On Kähler conformal compactifications of U(n) -invariant ALE spaces
aa r X i v : . [ m a t h . DG ] S e p ON K ¨AHLER CONFORMAL COMPACTIFICATIONS OF U ( n ) -INVARIANT ALE SPACES MICHAEL G. DABKOWSKI AND MICHAEL T. LOCK
Abstract.
We prove that a certain class of ALE spaces always has a K¨ahler con-formal compactification, and moreover provide explicit formulas for the conformalfactor and the K¨ahler potential of said compactification. We then apply this to givea new and simple construction of the canonical Bochner-K¨ahler metric on certainweighted projective spaces, and also to explicitly construct a family K¨ahler edge-cone metrics on CP , with singular set CP , having cone angles 2 πβ for all β > Introduction
This work is motivated by the conformal relationship between two well-knownK¨ahler metrics. The Burns metric on C blown-up at the origin is conformal in anorientation reversing manner to the Fubini-Study metric on CP minus a point. Infact, it extends smoothly to the one-point compactification, see [LeB88]. While thereis a conformal relationship between these metrics, it is important to note that theyare K¨ahler with respect to different complex structures. There are a wide variety ofK¨ahler metrics on noncompact spaces, and we investigate when relationships such asthis can occur in a more general situation.We consider K¨ahler metrics on noncompact manifolds which asymptotically looklike R n / Γ, for a finite subgroup Γ ⊂ SO( n ) acting freely on R n \ { } , with the metricinduced from the Euclidean metric. More precisely, we have the following definition. Definition 1.1.
We say that a complete Riemannian manifold (
X, g ) is asymptoticallylocally Euclidean (ALE) of order τ if there exists a finite subgroup Γ ⊂ SO( n ) whichacts freely on R n \ { } , a compact subset K ⊂ X , and a diffeomorphismΦ : X \ K → ( R n \ B (0 , R )) / Γ , satisfying (Φ ∗ g ) ij = δ ij + O ( r − τ ) ∂ | k | (Φ ∗ g ) ij = O ( r − τ − k )for any partial derivative of order k as r → ∞ , where r is the distance to a fixedbasepoint. We call Γ the group at infinity.Since ALE spaces have a group action at infinity, we should not expect them tocompactify to a smooth manifold. Instead, we look for compactifications to orbifoldswith isolated singularities modeled on R n / Γ for some Γ as above. We say that g isa Riemannian orbifold metric with isolated singularities on an n -manifold M if it is Research supported in part by the NSF RTG Grant DMS-1148490. a smooth Riemannian metric away from finitely many singular points, and at anysingular point the metric is locally the quotient of a smooth Γ-invariant metric on B n by the orbifold group Γ.We will also be interested in another class of singular metrics which is a general-ization of an orbifold metric having a higher dimensional singular set. Let M be asmooth n -manifold with a smoothly embedded ( n − p ∈ Σ choose coordinates ( y , y , x , . . . , x n − ) so that Σ is given by y = y = 0, and then change coordinates to a transversal polar coordinate system bysetting y = r cos θ and y = r sin θ . We say that g is a Riemannian edge-cone metric on ( M, Σ) with cone angle 2 πβ if it is a smooth Riemannian metric on M \ Σ, andaround any p ∈ Σ the metric can be expressed as g = dr + β r ( dθ + u i dx i ) + f ij dx i ⊗ dx j + r ǫ h, (1.1)where the f ij , which are symmetric in i and j , and the u i are smooth functions on Σ,and h is a symmetric two-tensor field with infinite conormal regularity along Σ. See[JMR11, AL13] for more details. Remark 1.2.
The definition of ALE space extends naturally to include Riemann-ian orbifold and edge-cone metrics. We will distinguish by describing the space asnonsingular or by the type of singularity as needed.1.1.
K¨ahler conformal compactifications.
Given a K¨ahler ALE space, we wantto conformally relate it to a compact K¨ahler manifold, orbifold, or edge-cone space.A conformal compactification of an ALE space (
X, g ), is a choice of a conformal factor u : X → R + such that u = O ( r − ) as r → ∞ , and we denote the compactified spaceby ( b X, b g ), where b X = X ∪{∞} is a compact orbifold and b g = u g . If ( X, g ) is an ALEspace with group at infinity Γ, then the conformal compactification naturally reversesorientation and has orbifold group Γ. However, if we reverse the orientation of ( b X, b g )the orbifold group will be the be the orientation reversed conjugate group Γ. By thiswe mean that the action is conjugate to that of Γ by an element of O( n ) \ SO( n ).The focus here will be on U( n ) -invariant K¨ahler ALE spaces . These are K¨ahlermanifolds ( X, J , g ) whose underlying Riemannian manifolds are ALE spaces, andwhose metrics satisfy the rotational symmetry condition that they arise from a smoothpotential function φ ( z ) on C n \ { } , where z = | z | + · · · + | z n | . The actual K¨ahlerALE space can then be obtained by taking the appropriate quotient of C n \ { } by Z k , given by ( z , · · · , z n ) ( e πi/k z , · · · , e πi/k z n ), and attaching a CP n − or smoothpoint at the origin so that the metric is complete. Note that this has a clear extensionto U( n )-invariant K¨ahler ALE spaces with edge-cone singularities along the CP n − , ororbifold points, at the origin. The underlying manifold of such ALE spaces obtainedby attaching a CP n − at the origin is the total space of O P n − ( − k ), the k th -power ofthe the tautological line bundle over CP n − .We will show that every U( n )-invariant K¨ahler ALE metric on O P n − ( − k ), includingthose with edge-cone singularities, admits a conformal compactification to a metricthat is K¨ahler with respect to a different complex structure . From here on, thiswill be referred to as a K¨ahler conformal compactification . This idea of a conformal ¨AHLER CONFORMAL COMPACTIFICATIONS 3 structure containing two representatives which are K¨ahler with respect to differentcomplex structures was examined in real 4-dimensions in [ACG13]. In arbitrary realeven-dimensions, the cohomogeneity-one situation, which is our focus, has certainstructural properties that we are able to exploit in order to obtain interesting results.We now state our main theorem.
Theorem 1.3.
Let g be a U( n ) -invariant K¨ahler ALE metric on O P n − ( − k ) , whicharises from the K¨ahler potential φ ( z ) . Then, there exists a U( n ) -invariant K¨ahlerconformal compactification given by the conformal factor u = (cid:16) z ∂φ∂ z (cid:17) − . Moreover, this is the unique such conformal factor up to scale.
Topologically, the conformal compactification of O P n − ( − k ) is the weighted projec-tive space CP n (1 , ··· , ,k ) , see Section 4.1. The proof of Theorem 1.3 is given in Section 2.In addition to finding the conformal factor, which gives a K¨ahler conformal compact-ification of any U( n )-invariant K¨ahler ALE space, we are actually able to modify theproof to be able to find the K¨ahler potential of the compactification. We state thisresult as the following corollary, which is also proved in Section 2. Corollary 1.4.
Let g be a U( n ) -invariant K¨ahler ALE metric on O P n − ( − k ) , whicharises from the K¨ahler potential φ ( z ) . Then the corresponding K¨ahler conformalcompactification, of Theorem 1.3, has K¨ahler potential b φ ( z ) = − Z (cid:16) z ∂φ∂ z (cid:17) − d z . Remark 1.5.
Both Theorem 1.3 and Corollary 1.4 extend to the situation where themetric has an edge-cone singularity.
Remark 1.6.
Consider, instead, a U( n )-invariant K¨ahler ALE space arising from theK¨ahler potential φ ( z ) and obtained by attaching a smooth point or orbifold point atthe origin. Applying the conformal factor u = (cid:0) z ∂φ∂ z (cid:1) − to this space, yields a metricthat is K¨ahler with respect to a different complex structure as well. However, thespace upon which this new metric lives would be the original space “flipped insideout” as this conformal factor compactifies at infinity, but blows up at the origin.When dealing with compactifications in general, a delicate issue arises with the reg-ularity of the metric b g because a priori it is only a C ,α -orbifold metric. However, withcertain geometric conditions, one does obtain a C ∞ -orbifold metric. For example, indimension four, if the ALE space is anti-self-dual then the conformal compactifica-tion is a C ∞ -orbifold metric [TV05, CLW08]. This was subsequently generalized toBach-flat metrics in [Str10], and to obstruction-flat metrics in [AV12].In our case of U( n )-invariant ALE spaces, if the conformal metric b g = u g admitsa Taylor expansion around the orbifold point, then it will be a C ∞ -orbifold metricbecause it would extend smoothly, in the orbifold sense, over the singular point sincethe expansion would be in terms of the variable ξ = | ξ | + · · · + | ξ n | . This amounts MICHAEL G. DABKOWSKI AND MICHAEL T. LOCK to checking that (cid:0) ∂φ∂ z (cid:1) − has a Taylor expansion at ALE infinity. It is also likely that aU( n )-invariant ALE space being K¨ahler scalar-flat is enough to ensure a C ∞ -orbifoldmetric in the compactification. This would be interesting to investigate in light ofpossible applications of Theorem 1.3 to a family of K¨ahler scalar-flat ALE metricson complex line bundles over CP n which are higher dimensional generalizations ofthe metrics that we introduce in Section 3, see [Sim91, Che09]. However, all of ourapplications of Theorem 1.3 are to anti-self-dual metrics, so the compactificationshere are already guaranteed to be C ∞ -orbifold metrics.1.2. Outline of paper.
K¨ahler ALE metrics play an important role in modern ge-ometry, and it is an interesting question to understand when they admit K¨ahler con-formal compactifications. Theorem 1.3 answers this question in the U( n )-invariantcase and Corollary 1.4 actually provides an explicit formula for the K¨ahler potentialof the compact orbifold metric in terms of the K¨ahler potential of the ALE space.We prove these results in Section 2. Next, in Section 3, we introduce a family ofK¨ahler scalar-flat ALE spaces, due to LeBrun, towards which the remainder of thiswork is focused. Finally, in Section 4 we provide a new and simple construction ofthe canonical Bochner-K¨ahler metric on a subclass of weighted projective spaces, aswell as explicitly construct a family of K¨ahler edge-cone metrics on ( CP , CP ).1.3. Acknowledgements.
The authors would like to thank Jeff A. Viaclovsky forsuggesting this problem as well as for his advice and many helpful suggestions.2.
Proofs of Theorem 1.3 and Corollary 1.4
Proof of Theorem 1.3.
Let g be a U( n )-invariant K¨ahler ALE metric on O P n − ( − k ) that arises from a K¨ahler potential φ ( z ) on C n \ { } with the standardcomplex structure J z , where z = | z | + · · · + | z n | . On a spherical shell centered atthe origin, where ∂ z and ¯ ∂ z are with respect to J z , consider the K¨ahler form ω = √− ∂ z ¯ ∂ z φ ( z ) . (2.1)Because of the U( n )-invariance, we can restrict our work to the z -axis where ω = √− h(cid:16) ∂φ∂ z + z ∂ φ∂ z (cid:17) dz ∧ d ¯ z + ∂φ∂ z n X i =2 dz i ∧ d ¯ z i i . (2.2)We search for a conformal compactification factor u ( z ), depending only on the theradial variable z , so that the conformal compactification ( b X, b g ), where b g = u g , isK¨ahler. Since u is rotationally symmetric, the metric b g would exhibit this symmetryas well around the orbifold point. Therefore, in some coordinate system ( ξ , · · · , ξ )on C n , with the standard complex structure J ξ and where { } corresponds to theorbifold point, we seek to find a K¨ahler potential b φ ( ξ ), a function of the radial variable ξ = | ξ | + · · · + | ξ n | , of some unknown conformal metric b g = u g . If such coordinatesand potential function exist, then letting ∂ ξ and ¯ ∂ ξ be with respect to J ξ , we could ¨AHLER CONFORMAL COMPACTIFICATIONS 5 similarly examine the restriction of the K¨ahler form b ω = √− ∂ ξ ¯ ∂ ξ b φ ( ξ )(2.3)to the the ξ -axis, which would be b ω = √− h(cid:16) ∂ b φ∂ ξ + ξ ∂ b φ∂ ξ (cid:17) dξ ∧ d ¯ ξ + ∂ b φ∂ ξ n X i =2 dξ i ∧ d ¯ ξ i i . (2.4)An inversion map would be necessary to relate the respective coordinate systems.One initially thinks to search for a map that is holomorphic on C n \ { } , in whichcase the pullback of b ω would be the K¨ahler form in ( z , · · · , z )-coordinates, and thenhope that this relationship could be be used to solve for u ( z ). However, there doesnot exist an inversion map that is holomorphic on all of C n \ { } when n >
1. Instead,to relate the two coordinate systems, let us choose the map ϕ : C n \ { } → C n \ { } defined by ϕ ( z , z , · · · z n ) = (cid:16) ¯ z z , z z , · · · , z n z (cid:17) = ( ξ , ξ , · · · , ξ n ) . (2.5)There is actually some flexibility in the choice of the inversion map, however we willsee that this choice in particular will greatly simplify our work.It is easy to check that ϕ = Id and that ϕ is holomorphic on the z /ξ -axes awayfrom the origin. Therefore, on these axes J ξ ◦ ϕ ∗ ( x ) = ϕ ∗ ◦ J z ( x ) , (2.6)where x ∈ T R n and C n is identified with R n as usual. Also, the pullback operatorsatisfies the usual commutativity properties with the ∂ and ¯ ∂ operators on the z /ξ -axes. However, this is not true away from these axes and even more importantly, since ϕ is not holomorphic on all of C n \ { } , the pullback operator does not commute withthe ∂ ¯ ∂ operator anywhere, even on the z /ξ -axes, because there are nonvanishingterms when one takes two derivatives. Therefore we cannot globally understand b ω in( z , · · · , z n )-coordinates directly in terms of the pullback, which is reasonable becauseour choice of the inversion map was somewhat arbitrary in that we do not reallyconsider what the complex structure J ξ is in ( z , · · · , z n )-coordinates. Although thisseems like a problem, we will now see that our choice of ϕ , because it is holomorphicjust on the z /ξ -axes, will be enough for us to find u and b φ in terms of the known φ .In the respective coordinate systems, write the K¨ahler forms in terms of the metricsand complex structures as ω ( x, y ) = g ( J z x, y ) and b ω ( v, w ) = b g ( J ξ v, w ) , (2.7)where x, y ∈ T R n and v, w ∈ T R n , and C n is identified with R n as usual in therespective coordinates systems. Even though the pullback of b ω will not globally bethe K¨ahler form, we will be able to see a useful relationship between ϕ ∗ b ω and ω . Since b g = u g , and ϕ ∗ commutes with the complex structure on the z /ξ -axes as in (2.6), MICHAEL G. DABKOWSKI AND MICHAEL T. LOCK we find the following relationship on the z -axis: ϕ ∗ b ω ( x, y ) = ϕ ∗ b g (cid:0) J ξ · , · (cid:1) ( x, y ) = b g (cid:0) J ξ ◦ ϕ ∗ ( x ) , ϕ ∗ ( y ) (cid:1) = b g (cid:0) ϕ ∗ ◦ J z ( x ) , ϕ ∗ ( y ) (cid:1) = ϕ ∗ b g (cid:0) · , · (cid:1)(cid:0) J z ( x ) , y (cid:1) = u g (cid:0) J z ( x ) , y (cid:1) = u ω ( x, y ) . (2.8)The restriction to the z -axis has greatly simplified our work, and we use this rela-tionship here to set up a system of ODEs, which we then solve to find the conformalfactor in terms of φ which is known. The pullback of b ω on the z -axis is ϕ ∗ b ω = √− h(cid:16) ∂ b φ∂ z + z ∂ b φ∂ z (cid:17) dz ∧ d ¯ z − ∂ b φ∂ z n X i =2 dz i ∧ d ¯ z i i . (2.9)It is essential to notice the minus sign appearing before the second term. Examiningthe equation ϕ ∗ b ω = u ω on the z -axis, from (2.8), using the formulas (2.9) and (2.2)for ϕ ∗ b ω and ω on this axis respectively, we arrive at the following system of equations u (cid:16) ∂φ∂ z + z ∂ φ∂ z (cid:17) = (cid:16) ∂ b φ∂ z + z ∂ b φ∂ z (cid:17) u (cid:16) ∂φ∂ z (cid:17) = − ∂ b φ∂ z . (2.10)By taking a derivative of the second equation in (2.10) we see that ∂ b φ∂ z = − u ∂u∂ z ∂φ∂ z − u ∂ φ∂ z . (2.11)Substituting this along with the second equation in (2.10) into the right hand side ofthe first equation from (2.10), we find that u (cid:16) ∂φ∂ z + z ∂ φ∂ z (cid:17) = − u ∂φ∂ z − u z ∂u∂ z ∂φ∂ z − u z ∂ φ∂ z , (2.12)which simplifies to ∂∂ z log( u ) = − ∂∂ z log (cid:16) z ∂φ∂ z (cid:17) . (2.13)Hence, the conformal factor is u = (cid:16) z ∂φ∂ z (cid:17) − . (2.14)Now, we examine the behavior of the conformal factor along the CP n − at theorigin. Since the spherical metric on S n − decomposes as g S n − = g CP n − + h , where g CP n − is the Fubini-Study metric on CP n − and h is the metric along the Hopf fiber,the U( n )-invariant K¨ahler ALE metric on O P n − ( − k ) descends via the Z k quotientfrom g = h ∂∂ z (cid:16) z ∂φ∂ z (cid:17)i(cid:16) d √ z + z h (cid:17) + (cid:16) z ∂φ∂ z (cid:17) g CP n − . (2.15) ¨AHLER CONFORMAL COMPACTIFICATIONS 7 Since this metric is defined along the CP n − at the origin, in other words when z = 0,we see that the conformal factor (2.14) extends smoothly over this exceptional orbit.Finally, we will show that u = O ( r − ) as r → ∞ to ensure that it is in fact aK¨ahler conformal compactification. First, on the z − axis change coordinates to realpolar coordinates by setting z = ( √ z , θ ) where z = | z | = O ( z ). Then, rewrite(2.2), the restriction of ω to the z -axis, in these coordinates ω = 2 (cid:16) ∂φ∂ z + z ∂ φ∂ z (cid:17) √ z ( d √ z ∧ dθ ) + √− ∂φ∂ z n X i =2 dz i ∧ d ¯ z i = ∂∂ √ z (cid:16) z ∂φ∂ z (cid:17) ( d √ z ∧ dθ ) + √− ∂φ∂ z n X i =2 dz i ∧ d ¯ z i . (2.16)Since ω = g ( J z · , · ) and is ALE of order τ , it must be asymptotic to the K¨ahler formof the standard Hermitian metric on C n . Examining the first term of ω restricted tothe z -axis we see that as r → ∞ d (cid:16) z ∂φ∂ z dθ (cid:17) = 12 d ( r dθ ) + O ( r − τ ) , (2.17)where r and r denote the radial distances in the standard Hermitian metric alongthe z -axis and on all of C n respectively. Therefore u = (cid:16) z ∂φ∂ z (cid:17) − = O ( r − )(2.18)as r → ∞ , which completes the proof.2.2. Proof of Corollary 1.4.
By substituting formula (2.14) for the conformal factorinto the second equation of (2.10), we see that ∂ b φ∂ z = − (cid:16) z ∂φ∂ z (cid:17) − , (2.19)so therefore b φ ( z ) = Z ∂ b φ∂ z d z = − Z (cid:16) z ∂φ∂ z (cid:17) − d z . (2.20) 3. LeBrun
U(2) -invariant K¨ahler ALE metrics
The remainder of this paper is dedicated to applications of Theorem 1.3. We willfocus on a family of K¨ahler scalar-flat ALE spaces, due to LeBrun. These arise in twoseemingly distinct ways. One way is by developing and solving a nonlinear ODE fora potential function, and the other is by using a hyperbolic analogue of the Gibbons-Hawking ansatz. For our purposes, it will be simpler to focus on the former of thetwo constructions of which we give a brief description in Section 3.1.These spaces are real four-dimensional, and there are certain geometric properties,unique to this dimension, which will come into play. Over an oriented Riemannianfour-manifold, the Hodge star operator restricted to two-forms ∗ : Λ → Λ satisfies ∗ = Id . This induces the decomposition of two-forms into the ± MICHAEL G. DABKOWSKI AND MICHAEL T. LOCK ∗| Λ . The Weyl tensor has a corresponding decomposition W = W + ⊕ W − , into theself-dual and anti-self-dual parts of the Weyl tensor respectively. The metric is calledanti-self-dual if W + = 0 and self-dual if W − = 0. Note that by reversing orientationa self-dual manifold is converted into an anti-self-dual manifold and vice versa. Fora K¨ahler metric in real four-dimensions, W + is determined by the scalar curvature,so a K¨ahler scalar-flat metric is necessarily anti-self-dual.3.1. LeBrun metrics.
In 1988 LeBrun explicitly constructed a nonsingular anti-self-dual K¨ahler scalar-flat ALE metric on the total space of the bundle O P ( − k ) forall integers k ≥
1, see [LeB88]. For k = 1 and k = 2, these are the well-known Burnsand Eguchi-Hanson metrics respectively [Bur86, EH79]. When k >
2, LeBrun showedthat these metrics have negative mass thereby providing an infinite family of counterexamples to the generalized positive action conjecture [HP78], hence they are knownas the
LeBrun negative mass metrics . These metrics can also be reworked in a way asto give a 1-parameter family of anti-self-dual K¨ahler scalar-flat ALE edge-cone metricson O P ( − CP at the origin, with cone angles 2 πβ for all β >
0. These will be referred to as the
LeBrun edge-cone metrics . We briefly describeLeBrun’s method of construction to make it clear that Theorem 1.3 is applicable. Fora more thorough description see [LeB88, LeB91, LNN97, Via10, AL13].Consider a K¨ahler potential φ ( z ) on C \{ } , where z = | z | + | z | . The (1 , ω = √− ∂ ¯ ∂φ, (3.1)is the K¨ahler form of a metric on a spherical shell centered at the origin. Recall thatany K¨ahler metric satisfies R ω ∧ ω = ρ ∧ ω, (3.2)where R is the scalar curvature and ρ the Ricci form. Since LeBrun searched forK¨ahler scalar-flat metrics, he used the rotational symmetry of the situation to solve0 = ρ ∧ ω, (3.3)and thereby obtained a family of potential functions { φ β ( z ) } β ∈ R + . In fact, he obtaineda wider family of potential functions, but for our purposes and in the interest claritywe restrict our attention to these. For each φ β ( z ), LeBrun defined a new radialcoordinate r = r z ∂φ β ∂ z , (3.4)and showed that the corresponding metric is g LB ( β ) = dr β − r + − βr + r h σ + σ + (cid:16) β − r + 1 − βr (cid:17) σ i , (3.5)where r is the radial distance from the origin and σ , σ , σ are the usual left-invariantcoframe on SU(2) = S . Since for a K¨ahler metric in four-dimensions the self-dualpart of the Weyl curvature tensor is determined by the scalar curvature, and thesemetrics are K¨ahler scalar-flat, they are anti-self-dual. ¨AHLER CONFORMAL COMPACTIFICATIONS 9 This metric is clearly singular at the origin. However, by redefining the radialcoordinate as ˜ r = β − ( r −
1) and attaching a CP at ˜ r = 0, one sees that g LB ( β ) = d ˜ r + ( σ + σ ) + β ˜ r σ + ˜ r h ( β − r + 1 d ˜ r + β ( σ + σ ) + (cid:16) − ββ ˜ r + 1 (cid:17) β ˜ r σ i , (3.6)is in fact a K¨ahler scalar-flat ALE edge-cone metric on ( O P ( − , CP ) with coneangle 2 πβ , where the singular set is the CP at the origin. Moreover, due to theconstruction, it is clearly U(2)-invariant. These are the LeBrun egde-cone metrics .Note that when β = 1 this is the Burns metric.When β = k is a positive integer, LeBrun obtained a nonsingular ALE metric.Consider the metric g LB ( k ) for any positive integer k . By taking the Z k quotient of C \ { } generated by ( z , z ) ( e πi/k z , e πi/k z ) , (3.7)which is rotation in the fiber, it is clear that this metric extends smoothly over the CP at ˜ r = 0. Therefore g LB ( k ) defines a nonsingular U(2)-invariant K¨ahler scalar-flatALE metric on the total space of O P ( − k ). These are the LeBrun negative massmetrics . It is important to distinguish that only when β is a positive integer can weobtain a nonsingular metric.4. Explicit K¨ahler conformal compactifications
Here we use Theorem 1.3 to examine K¨ahler conformal compactifications of theLeBrun U(2)-invariant K¨ahler ALE spaces introduced in Section 3. Since these spacesare K¨ahler scalar-flat they are necessarily anti-self-dual and will therefore compactifyto C ∞ -orbifold metrics, recall [TV05, CLW08]. In Section 4.1 we provide a newand simple construction of the canonical Bochner-K¨ahler metric on CP , ,k ) for anypositive integer n . Then, in Section 4.2, we explicitly construct a family of extremalK¨ahler edge-cone metrics on ( CP , CP ) having cone angles 2 πβ for all β >
0. Also, inRemark 4.2, we note a interesting relationship between the conformal compactificationfactor and an explicit formulation of the Green’s function for the conformal Laplacianon an orbifold.4.1.
Bochner-K¨ahler metrics on weighted projective space.
Given relativelyprime integer weights 1 ≤ p ≤ p ≤ · · · ≤ p n , the complex n -dimensional weightedprojective space CP n ( p ,p , ··· ,p n ) is the quotient S n +1 /S , where S acts by( z , z , · · · , z n ) ( e ip θ z , e ip θ z , · · · , e ip n θ z n ) , (4.1)for 0 ≤ θ < π . This has the structure of a compact complex orbifold with thenumber of singular points corresponding to the number of weights greater than 1.Bryant proved that every weighted projective space admits a Bochner-K¨ahler metric[Bry01]. When p = · · · = p n = 1, this metric is just the Fubini-Study metric on CP n .Later, David and Gauduchon gave a direct construction of these metrics and used anargument due to Apostolov to show that this metric is the unique Bochner-K¨ahler metric on a given weighted projective space [DG06]. Therefore, we refer to it as the canonical Bochner-K¨ahler metric . See also [GL88] for earlier related work. We do notdiscuss the construction of [DG06] here, because it involves sophisticated techniquesin complex geometry, and the goal here is to focus on real 4-dimensions and providea simple construction for the canonical Bochner-K¨ahler metric on CP , ,k ) for anypositive integer k . In real 4-dimensions, the Bochner tensor is exactly the anti-self-dual part of the Weyl tensor, so Bochner-K¨ahler metrics are the same as self-dualK¨ahler metrics. It is interesting to remark that these metrics are in fact extremalK¨ahler, see [Der83].Topologically CP , ,k ) = b O P ( − k ). Joyce proved that there is a quaternionic metricon CP , ,k ) which is conformal to the LeBrun negative mass metric on O P ( − k ), see[Joy91]. This metric is necessarily the canonical Bochner-K¨ahler metric, however hedoes not find an explicit conformal factor or construction of the metric. Here, usingTheorem 1.3, we do just that. Theorem 4.1.
The canonical Bochner-K¨ahler metric on CP , ,k ) is a K¨ahler confor-mal compactification of the LeBrun negative mass metric on O P ( − k ) , and is explicitlygiven by g BK ( k ) = dr ( r + k − r −
1) + 1 r h σ + σ + (cid:16) k − r + 1 − kr (cid:17) σ i . Proof.
Recall the LeBrun negative mass metric g LB ( k ) on O P ( − k ) from Section 3.1.This anti-self-dual K¨ahler ALE metric arises from the U(2)-invariant K¨ahler potential φ k ( z ) on C \ { } , and thus satisfies the conditions of Theorem 1.3. Therefore b g LB ( k ) = (cid:16) z ∂φ k ∂ z (cid:17) − g LB ( k ) (4.2)is a self-dual K¨ahler orbifold metric on b O ( − k ) = CP , ,k ) . Since the canonicalBochner-K¨ahler metric is the only such metric on CP , ,k ) , it must be b g LB ( k ) . Now, re-calling the coordinate change (3.4) for the radial variable r , we see that the conformalfactor (cid:0) z ∂φ k ∂ z (cid:1) − is exactly r − hence g BK ( k ) = b g LB ( k ) = r − g LB ( k ) . (4.3) (cid:3) Remark 4.2.
Notice that u − = r is the Green’s function, associated to the orbifoldpoint, for the conformal Laplacian on ( CP , ,k ) , g BK ( k ) ). In general, such a Green’sfunction on a compact Riemannian orbifold is only guaranteed to exist if the Yamabeconstant is nonnegative, and even when one is known to exist it is rare to know itexplicitly. It is useful when these functions exist because, given a compact Riemannianorbifold ( M, g ) with a Green’s function G for the conformal Laplacian associated toa point p ∈ M , one can obtain a scalar-flat ALE space as the “conformal blow-up”( M \ { p } , G g ) . Here a coordinate system at infinity arises from inverted normal coordinates around { p } . ¨AHLER CONFORMAL COMPACTIFICATIONS 11 Extremal K¨ahler edge-cone metrics on ( CP , CP ) . Extremal K¨ahler met-rics were first introduced by Calabi in an effort to obtain a canonical metric in agiven K¨ahler class [Cal82, Cal85]. Given a compact complex manifold with a K¨ahlermetric, Calabi fixed the deRahm cohomology class of this metric and then consideredthe functional of the L -norm squared of the scalar curvature on the set of K¨ahlerforms in this class. Extremal K¨ahler metrics are the critical points on this functional.Calabi showed that a K¨ahler metric is extremal K¨ahler if and only if the gradient ofits scalar curvature is the real part of a holomorphic vector field, so in a way thesemetrics can be viewed as a generalization of constant scalar curvature K¨ahler metrics.Abreu constructed a 1-parameter family of U(2)-invariant extremal K¨ahler edge-cone metrics on ( CP , CP ) having cone angles 2 πβ for all β >
0, see [Abr01]. Theyare also self-dual. Using the work of Derdzinski [Der83], which we discuss in moredetail below, Abreu then showed that, when the cone angle is restricted to 0 < β < CP , CP ). In [AL13],Atiyah-LeBrun gave an independent construction of these Einstein metrics. Theybegin by using a hyperbolic analogue of the Gibbons-Hawking ansatz to constructmetrics on the total space of a U(1)-bundle over hyperbolic 3-space minus a point.(These metrics are conformal to the metrics g LB ( β ) .) Then, they used results from[LNN97, Hit95] to solve for the explicit conformal factor that yields these Einsteinedge-cone metrics. The restriction of 0 < β < CP , CP ),having cone angles 2 πβ for all β >
0, in a very simple way by applying Theorem 1.3to the LeBrun edge-cone metrics discussed in Section 3.1. It is almost exactly howwe obtained the canonical Bochner-K¨ahler metrics on CP , ,k ) above.Recall the family of LeBrun edge-cone metrics g LB ( β ) on ( O P ( − , CP ), with coneangles 2 πβ for all β >
0, from Section 3.1. By Theorem 1.3, for all β > b g LB ( β ) = (cid:16) z ∂φ β ∂ z (cid:17) − g LB ( β ) (4.4)to a self-dual K¨ahler edge-cone metric on ( CP , CP ) with cone angle 2 πβ . Changingcoordinates to the radial variable r as in (3.4), we see that b g LB ( β ) = r − g LB ( β ) so on( CP , CP ) this metric is explicitly b g LB ( β ) = dr ( r + β − r −
1) + 1 r h σ + σ + (cid:16) β − r + 1 − βr (cid:17) σ i . (4.5)In real four dimensions, Derdzinski showed that the gradient of the scalar curvatureof self-dual K¨ahler metric is the real part of a holomorphic vector field [Der83], hencethe metric itself is necessarily extremal K¨ahler. Therefore, b g LB ( β ) is extremal K¨ahler,and thus we have completed the construction of the desired family U(2)-invariantextremal K¨ahler edge-cone metrics. It is important to distinguish that in Section 4.1 we began with the nonsingu-lar ALE metric g LB ( k ) on O P ( − k ) and compactified to obtain an orbifold with iso-lated singularities, while here we started with the edge-cone ALE metric g LB ( β ) on( O P ( − , CP ) and compactified to obtain a compact manifold with an edge-conesingularity.Now, we examine the conformal relationship between the b g LB ( β ) and (locally) Ein-stein metrics. The procedure discussed below was followed in [Abr01] to obtain theEinstein edge-cone metrics mentioned earlier, however we expand upon this to whenthe metric is not globally conformally Einstein, which is when β ≥
2. From ourformulation, the reader is able to obtain a clear picture of the geometry. Theseresults have been obtained previously so we only give a brief description here, see[PP90, Hit95, CP00, AL13] for more thorough discussions. However, the previous re-sults in this direction relied upon twistor theory or finding the solution to an equationinvolving a conformal change in Ricci curvature, while we will need only to computethe scalar curvatures of the aforementioned family of extremal K¨ahler metrics.Derdzinski showed that a self-dual K¨ahler metric g is conformal to a self-dualHermitian Einstein metric on M ∗ := { x ∈ M : R ( x ) = 0 } , given by ˜ g = R − g , where R is the scalar curvature [Der83]. The conformal metric ˜ g is no longer K¨ahler unless R is constant. Therefore, we wish to compute the scalar curvature of the metrics b g LB ( β ) on ( CP , CP ), which we denote by b R . (We suppress the particular β to whichthis is with respect because it will be clear from the context.) Since g LB ( β ) is scalarflat, the well-known formula for the scalar curvature of a conformal metric reduces tothe equation b R = − u ∆ g LB ( β ) u, (4.6)where u = (cid:0) z ∂φ β ∂ z (cid:1) − = r − , so we can compute that b R = 24 h (2 − β ) + 2( β − r i . (4.7)To see where this metric is conformally Einstein, we examine if and when b R = 0.Away from where the scalar curvature vanishes, these metrics will be conformallyEinstein with Einstein constant 6 β (2 − β ). We have the following cases:(1) When 0 < β <
2, the scalar curvature b R is everywhere positive. Therefore,the metric b R − b g LB ( β ) is a self-dual Einstein edge-cone metric, with positiveEinstein constant, on ( CP , CP ) with cone angle 2 πβ . It is easy to check that,after scaling, this construction gives the Einstein metrics found independentlyby Atiyah-Lebrun and Abreu discussed earlier. Only for this range of coneangles does the scalar curvature not vanish somewhere, hence they are theonly globally conformally Einstein metrics.(2) When β = 2, the scalar curvature b R vanishes at the point of compactification,so the metric is conformally Einstein by a factor of b R − = r / , with vanish-ing Einstein constant, away from this point. Notice that the conformal factor ¨AHLER CONFORMAL COMPACTIFICATIONS 13 is a scalar multiple of the inverse of the K¨ahler conformal compactificationfactor, so the conformal metric will be a scaled version of the original ALEmetric. This is the Eguchi-Hanson metric on O P ( − CP at the origin of the usual space [EH79].(3) When β >
2, the scalar curvature b R vanishes along the hypersurface definedby r = 2( β − / ( β − b R − . The piece containing the singu-lar set CP is diffeomorphic to O P ( − β = k is a positive in-teger, the corresponding AHE edge-cone metric has a quotient as in (3.7) to anonsingular self-dual (locally) AHE metric on O P ( − k ) with boundary a lensspace. These are the well-known Pedersen-LeBrun metrics, see [Hit95, CS04].The piece containing the point of compactification is diffeomorphic to the 4-ball, and on this we obtain a family of smooth self-dual AHE metrics. Afterchanging variables and rescaling, we see that these are in fact the well-knownPedersen metrics [Ped86]. References [Abr01] Miguel Abreu,
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