uhlenbeck [SaUh], Uhlenbeck [Uh], Sedlacek [Se],Brezis-Coron [BrCo], Bahri-Coron [BaCo], and so on.
As a bi-product of the convergence theorem, there appears naturally a family of
noncompact Ricci-flat manifold, called ALE spaces. (It corresponds to a “bubble”
in the case of harmonic maps or Yang-Mills connections.) The ALE stands forasymptotically locally Euclidean and means that the metric approximates the standard metric on R4 /Γ in the end. The ALE spaces with hyper-Kähler structures already appeared in the very diﬀerent context. Physicists, Eguchi-Hanson [EgHa] andGibbons-Hawking [GiHa] constructed ALE spaces corresponding to cyclic groups.
On the other hand, Hitchin [Hi] constructed the same space by using the twistor
method. His construction suggested us a relationship to a deformation theory of
the simple singularity C2 /Γ. Thus he conjectured that there exist similar spaces
corresponding to other finite subgroup of SU(2) (i.e., the binary polyhedral groups).
This conjecture was solved aﬃrmatively by Kronheimer using the method called
hyper-Kähler quotients. His construction gives a hyper-Kähler structure on eachfiber of the semi-universal deformations and their simultaneous resolutions, whichwere constructed by Brieskorn [Br] and Slodowy [Sl] by using nilpotent varieties of
Lie algebras.
ALE spaces with hyper-Kähler structures can be considered as analogues of K3
surfaces in several points. The author [Na2] studied the moduli space of anti-selfdual connections motivated by the study of stable holomorphic vector bundles overK3 surfaces by Mukai [Mu]. The moduli space has a hyper-Kähler structure, andbecomes again an ALE space when its dimension is 4. But the base space and themoduli space are not homeomorphic to each other in general, in contrast with thecase of K3 surfaces. This result was further developed by Kronheimer-Nakajima[KrNa]. We gave an analogue of the ADHM construction for the anti-self-dual
connections on S 4 . Our theory relates to the representations of quivers on theextended Dynkin diagrams. All these results show us the richness of the geometryof ALE spaces.
In this article we shall explain about the above results, but we cannot write
about moduli spaces of anti-self-dual connections because of limitations of space. I
hope other occasion. In §1, we shall introduce the convergence theorem. In §2, weshall see how ALE spaces bubble oﬀ. Then in §3, we shall explain the constructionof ALE spaces with hyper-Kähler structures. You can read §3 independent of
the previous sections, though §2 depends on §1. The statements of theorems areanalytic, but we must use the languages of the algebraic geometry to give examples.
Please read only the part which the reader has an interest.
1. A convergence theorem for Einstein metrics
As we said in the introduction, the Einstein metric is a solution of a PDE describing the space-time. Here we consider a sequence of Einstein metrics and study
its convergence. This means that we study the situation when the space-time ischanging and broken. Such convergence theorems appear very often in the studyof elliptic partial diﬀerential equations, and actually our convergence theorem isstrongly motivated by Sacks-Uhlenbeck’s theorem on harmonic maps [SaUh] andUhlenbeck’s compactness theorem for Yang-Mills connections [Uh1, 2]. So we firstrecall the case of harmonic maps, which seems simplest, then proceed to the caseof Einstein metrics.
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Theorem 1.1 [SaUh]. Let (X, g) be a compact 2-dimensional Riemannian manifold, and let (M, h) be a compact Riemannian manifold of arbitrary dimension. Let{fi : X → M } be a sequence of harmonic maps withdef.

∫
|dfi |2 dVg ≤ E < ∞,E(fi ) =X

where E is a positive constant independent of i. Then there exists a subsequence{j} ⊂ {i} satisfying the following properties:(1) There exists a finite set S = {x1 , . . . , xk } in X such that fj converges to a∞
map f∞ in Cloc
(X \ S).
(2) The limit map f∞ : X \ S → M extends smoothly to the whole space X.
The second statement is deduced from the following removable singularities theorem.
Theorem 1.2 [SaUh]. Let D be the 2-disk. If a harmonic map f defined overD \ {0} satisfies∫|df |2 < ∞,D\{0}it extends smoothly across the singularity.
Now give an example of convergence. Take the Riemann sphere S 2 = C ∪ {∞}for both (X, g) and (M, h), and consider the sequence of rational functions given by{fi (z) = iz}2 . The energy E(fi ) is independent of i. When i goes to infinity, fi (z)converges to ∞ if z ̸= 0, but fi does not converge uniformly in the neighbourhoodof 0. The limit is the constant map, so of course extends to the whole space S 2 .
If we consider the energy density |dfj |2 dV as a measure on X in Theorem 1.1, it
converges to∑|df∞ |2 dV +ax δx ,
x∈S

where δx is the Dirac measure at x and ax is a positive constant. That is to say,the energy density becomes concentrated around S and goes to the Dirac measurein the limit. (Remark: We have a similar result when the dimension of X is greaterthan 2. But the singular set S becomes n − 2-dimensional, where n = dim X.)
When the target manifold is isometrically embedded in the Euclidean space, theharmonic map equation can be written as ∆f +Π(df, df ) = 0, where Π is the second
fundamental form. It is a non-linear elliptic PDE of second order and the EulerLagrange equation for the energy functional. When we study analytic aspects of
the harmonic map equation, it is natural to introduce the function space3 W 1,2 .
It is quite essential in the proof of Theorem 1.1, especially for the finiteness of thesingular set, that the p = 2 is the critical exponent in 2-dimension. (Here we say
p is the critical exponent, if any function in the Sobolev space W 1,q is continuous
if q > p, but not continuous when q = p.) It is also important that the harmonic
2 A holomorphic map between Kähler manifolds is harmonic.

In fact, it gives a minimum of

the energy in its homotopy class.
3

def.

∫

W 1,p = {f |∫|f |p dV +X

|df |p dV < ∞}X

3

map equation has the ‘scaling invariance’ in 2-dimension. For an Einstein metricg on a 4-manifold X, the counterpart to the energy for the harmonic map is thesquare of the L2 -norm of the full curvature tensor∫|Rg |2 dVg .
X(In fact, the Einstein metric gives the minimum of this functional defined on theset of metrics on X.) The curvature tensor is given by the second order diﬀerentialof the metric, so one must use W 2,2 as the function space. In 4-dimension, theexponent p = 2 has a similar property as above concerning W 2,q . So it is natural
to expect a similar convergence theorem. Actually we have the following:
Theorem 1.3 [An1, Na1, BaKaNa]. Let {(Xi , gi )} be a sequence of pairs of 4dimensional compact manifolds and Einstein metrics with
Ric gi = kgi , diam(Xi , gi ) ≤ D, vol(Xi , gi ) ≥ V, the Euler number of Xi ≤ R,
where k, D, V , R is constants independent of i. We also assume that k is normalized to be 0 or ±1. Then there exists a subsequence {j} ⊂ {i} with the following
properties:(1) {(Xj , gj )} converges to a compact metric space X∞ in the following sense 4 :If we remove a finite set S = {x1 , . . . , xk } from X∞ , a C ∞ -manifold structure andan Einstein metric g∞ are defined over there. There exists a diﬀeomorphism Fj∞from X∞ \ S into Xj such that Fj∗ gj converges to g∞ in the Cloc
(X∞ \ S)-topology.
(2) The manifold structure and the Einstein metric g∞ on X∞ \ S extends tothe whole X as an orbifold structure and an orbifold Einstein metric.
We mean the orbifold structure and the orbifold metric in the following sense:(a) X∞ \ S is a C ∞ -manifold and the restriction of g∞ is a Riemannian metric.
(b) For each singular point xn , there exists a neighbourhood Nn such thatNn \ {xn } is diﬀeomorphic to B 4 \ {0}/Γ, where B 4 is a 4-dimensional unit balland Γ is a finite subgroup of O(4) acting freely on B 4 \ {0}. And if we pull backthe metric g∞ to B 4 \ {0}, it extends smoothly across the singular point 0. (Γ may
depends on the singular point xn .)As in the case of harmonic maps, we have the removable singularities theorem,
but we omit the statement because the conditions are complicated.
The same result under the further assumption on the lower bound of the injectivity radius was independently obtained by Gao [Ga]. In this case, S is an empty set,
and the situations like example 1.5 given later do not appear under his assumption.
Remark 1.4. Since we do not give the proof of Theorem at all, the reader may
be diﬃcult to understand the meaning of the conditions. The conditions are used
to derive the estimates of the isoperimetric constants, are indispensable to obtain
the a priori estimates for curvatures. From a position of the user of theorem, theconditions are quite natural as:(1) The assumption k to be 0 or ±1 is always satisfied if you multiply the metricby an appropriate positive constant. So this condition is not essential.
4 In fact, the convergence is with respect to the Hausdorﬀ distance introduced by Gromov. We

do not talk about the Hausdorﬀ distance here, but note that the recent development of the theoryof the Hausdorﬀ convergence is quite essential in the proof of Theorem 1.3.

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(2) When k = 1, the condition diam(Xi , gi ) ≤ D follows from Myers’ theorem.
(3) For a 4-manifold admitting an Einstein metric, the Euler number is given by
the (universal) constant multiple of the square integral of the curvature tensor.
(4) The assumption dim Xi = 4 is not come from the fact the dimension5 of
the space-time is equal to 4. It comes from the technical reason. In higher dimension, the author conjectures that the sequence converges outside a singular set ofcodimension 4 if the L2 -norm of the curvature is uniformly bounded.
As in harmonic map case, the measure |Rgj |2 dVgj is converges to|Rg∞ |2 dVg∞ +∑a x δxx∈S

in a certain sense. (Since the spaces are changing, the usual convergence does notmake sense.) The second term will be explained in the next section. Recall thatthe curvature measures how the space curves. As j → ∞, the spaces bend aroundS, and become singularities in the limit. But the singularities are quite simple as
finite quotients of manifolds. So it is nothing fearful !
We now give examples.
Example 1.5 (Page[Pa], R.Kobayashi-Todorov[KorTo]). Let X∞ be an orbifold
given by the Z/2Z-quotient of the complex 2-torus T = C2 /Z4 , where −1 ∈ Z/2Z
acts on T by
(z1 , z2 ) mod Z4 7→ (−z1 , −z2 ) mod Z4 .
The flat metric on T descends to an orbifold metric g∞ on X∞ , and it is an orbifold
Einstein metric. Moreover X∞ has a structure of a complex manifold (with singularities) since the Z/2Z-action is holomorphic. Let us take the minimal resolution
π: X → X∞ . The singularities are sixteen simple singularities of type A1 . The
minimal resolution X is called a Kummer surface, and an example of K3 surfaces.
Let S = {x1 , . . . , x16 } be the singular set, and let E1 , . . . , E16 be the exceptional
sets. These are complex submanifold of X biholomorphic to CP1 with the selfintersection number −2. By the solution of the Calabi conjecture by Yau [Ya] wehave a unique Ricci-flat Kähler metric in each Kähler class6 . Take a Kähler class,so a Ricci-flat Kähler metric gi as follows:(1) The volume of X with respect to gi is equal to 1.
(2) The volume of the exceptional set En is equal to 1/i for n = 1, . . . , 16.
It can be shown that the metric gi converges to π ∗ g∞ over X\∪En , but the condition(2) forces the metric becomes degenerated along En as i → ∞. Since X \ En is
diﬀeomorphic to X∞ \S via the map π, this gives an example of Theorem 1.3. Notethat the limit object is (X∞ , g∞ ), not π ∗ g∞ . The limit metric π ∗ g∞ is degenerate
along En and the distance between two points in En becomes 0. Hence it is morenatural to collapse En to a point ! (See Figure 1.1.)

Figure 1.1. The behaviour of the metrics around the singular point x1
5 I mean the dimension of the space-time in relativity, not in the superstring theory.
6 An Einstein metric with Ric = 0 is called a Ricci-flat metric.

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The moduli space of polarized K3 surfaces (i.e., pairs of K3 surfaces and Kählerclasses with the volume 1) is known to be an open dense subset KΩ ofKΩ = Γ\SOo (3, 19)/SO(2) × SO(19),where Γ is the group of automorphisms preserving the intersection form on H 2 (X; Z).
If we add singular K3 surfaces as above to the moduli space, we get the whole space
KΩ. Thus Theorem 1.3 gives a natural completion of the moduli space (see [KorTo,An2] for detail).
Similarly by using the existence of Einstein-Kähler metric7 obtained by Aubin
and Yau, one can also give examples with k = −1. Consider a deformation of projective surfaces of general type π: X → ∆ such that Xt = π −1 (t) satisfies c1 (Xt ) < 0(namely the canonical bundle is ample) when t ̸= 0, and the central fiber X0 satisfies c1 (X0 ) ≤ 0 (the canonical bundle is nef), but c1 (X0 ) ≮ 0. When t ̸= 0, Xt hasa unique Einstein-Kähler metric, but X0 does not. The limit of the Einstein-Kähler
metric as t → 0 is the orbifold Einstein-Kähler metric on the canonical model ofX0 , that is the surface obtained by collapsing all (−2)-curves in X0 (see Tsuji [Ts]).
When algebro-geometers talk about the “degeneration”, they also consider thecases which are not covered by Theorem 1.3, namely the diameter of (Xi , gi ) may
diverge. (For example, in the above case of K3 surfaces, consider a sequence whichcorresponds a divergent sequence in KΩ.) We do not understand what happens
diﬀerential geometrically when the diameter goes to infinity. (But see the work of
R.Kobayashi [Kor1].) If we consider the moduli space of constant curvature metrics
on Riemann surfaces, the limit space is always smooth under the assumption on
the diameter. The stable curve does not appear ! On the other hand, when k = 1,
i.e., when we consider Einstein-Kähler metrics on del Pezzo surfaces8 , the diametercondition follows from Myers’ theorem as remark above, so the all conditions in
Theorem 1.3 are satisfied. Thus we get the following:
Corollary 1.6. Fix a underlying diﬀerentiable structure X of the del Pezzo surface
X. Let M(X ) be the moduli space of pairs of complex structures with c1 > 0 and
Einstein-Kähler metrics on them. Then adding orbifolds, we can compactify M(X ).
Consider only the complex structures, we have a natural map Φ from M(X ) to
the moduli space of complex structures H(X ). By Bando-Mabuchi’s uniquenesstheorem of Einstein-Kähler metric [BaMa], the map Φ is injective. In fact, Tian
proved the following:
Theorem 1.7 [Ti]. If the automorphism group of the del Pezzo surface X is reductive 9 , X has an Einstein-Kähler metric. In other words, the map Φ is bijective
unless X is either one or two points blowing up of CP2 .
The proof is given by showing that the image of Φ is nonempty, both open andclosed in H(X ). It is not so diﬃcult to show the openness. The closedness is
7 We say an Einstein-Kähler metric the Einstein metric which is Kählerian at the same time.

Then the first Chern class c1 of the manifold is given by k[ω] ([ω] denotes the Kähler class). In
particular, if the complex manifold has an Einstein-Kähler metric, it holds either c1 > 0, = 0, or< 0.
8 A complex surface X with c (X) > 0 is called a del Pezzo surface. It is known that such a X
1is either CP2 , CP1 × CP1 , or blowing up of CP2 at r generic points (1 ≤ r ≤ 8).
9 Matsushima’s theorem says if a del Pezzo surface admits an Einstein-Kähler metric, its automorphism group is reductive.

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shown by using Theorem 1.3. Suppose that (Xi , gi ) is a sequence in M(X ) andΦ(Xi , gi ) ∈ H(X ) converges. We want to show that (Xi , gi ) also converges and itholds
lim Φ(Xi , gi ) = Φ( lim (Xi , gi )).
i→∞

i→∞

The diﬃculty relates to the Hausdorﬀ-ness of the moduli space H(X ). If the jumping of the complex structure would occur, one cannot expect the above equality.
Moreover the following problem is still open.
Problem 1.8. When does a del Pezzo surface with quotient singularities have an
Einstein-Kähler orbifold metric and lies in the boundary of the compactification of
the moduli space M(X ) given in Theorem 1.3 ?
The problem should relate to the Chow stability, but things are not clear yet.10
2. Bubbling out of ALE spaces
In order to study the convergence in more detail, we “blow up” the metrics
around the singular points by rescaling. This is a standard technique widely usedin problems of weak convergence. As before, first study the case of harmonic maps.
Let fj be a sequence of harmonic maps as in Theorem 1.1, and suppose thatfj does not converges at x, i.e., x ∈ S. Then |dfj (x)| goes to the infinity as
j → ∞. Take a normal coordinate system (x1 , x2 ) defined on a neighbourhood
U of x. Let xj be a point at which |dfj | attains the maximum in U . We may
assume that {xj } converges to x. Define a new coordinate system by (y1 , y2 ) =(|dfj (xj )|x1 , |dfj (xj )|x2 ) and change the metric as |dfj (xj )|2 g. Remark that the
maximum of the energy density of fj in U is normalized to be 1 with respect tothe new metric. The neighbourhood {(x1 , x2 ) | |x1 |2 + |x2 |2 < δ} of x is written
as {(y1 , y2 ) | |y1 |2 + |y2 |2 < |dfj (xj )|2 δ}, in the new coordinates, and converges to{(y1 , y2 ) ∈ R2 } as j → ∞. The rescaled metrics converge to the standard metricon R2 . We regard the map fj defined on the y-plane, and denote by fej . Then theenergy of the fej is bounded from the above independent of j. Then one can apply
Theorem 1.1 to fej . In this case, we have a uniform bound on the energy density,
so the concentration of the energy does not happen and we get the following:
Theorem 2.1 [SaUh]. There exists a subsequence of {fej }, also denoted by {fej },which converges to a harmonic map fe∞ defined on the whole plane R2 with finite
energy.
Noticing that R2 and S 2 \ {p} are conformal to each other, and that the energy
and the harmonicity are preserved under a conformal transformation, we can regard
fe∞ as a finite energy harmonic map defined on S 2 \ {p}. Then by the removablesingularities theorem, fe∞ can be extended to the whole S 2 . Also note that fe∞
depends on x ∈ S.
Let use the same technique in the case of Einstein metrics. Let {(Xj , gj )} bea sequence of Einstein metrics as in Theorem 1.3, and take x ∈ S. The absolutevalue of the curvature |Rgj | diverges to infinity at x. Let rj be the value of |Rgj |at x. Since x is a point in X, not in Xj , we cannot talk about |Rgj | at x. Precisely
10 There are progress to this problem after writing the original manuscript. See the references

added in translation.

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speaking, we mean |Rgj (xj )|, where xj is a point in Xj and {xj } converges to x ∈ X
in a certain sense (we do not give the precise meaning here to avoid the technicalcomplexity). Now we multiply the metric gj by rj . Then the absolute value ofthe curvature of the new metric rj gj is normalized to be 1. This rescaling procedure means “viewing by the microscope”. Since the curvatures become uniformly
bounded, we can expect the convergence of the manifolds. But the diameter goesto infinity, so the limit space becomes a noncompact manifold. In fact, we have thefollowing:
Theorem 2.2 [Na1]. Consider the sequence {(Xj , rj gj , xj )} of the pairs of Riemannian manifolds and points on it. There exists a subsequence, also denoted by{(Xj , rj gj , xj )}, which converges 11 to a complete Ricci-flat manifold (M, h, o) having the∫following properties.
|Rh |2 dVh < ∞,(1)M(2) vol B(o; r) ≥ V r4 for some o ∈ M , V > 0,
where B(o; r) denotes the ball of radius r centered at o. More precisely, there exists
a diﬀeomorphism Gj from M to a neighbourhood of xj such that G∗j (rj gj ) converges∞
to h in Cloc
(M ).
We call these phenomena, such as a map from S 2 being cut oﬀ from sequences ofharmonic maps, a part of the manifold being torn oﬀ in our case, “bubbling out”.
In the case of harmonic maps, one can regard the limit map fe∞ as a map from2
S through the conformal compactification of R2 . The Einstein equation is notpreserved under the conformal transformation, so one cannot apply the removablesingularities theorem in the same way. However one can modify its proof to thissituation to show the following (in fact, the proof becomes more diﬃcult).
Theorem 2.3 [BaKaNa]. If a 4-dimensional complete Ricci-flat manifold satisfies
the conditions (1) and (2) in Theorem 2.2, then it is ALE of order 4.
We say a 4-dimensional Riemannian manifold to be ALE of order τ if there exists
a compact set K ⊂ X and a finite subgroup Γ ⊂ SO(4) and a diﬀeomorphism(coordinates at infinity) X: X \ K → (R4 \ BR )/Γ such that the following holds inthe coordinates X:
p timesz }| {|∂ · · · ∂(gij (x) − δij )| = O(|x|−p−τ )for x ∈ (R4 \ BR )/Γ, p = 0, 1, 2, 3, · · · .

In the orbifold, the distance sphere around the singular point of suﬃciently small
radius is diﬀeomorphic to S 3 /Γ, and the distance sphere of suﬃciently large radius
is diﬀeomorphic to S 3 /Γ in the ALE space. Thus one can understand the ALE
space is the counterpart to the orbifold in the category of noncompact manifolds.
Remark 2.4. One cannot drop the condition (2) in Theorem 2.3. There exists a
Ricci-flat Kähler metric, called the Taub-NUT metric, defined on C2 . This metrichas an asymptotic behaviour called ALF, and the volume of ball of radius r grows
as O(r3 ) (see [EGH]). We do not have examples of Ricci-flat manifolds satisfying
the condition (2) but not satisfying (1).
11 with respect to the pointed Hausdorﬀ convergence8See what happens in Example 1.5. Take a point yn in an exceptional set En in
X which corresponds to a singular point xn in X∞ . If the value of |Rgi | at yn isequal to ri , (X, ri gi ) converges to an ALE Ricci-flat Kähler manifold (M, h). This
M is called the Eguchi-Hanson space and biholomorphic to a cotangent bundle
T ∗ CP1 of the complex projective line. (T ∗ CP1 is the minimal resolution of thesimple singularity of type A1 . The exceptional set is the zero section.) In this casethe limit space is independent of n. In general, when K3 surfaces degenerate to an
orbifold with simple singularities, one can obtain a Ricci-flat Kähler metric on theminimal resolution of each singularity in the same way (see [Kor2]).
In Theorem 2.2, 2.3, we rescale the metric so that the curvatures are uniformly
bounded. We lose some informations and need more detailed studies in general.
This is because several ALE spaces may blow up from one singular point x ∈ S ⊂X∞ . An accurate version of the convergence theorem which includes the above
situations was obtained by Bando [Ba]12 . In order to explain his result, we needsome terminology and symbols. We say an orbifold metric is ALE when there isno singular set outside a compact set, and there exists a coordinate system X at
infinity as usual ALE Riemannian manifolds. For such an ALE orbifold (M, h), we
denote by S(M ) the set of singular points. By definition, S(M ) is a finite set. We
denote by xnj the point in Xj which converges to the singular point xn in X∞ inthe same sense as before. (See the explanation above Theorem 2.2).
We see a neighbourhood of xnj by a microscope. In this case, we set the magnification rj so that∫
|Rgj |2 dVgj = ε,n √1 )
B(xnj ;R)\B(xj ;rjwhere ε, and R is a (suﬃciently small) positive constant. Here B(x; r) denotes themetric ball of radius r centered at x. The similar convergence theorem as 1.3 canbe applied to the magnified Riemannian manifold (Xj , hj ) = (Xj , rj gj ). Since we
have
∫
√
n
B(xn
j ; rj R)\B(xj ;1)|Rhj |2 dVhj = ε√
in this case, the concentration of the curvatures does not happen in B(xnj ; rj R) \B(xnj ; 1), hence the limit (M, h) of (Xj , hj ) is an ALE orbifold which does nothave singularities outside the ball of radius 1. If we want to know the situationof the convergence, we apply the same argument to (Xj , hj ) in stead of (Xj , gj ).
Namely, we multiply hj again around a singular point of (M, h). This means themagnification is reset bigger. We make the magnification bigger as a singular point
occur in the limit ALE Ricci-flat orbifold in this way, then finally no singular points
appear and the procedure ends in the finite step. We adjust the following form.
The statement becomes complicated, so we give the figure 2.1 which describe thecircumstances.

Figure 2.1

12 This paper was dedicated to the late cartoonist Osamu Tezuka.

9Theorem 2.5 [Ba]. (1) In the same notation as in Theorem 1.3, (taking a further
subsequence {j} ⊂ {i} if necessary), there exist a family of pointed ALE Ricciflat orbifolds and sequences of positive numbers which has a recursive relation as
follows:
1 There exist a pair (M n , hn , on ) of an ALE Ricci-flat orbifold and a point
corresponding to each singular point xn ∈ S in X∞ (n = 1, 2, · · · , #S) and adivergent sequence {rjn }j=1,2,··· of positive numbers such thatlim (Xj , rjn gj , xnj ) = (M n , hn , on ).

j→∞

Here lim is the convergence in the sense as in Theorem 2.2. Moreover the fundamental group of the end of M n and the group corresponding to the singular point
xn are isomorphic and their actions on R4 are the same.
k − 1 ⇒ k Suppose that ALE Ricci-flat orbifold M n1 ,n2 ,··· ,nk−1 is defined andhas a singular point y n1 ,n2 ,··· ,nk ∈ S(M n1 ,n2 ,··· ,nk−1 ) (nk = 1, 2, · · · , #S(M n1 ,n2 ,··· ,nk−1 )).
There exist a pair (M n1 ,n2 ,··· ,nk , hn1 ,n2 ,··· ,nk , on1 ,n2 ,··· ,nk ) of an ALE Ricci-flat orbifold and a point, a sequence {rjn1 ,n2 ,··· ,nk }j=1,2,··· of positive numbers, and a sequence {xnj 1 ,n2 ,··· ,nk }j=1,2,··· of points in Xj such thatlim (Xj , rjn1 ,n2 ,··· ,nk gj , xnj 1 ,n2 ,··· ,nk ) = (M n1 ,n2 ,··· ,nk , hn1 ,n2 ,··· ,nk , on1 ,n2 ,··· ,nk ),j→∞rjn1 ,n2 ,··· ,nk

lim n ,n ,··· ,nk−1
j→∞ r 1 2j

= ∞.

Moreover the fundamental group of the end of M n1 ,n2 ,··· ,nk and the group corresponding to the singular point y n1 ,n2 ,··· ,nk are isomorphic and their actions on R4
are the same.
(2) This recursive procedure ends in a finite step, namely the ALE orbifold
n1 ,n2 ,··· ,nkMbecomes nonsingular for some k. Moreover for the suﬃciently large jand suﬃciently small R, the ball B(Xjn ; R) is diﬀeomorphic to a manifold which is
obtained from M n1 ,n2 ,··· ,nk \ S(M n1 ,n2 ,··· ,nk )’s with n1 = n by attaching a neighbourhood of y n1 ,n2 ,··· ,nk and the end of M n1 ,n2 ,··· ,nk .
(3) The measure |Rgj |2 dVgj converges to|Rg∞ | dVg∞ +2#S∑an δxn ,n=1

where an is given by
an =

∑ ∫
(M,h)

|Rh |2 dVh .
M

Here the summation runs over the set of ALE orbifolds (M, h) = (M n1 ,n2 ,··· ,nk , hn1 ,n2 ,··· ,nk )with n1 = n.
For a compact Einstein 4-manifold, the L2 -norm of the curvature gives its Euler
number. For an ALE Einstein manifold (M, h) we have∫1
1
|Rh |2 dVh = the Euler number of M −.
2
8π M
the order of Γ
(There are similar formulae for compact orbifolds and ALE orbifold.) We have thefollowing. (For the definition of hyper-Kähler structure, see Sect. 3.)
10Proposition 2.6. Under the same situation as in Theorem 1.3, suppose that
(Xi , gi ) has a Kähler (resp. hyper-Kähler) structure. Then the limit space (X∞ , g∞ )and the space (M n1 ,n2 ,··· ,nk , hn1 ,n2 ,··· ,nk ) appeared in Theorem 2.5 also have a Kähler(resp. hyper-Kähler) structure in the sense of orbifolds.
3. ALE spaces with hyper-Kähler structures
As we said in the previous section, the ALE spaces bubble out when Einstein
metrics degenerate to an orbifold metric. In this way, one obtain existence of ALE
spaces. In this section, we shall give a more concrete way to give ALE spaces
with hyper-Kähler structures. This is a result by Kronheimer [Kr1, 2]. His result
is closely related to the theory of simple singularities. A simple singularity is aquotient space C2 /Γ, where Γ is a finite subgroup of SU(2), and has been studied
for a long time. The classification is known, and it can be realized as a complex
hypersurface in C3 . The dual graph of the exceptional set of its minimal resolution isa Dynkin graph with no double edges (i.e., of type A, D, E). (See the table below.)
By the theory of Brieskorn [Br] and Slodowy [Sl], one can construct the simple
singularity using nilpotent varieties of the corresponding simple Lie algebra. One
can also describe the semi-universal deformation and its simultaneous resolution
in terms of the Lie algebra. Kronheimer succeeded to construct these varieties
using so called quivers. Moreover his construction also showed that each fiber hasa hyper-Kähler structure.
rootsystemgroup

Dynkin graphhypersurfaceAnthe cyclic group of order n + 1

xn+1 + yz = 0Dn

the binary dihedral groupof order 4(n − 2)E6the binary tetrahedral groupE7

the binary octahedral groupE8

the binary icosahedral group◦−◦−◦−· · ·−◦◦◦−◦−· · ·−◦⧸⧹◦◦|◦−◦−◦−◦−◦◦|◦−◦−◦−◦−◦−◦◦|◦−◦−◦−◦−◦−◦−◦

x2 + y 2 z + z n−1 = 0
x2 + y 3 + z 4 = 0
x2 + y 3 + yz 3 = 0
x2 + y 3 + z 5 = 0

A hyper-Kähler structure on a Riemannian manifold (M, g) is, by definition, a
triple of almost complex structures (I, J, K) which satisfies a quaternion relation
IJ = −JI = K and is parallel with respect to the Levi-Civita connection of g,i.e., ∇I = ∇J = ∇K = 0. We call a Riemannian manifold with a hyper-Kähler
structure simply the hyper-Kähler manifold. The holonomy group of a hyperKähler manifold is contained in Sp(n), where n = 14 dim M . Each almost complex
structure I, J, or K defines a Kähler structure on (M, g). When the manifold
is 4-dimensional and simply-connected, a hyper-Kähler structure is nothing but a
Ricci-flat Kähler structure thanks to the isomorphism Sp(1) = SU(2). For example,
a Ricci-flat Kähler metric on a K3 surface gives us a hyper-Kähler structure. Taking
a particular almost complex structure I and regarding (M, g) as a Kähler manifold,
we have a nowhere vanishing closed (2, 0)-form ωJ +iωK , where ωJ (resp. ωK ) is theKähler form associated with J (resp. K). Namely we have a holomorphic symplectic
form. Conversely if a compact Kähler manifold has a holomorphic symplectic form,
11there exists a Ricci-flat Kähler metric by the solution of Calabi conjecture, andhence there exists a hyper-Kähler structure.
On the other hand, Hitchin et al. [HKLR] introduced a notion of hyper-Kähler
quotients which is an analogue of Marsden-Weinstein quotients for symplectic manifolds, therefore gave a diﬀerent method to construct hyper-Kähler manifolds. Recall
their result.
Suppose that a Lie group G acts on a hyper-Kähler manifold (M, g, I, J, K)
preserving the hyper-Kähler structure. Let denote the Lie algebra of G by g, andits dual space by g∗ . G acts on g∗ by the coadjoint action. A G-equivariant mapµ = (µI , µJ , µK ): M → R3 ⊗ g∗ is said to be a hyper-Kähler moment map if wehave
I grad⟨µI , ξ⟩ = J grad⟨µJ , ξ⟩ = K grad⟨µK , ξ⟩ = ξ ∗for any ξ ∈ g. Here ⟨ , ⟩ is a natural pairing between g and g∗ , and ξ ∗ is a vector
field generated by ξ. Let define
Z = {z ∈ g∗ | Ad∗g (z) = z for any g ∈ G.}
Then µ−1 (ζ) is invariant under the G-action for ζ ∈ R3 ⊗ Z. So we can consider
the quotient space Xζ = µ−1 (ζ)/G.
Theorem 3.1 [HKLR]. Suppose that the G-action on µ−1 (ζ) is free. Then thequotient space Xζ is a C ∞ manifold and has a Riemannian metric and a hyperKähler structure induced from those on M .
As an application of the above hyper-Kähler quotient construction, we give ALE
spaces. ∑Fix a root system of type A, D, or E. Let θ1 , . . . , θr be its simple root. Let
θ0 = − i ni θi be the negative of the highest root. Set n0 = 1 for convenience.
Draw the extended Dynkin diagram, and put a complex vector space of dimension
ni on each vertex i. Consider linear maps fij : Cnj → Cni and fji : Cni → Cnj whenthe vertices i and j are joined by an edge. See the figure for D5 .
Figure 3.1
Let M be the linear space of all such linear maps fij . Put a hermitian metricon each Cni and also the induced metric on M . To make M a quaternion vector
space, we introduce J as follows. For joined vertices i and j, we have two linear
maps from Cni to Cnj and Cnj to Cni . Choose one of them and fix henceforth.
When i → j is chosen, we define J by∗∗
J(fij , fji ) = (−fji
, fij
),

where ∗ means the adjoint. Then J satisfies J 2 = −1 and is anti-linear with respect
to the complex structure. Hence M has a structure ofvector space.
∏ quaternion
ni 13
There is a natural action on M of a Lie group G = i̸=0 U(C ) This action
preserves the metric and quaternion linear. Let µ be the hyper-Kähler momentmap vanishing at the origin, which exists uniquely. Let Z be as above, and takeζ ∈ R3 ⊗ Z. Then the space Xζ = µ−1 (ζ)/G has a hyper-Kähler structure on its
nonsingular part. More precisely Kronheimer showed the following:
13 The action of∏U(Cni ) is not appropriate, since c Id ∈12∏

U(Cni ) fixes any element in M .

Theorem 3.2 [Kr1]. For a generic ζ, the G-action on µ−1 (ζ) is free and thequotient space Xζ is a nonsingular 4-dimensional hyper-Kähler manifold. MoreoverXζ is diﬀeomorphic to a minimal resolution of C2 /Γ and the metric is ALE.
He had a precise description of Xζ even for non generic ζ, for example, Xζ =C /Γ if ζ = 0. Write ζ = (ζI , ζJ , ζK ), and consider the following family of spaces
with ζI = 0∪
X(0,ζJ ,ζK )2

ζJ +iζK ∈C⊗Z

and the natural projection to C ⊗ Z. Then it is a semi-universal deformation of
C2 /Γ pulled back to the Weyl group covering. For generic ζI , one can define aholomorphic map
X(ζI ,ζJ ,ζK ) → X(0,ζJ ,ζK ) ,which is a simultaneous resolution of singularities. When a sequence of genericparameters converges to a non generic parameter, corresponding Einstein metrics
converges to an ALE orbifold as in §§1,2.
The geometric meaning of the parameter ζ can be explained in terms of period
maps. The exceptional set of the resolution of C2 /Γ is a union of CP1 ’s and theirintersection matrix is −1 times Cartan matrix of the corresponding Lie algebra.
Then for generic ζ, one can identify H 2 (Xζ ; R) with the Cartan subalgebra h, andan element Σ in H2 (Xζ ; Z) with Σ·Σ = −2 corresponds to a root. The set (R3 ⊗Z)◦of generic ζ is simply-connected, so the local system H 2 (Xζ ; R) can be trivialized.
So taking cohomology classes of three Kähler forms ωI , ωJ , ωK on Xζ , we candefine a period map
P : (R3 ⊗ Z)◦ → R3 ⊗ h.
Then Kronheimer showed that P is induced from an isomorphism between Z andh, and ζ ∈ R3 ⊗ Z is generic (i.e., Xζ is smooth) if and only if
P (ζ) ∈/∪

R3 ⊗ πθ ,θwhere πθ is a hypersurface defined by the root θ.
He also showed the following classification theorem:
Theorem 3.3 [Kr2]. An ALE spaces with a hyper-Kähler structure is isomorphic
to the one constructed by the above construction.
As above, the ALE space is well-understood when it has a hyper-Kähler structure. But it seems also important to study the spaces with Kähler structures in
conjunction with the algebraic geometry. In this case the ALE space is a cyclic
quotient of the ALE space with hyper-Kähler structure [Ba]. Using the above
classification theorem and an argument in [Ti], we can show the following:
Theorem 3.4. Let (X, g) be a nonsingular ALE Ricci-flat Kähler 4-manifold.
Then the one of the followings holds.
(1) (X, g) has a hyper-Kähler structure compatible with the Kähler structure.
e of type An by the(2) (X, g) is a quotient of an hyper-Kählerian ALE space X
cyclic group Z/rZ. And n + 1 is a multiple of r.
13

As we saw in §§1,2, an ALE space with a Kähler structure bubbles out when
Einstein-Kähler metrics converges to an orbifold. For example, consider a deformation of algebraic surfaces π: X → ∆ and suppose that Xt = π −1 (t) has an
Einstein-Kähler metric gt when t ̸= 0. If the diameter of (Xt , gt ) is estimated fromthe above independent of t, the convergence theorem gives us an Einstein-Kähler
orbifold as a limit. The above theorem together with the relation between the singularity and the bubbling oﬀ ALE spaces, the singularity of the orbifold must be asimple singularity or a cyclic quotient of a simple singularity of type An . Strikingly,
the similar result is obtained by using the minimal model theory (see Kawamata[Ka]). If the total space X has only terminal singularities and the central fiber X0
has only orbifold singularities, then the singularities in the central fiber are of typejust mentioned.
At the last, we give the following problem:
Problem 3.5. Are there ALE Ricci-flat 4-manifolds without Kähler structures ?
We do not have such examples at this moment.
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Translated by HIRAKU NAKAJIMA

Aramaki, Aoba-ku, Sendai 980, Japan
E-mail address: nakajima@math.tohoku.ac.jp

16