ALE Ricciflat Kähler metrics and deformations of quotient surface singularities
Abstract.
Let be an isolated quotient singularity with a finite subgroup. We show that for any Gorenstein smoothings of a nearby fiber admits ALE Ricciflat Kähler metrics in any Kähler class. Moreover, we generalize Kronheimer’s results on hyperkähler manifolds [14], by giving an explicit classification of the ALE Ricciflat Kähler surfaces.
We construct ALF Ricciflat Kähler metrics on the above nonsimply connected manifolds. These provide new examples of ALF Ricciflat Kähler manifolds, with cubic volume growth and cyclic fundamental group at infinity.
Contents
1. Introduction
Orbifold Kähler spaces appear naturally in the study of the moduli space of metrics. In this paper we consider quotient orbifolds of the form with a finite subgroup acting freely on There are two techniques to associate a smooth manifold to a singular space: one is resolving the singularity by introducing an exceptional divisor and the other is deforming the orbifold to a smooth manifold.
If we consider the resolution of the singularity, then there are two cases which behave quite differently. The first case is when is a subgroup of In this situation the singularities are called rational double points, and they are classified to be of type corresponding to the tree configuration of selfintersection rational curves which appear in the minimal resolution. We call the underlying differential manifold of the minimal resolution an or manifold. These manifolds are shown to support a hyperkähler structure. They were extensively studied and classified by EguchiHanson [7], Hitchin [9], GibbonsHawking [8], Kronheimer [13, 14], Joyce [11] and others. They have proved the existence of asymptotically locally Euclidean (ALE) Ricciflat Kähler metrics on these manifolds. In the second case, when is a subgroup of rather than the minimal resolution has nontrivial canonical line bundle, and hence it does not admit Ricciflat Kähler metrics. In the special case when the group is cyclic Calderbank and Singer [5] showed that the minimal resolution admits scalar flat curvature Kähler metrics, which are ALE. To the author’s knowledge, little is known for other groups.
The second method of obtaining a smooth manifold is by deforming the singularity. There are again two cases which arise. The first situation, when the subgroup is a subgroup of yields in fact the hyperkähler spaces, but endowed with a different complex structure. Only for a subgroup of the resolution and the smoothed manifolds are diffeomorphic. More details on the results on hyperkähler manifolds will be described in sections 3 and 4. For a subgroup of some power of the canonical line bundle of the smoothing of the singularity is trivial and we can prove that the manifold admits a best metric which is ALE Ricciflat Kähler:
Theorem A.
Let be an isolated quotient singularity of complex dimension Assume there exists a oneparameter Gorenstein smoothing where is a disk around the origin and Then for arbitrary small the manifold admits an ALE Ricciflat Kähler metric . More precisely, in any Kähler class of there exists a unique ALE Ricciflat Kähler metric.
We recall that a normal complex space is Gorenstein if it is CohenMacaulay and a multiple of the canonical divisor is Cartier [12, 3.1].
In the case of a resolution, we know that the complex structure at infinity is the canonical one. When we consider deformations we need to give a more precise definition of what an ALE Kähler manifold is:
Definition 1.1.
Let be a finite subgroup of acting freely on and let be the Euclidean metric and the canonical complex structure on We say that a Kähler manifold is an ALE Kähler manifold asymptotically to if there exists a compact subset and a map which is a diffeomorphism between and the subset for some fixed such that and .
In the study of hyperkähler 4manifolds, a key ingredient used both in Hitchin’s ([9]) and Kronheimer’s ([14]) proofs was the fact that the complex surface is birational to a deformation of a rational double point singularity. We obtain a similar result in the Ricciflat case:
Theorem B.
Let be a smooth ALE Ricciflat Kähler surface, asymptotic to with a finite subgroup of acting freely on . The complex manifold can be obtained as the minimal resolution of a fiber of a oneparameter Gorenstein deformation of the quotient singularity Given any Kähler class , then is the unique ALE Ricciflat Kähler metric in that class. In particular, if is not simply connected then its fundamental group is finite and cyclic, its universal cover is an type manifold, and the decktransformations are explicitly described.
Remark 1.2.
If is simply connected then is hyperkähler and the theorem is in fact a reformulation of Kronheimer’s classification result. In the case when is non simply connected, then its universal cover can not be a or type manifold.
Remark 1.3.
The ALE is a necessary condition in the above classification and it is essential for the proof of the theorem B.
In section 5, we show that the above nonsimply connected manifolds also admit complete asymptotically locally flat (ALF) Ricciflat Kähler metrics and we also give examples of nonsimply connected manifolds with infinite topology which have complete Ricciflat Kähler metrics, by emulating a construction of Anderson, Kronheimer and LeBrun [1].
If one would like to generalize the dimensional results to higher dimensions, then there are several approaches which are used for obtaining good metrics on resolutions of singularities. There is no analog of the deformation part, as Schlessinger Rigidity Theorem says that quotient singularities with singularity of codimension three or more have no nontrivial deformations [18].
2. Smoothings of quotient singularities
In this section we give a short summary of the Kollár and ShepherdBarron’s results [12] on the classification of isolated quotient singularities which admit a smoothing.
Definition 2.1.
A flat, surjective map where is an open neighborhood of is called a oneparameter Gorenstein smoothing of a normal variety if and the following conditions are satisfied.

is Gorenstein

is smooth for every
For a definition of a Gorenstein variety and more details on the algebraic geometry aspects see [12, Def. 3.1]. In their paper, Kollár and ShepherdBarron prove the following classification theorem:
Theorem 2.2.
[12, 3.10] An isolated quotient surface singularity which admits a oneparameter Gorenstein smoothing is either a rational double point or a cyclic singularity of type for and relatively prime.
From a topological point of view the two types of singularities are distinguished by the fundamental group of a smoothing. The smoothing of a rational double point is simply connected, while the deformations of the second type have finite cyclic fundamental group. On the algebraic geometric side of the picture this is reflected on the canonical line bundle being trivial, or nontrivial torsion, respectively.
As the rational double point singularities and their preferred metrics are well understood [9, 13], we focus our attention on the second type of orbifolds. A singularity of type is a quotient singularity where is the finite cyclic group of order which acts on diagonally with weights i.e. if and then If we consider the subgroup of order then its quotient singularity is of type which is the rational double point singularity. The quotient singularity can be biholomorphically embedded as a hypersurface in by the following map: On there is an induced action of the quotient group As this group is cyclic of order , we can represent its action by the action of a unit complex number of order as follows:
The deformations of type singularities are known. They are given by where The action of the group can be extended trivially on the coordinates of
Then, there is a special family of deformations which is invariant and it is given by the equation
where are linear coordinates on Let and be the quotient map of the projection
Proposition 2.3.
[12] The map is a Gorenstein deformation of the cyclic singularity of type Moreover, every oneparameter Gorenstein deformation of a singularity of type is isomorphic to the pullback through of some germ of a holomorphic map
If we consider the case of oneparameter Gorenstein deformations which are equivariant under some free action of the cyclic group of order , and such that the fibers have isolated singularities, then we can ask if we can obtain a similar classification result. This situation was studied by Manetti in [15] by using Catanese’s explicit classification [6] of the automorphism group of rational double point singularities. He proves the following:
Proposition 2.4.
Let be a rational double point singularity and let be an action of the finite cyclic group of order on coming from a subgroup of Moreover, there exists a oneparameter deformation of which is invariant, where the action extends trivially to the coordinate. Assume further that the acts freely on Then the singularity is a singularity of the form where are relatively prime integers, and the deformation family is of the form described in Proposition 2.3.
3. ALE Ricciflat Kähler metrics via the twistor space
On the minimal resolution of a rational double point singularity, ALE hyperkähler metrics (in particular Ricciflat Kähler metrics) where constructed by Hitchin [9], Kronheimer [13, 14], and others. In this section we show how their results can be rewritten in the setup of Theorem B and we analyze in detail the deformations of the second type of singularities from Theorem 2.2. As we observed in Proposition 2.3, these singularities and their deformations are closely related to the type singularities. We start first by reviewing Hitchin’s construction of hyperkähler metrics on deformations of these orbifolds. This is done via the twistor space construction.
The Penrose twistor correspondence [2] is a onetoone correspondence between halfconformally flat manifolds and complex folds, the twistor spaces, which satisfy certain properties. We will give a short summary of the twistor space of a hyperkähler manifold , by following Hitchin’s exposition in [9]. The hyperkähler metrics are antiselfdual if is endowed with the orientation induced by any of the complex structures. In our particular case, we have the following version of the Penrose correspondence between a simply connected manifold endowed with a hyperkähler metric and the complex fold which has the following properties:

is the total space of a holomorphic fibration

There exists a parameter family of sections of each with normal bundle

There exists a nonvanishing holomorphic section of .

There exists a real structure, i.e. a free antiholomorphic involution, on which is the extension of the antipodal map on , such that and are real, and is fibered by the real sections of the family.
The space of real sections recovers the manifold and the conformal class of the metric. We use property to fix a metric in the conformal class. Property is specific for hyperkähler manifolds. For each the projection from to the space of real sections identifies with which induces a complex structure on such that the metric is Kähler.
We give now a short account of the twistor space associated to a smoothing of an type singularity with a Ricciflat Kähler metric, by following Hitchin’s presentation in [9]. A smoothing of an singularity is biholomorphic to for some constants In order to construct we need a family of complex structures on , parametrized by Hitchin [9] considers the manifold
(3.1) 
as a first approximation, where is an affine coordinate and are holomorphic functions. As we need to define a family of complex structures, we need to consider the extension of the coordinate to homogeneous coordinates on then are going to be extended to sections of holomorphic line bundles on while are local coordinates on certain line bundles. Using the properties Hitchin shows that need to be local coordinates on , respectively, while each is a section of Hence is a hypersurface in the vector bundle
The hypersurface has at most rational double point singularities, which can be resolved [9] to obtain a manifold which is the twistor space of . Moreover, all the structures defined on (properties 14) lift to corresponding structures on
On Hitchin constructs a natural real structure induced from a real structure on (property , and a nonvanishing holomorphic section (property 3). The last ingredient missing is a parameter family of sections. First, we make the assumption, [9], that the equation 3.1 factorizes as:
(3.2) 
where As a function on the affine parameter the polynomials can be written as Let be a section of and let If we require that and are invariant, then we need to consider satisfying , or equivalently and We can solve the equation 3.2 by a simple factorization:
where is the discriminant of and are constant coefficients which must satisfy The reality condition on the polynomials and implies that
(3.3) 
By Penrose correspondence the subspace of real sections , can be identified with endowed with a conformal structure. As we saw, the real sections are parametrized by and the angular coordinate associated to . Moreover, each conformal class is determined by a choice of the real quadratic polynomials , or equivalently points in On this real submanifold, the conformal structure is given by [9, eq.4.4]:
(3.4) 
where
To recover the metric on , first we assume that is identified with the smooth hypersurface corresponding to i.e. where for We have local coordinates whenever In terms of these local coordinates we have the following identifications:
(3.5) 
In these local coordinates, the metric corresponding to the twistor space data is the following [9]:
(3.6) 
where
and is defined implicitly by
(3.7) 
An easy computation shows that the metric extends smoothly at and that the above metric is asymptotically locally Euclidean and complete.
The Kähler form associated to the metric is given by:
Moreover, Hitchin shows that we can choose a distinguished basis , of and then the Kähler class of is given by the Poincaré dual of .
The condition that for is equivalent to the fact the complex surface is smooth. If this condition is not satisfied then both and have singularities above , which Hitchin shows that can be resolved by introducing exceptional divisors. If the corresponding ’s are distinct then there are corresponding second homology classes which can be represented by exceptional rational curves. Equal corresponding ’s will yield an orbifold ALE Ricciflat Kähler space
We conclude that the metric is uniquely determined by the complex constants which are given by the complex structure and the real numbers which determine the Kähler class of The Euclidean translation along the axis induces equivalent metrics. We have the following reformulation of Hitchin’s result:
Theorem 3.1 (Hitchin [9]).
Let be a quotient singularity of type Let be the fiber of a Gorenstein deformation of and its minimal resolution. For any such and any Kähler class there exists a unique ALE Ricciflat Kähler metric such that its Kähler form is in the class
Remark 3.2.
We return to the case when and consider the singular manifold and a smoothing of the form This deformation is invariant, where the group acts by:
(3.8) 
We would like to find which of the Ricciflat Kähler metrics constructed above on are also invariant, and for this we need to extend the action to the twistor space associated to
First, we study the special case of and later discuss the situation of arbitrary For after a change in coordinates, we can assume to be given by the equation This equation can be factored as where Hence, if we are looking for the twistor space associated to where the fixed complex structure is obtained at , then we are looking for a twistor space which has a first approximation defined by where Next we need to extend the action 3.8 to such that is invariant. Let denote the action on the affine coordinate. The equation becomes under the action of
or equivalently
Requiring to be invariant implies which gives the action on and also As and are relatively prime, we have that and we can assume This fixes the position of the points on as vertices of a regular polygon with sides, in the plane centered at the origin. Moreover, using the identification 3.5 of the local coordinates, we can identify the exact action on as follows:
(3.9) 
To identify the action on the coordinate, we only need to observe that the defining implicit equation 3.7 is invariant under the action, and as is a real number, the action is trivial.
An easy computation shows that is invariant under the action, while Hence is invariant under the free action of and induces a smooth, complete ALE Ricciflat Kähler metric on The quotient metric is locally hyperkähler, but not globally hyperkähler, as is a deformation of the manifold where but is not a subgroup of and is the canonical quotient metric on In terms of our construction of twistor spaces, this translates in the fact that the group acts nontrivially on (the complex structures on ) as a rotation by a factor of fixing the initial complex structure as well as
In the case the quotient manifold has Euler characteristic equal to is a rational homology ball with infinity asymptotic to Moreover, proposition 2.3 tells us that there exists a unique smoothing up to biholomorphisms. We have showed: and
Proposition 3.3.
The rational homology ball with fundamental group and infinity of the form constructed above admits a unique ALE Ricciflat Kähler metric.
Given an arbitrary for the defining polynomial of we can consider the following factorization where and has the smallest possible argument. As we assume that is smooth, then the sets and contain distinct numbers. The defining polynomial decomposes furthermore as:
(3.10) 
In particular, we have .
The same arguments, as in the case give us the position of the points in as the vertices of regular polygons with sides, each polygon in a horizontal plane constant, , centered at the origin of the plane and with one of the vertices at The actions on and are the same as in the simple case 3.9.
The group action rearranges the terms in the formula of and after factorizing out we obtain a similar formula for i.e. is invariant under the action, while Hence, the Hitchin’s metrics, given by the equation 3.6, are invariant for the above choice of points. The fact that the coordinates of the vertices satisfy the conditions translates to the fact that the Kähler class of the metric is invariant. Moreover any invariant Kähler class must satisfy these conditions.
We are now ready to give the proof of the main theorem:
Proof of Theorem A.
From Theorem 2.2 we know that there are only two types of isolated quotient singularities which admit a Gorenstein smoothing. For the first type of singularities the existence of ALE Ricciflat Kähler metric in any Kähler class was proved by Hitchin [9], for the singularities, and by Kronheimer [13, 14] in general (according to the reformulation of their results in Theorem 3.1 and Remark 3.2).
In the case of a singularity, we saw that a smoothing of is the quotient of a manifold, see Proposition 2.3. In this section we looked at the explicit description of Hitchin’s metrics on these deformations and proved that the Kähler metric is also group invariant as long as it satisfies the condition that its Kähler class is invariant.
4. Classification of ALE Ricciflat Kähler metrics on surfaces
The simply connected ALE Ricciflat Kähler surfaces are classified by Kronheimer [14]. These are the ALE hyperkähler surfaces, and their metrics are explicitly described. When the manifolds are not simply connected, the author could not find such a classification in the literature. The present paper is intended to fill in this gap, by giving a complete description of the ALE Ricciflat Kähler, with emphasis on the nonsimply connected case.
In this section we are going to use the Kronheimer’s approach to classify the ALE Ricciflat Kähler surfaces . We will see that the 4manifold and the metric are determined by the topology of the asymptotic end and the Kähler class of the metric, respectively. We prove this by starting with some easy lemmas.
Lemma 4.1.
Let be an ALE Ricciflat manifold then the fundamental group of is finite.
Proof.
This is an immediate consequence of the fact that is asymptotically locally Euclidean and the Volume Comparison Theorem on Ricciflat manifolds. The Volume Comparison Theorem for the universal cover tells us that must have volume growth less than that of and as is ALE we have that the universal cover must be a finite covering of . Hence is finite. ∎
Lemma 4.2.
Let be an ALE Ricciflat manifold, then its universal cover has only one end, in particular is ALE.
Proof.
If is disconnected at infinity, then contains a line ([17, Ch.9]), and as the Splitting Theorem tells us that is isometric to a product In particular the dimensional manifold is Ricciflat, and hence flat. But as is simply connected, we have is the Euclidean space, and so is In particular, has only one end at infinity. ∎
Remark 4.3.
The lemma actually proves a stronger result. If the asymptotic models of and are those of and with finite subgroups of and we denote the fundamental group by then is an extension of by
Proposition 4.4.
The universal cover of an ALE Ricciflat Kähler manifold is an ALE hyperkähler manifold, and is obtained by taking the quotient of by a finite cyclic group of automorphisms.
Proof.
Lemma 4.2 tells us that the universal cover is ALE Ricciflat Kähler manifold, and as is simply connected, it is hyperkähler [10]. In particular, as the metrics are compatible with the complex structure, the asymptotics of are given by finite subgroups of respectively.
Let be the normal subgroup of given by Then First we will show that the two subgroups are equal to each other. As is a subgroup of , it has an associated covering with the pullback metric. As is the universal cover of we have a covering Moreover, has one end at infinity and the group of deck transformations is given by As is a hyperkähker manifold, the manifold will be hyperkähler if acts trivially on the canonical bundle But any element in can be identified with an element in and can be represented be a loop in the asymptotic model, As the loop acts trivially on Hence is also a hyperkähler manifold. But Kronheimer [14] classified all ALE hyperkähler manifolds, and showed that they are, in particular, simply connected. Hence and
This implies that is a subgroup of So, is a finite cyclic group.
∎
Remark 4.5.
In particular, if the manifold is not simply connected then it is not hyperkähler, just locally hyperkähler, as the group acts nontrivially on the canonical bundle Moreover, is a spin manifold iff the order of is odd.
By Proposition 4.4, classifying the nonsimply connected ALE Ricciflat Kähler manifolds is equivalent to classifying the finite cyclic free groups of isometries on hyperkähler manifolds.
In the remaining part of this section, we prove that the manifold admits a equivariant degeneration to the quotient orbifold This will lead to an explicit description of and it will give us a complete classification of the ALE Ricciflat Kähler manifolds. To do this we go back to Kronheimer’s paper [14] and study the action on the twistor space
As we mentioned in the previous section, to any halfconformally flat (spin) manifold we can associate a complex threemanifold its twistor space. If the manifold is scalar flat Kähler, then the metric is antiselfdual (ASD), and its twistor space is the projective bundle where is the bundle of selfdual spinors. If we consider the manifold with the reversed orientation , then is a selfdual manifold, and to it we associate the same twistor space which now is defined as where is the bundle of antiselfdual spinors on
Given an ALE Ricciflat hyperkähler manifold Kronheimer [14] constructs orbifold compactifications of both and and . Let be a neighborhood of infinity in such that is a neighborhood of infinity in and let be the universal cover of . If are invariant local coordinates at infinity in then is a smooth chart if we consider the coordinates Moreover, if is a smooth invariant function, equal to outside some compact set, then the metric extends to a class C metric on [14], which is invariant. Then defines Riemannian orbifold structures on the compactifications and The manifold is obtained from by taking the quotient by the group which acts freely on and fixes on .
We now construct a compactification of the twistor space by defining the twistor space for an orbifold space. As is in the same conformal class as it is ASD in coordinates. But if we consider coordinates, as the transition map is reversing the orientation, the metric is be selfdual. To leading order, acts linearly on both and coordinates, by the same representation , and so does its subgroup Moreover (see [14, Lemma 2.1]), in coordinates the action of on is trivial, but nontrivial on For the neighborhood of we consider the twistor space As the metric is both and invariant, the actions of and extend to holomorphic actions on Moreover, if is the twistor line corresponding to the then and act on by the standard action of on [14]. Let We define the compactifications of the twistor spaces to be and with the complex structures given by quotients of in the neighborhoods of the twistor lines at infinity. The complex fold is obtained as the quotient where acts freely on and nontrivially on
As is hyperkähler, there is a holomorphic projection The sphere gives a family of complex structures on to each point there is an associated complex structure on given by the complex structure on the fiber Note that the compactification is not hyperkähler and the projection does not extend to the compactification of the twistor space.
Kronheimer’s description of the twistor space of
On there is a preferred line bundle By Hartog’s theorem this extends to a line bundle on which can be restricted to a line bundle on
To understand the twistor space Kronheimer considers the following graded rings:
and the ideal generated by the pullbacks of the two sections of The two sections and are associated to the homogeneous coordinates ofn There is a relation between the graded rings given by [14, Prop. 2.3]:
(4.1) 
If we look at the affine varieties associated to the coordinate rings, the above sequence can be interpreted as saying that there exists a fourdimensional affine variety and a fibration such that i.e. :
(4.2) 
Moreover [14], is a deformation of and it can be considered a subset of On there is an induced action given by the grading on the rings such that is equivariant, and the action of on is the usual multiplication. Let and where The map induces a projection . Then, and are orbifold models of the twistor spaces