Degenerations of \mathbf{C}^n and Calabi-Yau metrics
aa r X i v : . [ m a t h . DG ] A p r DEGENERATIONS OF C n AND CALABI-YAU METRICS
G ´ABOR SZ´EKELYHIDI
Abstract.
We construct infinitely many complete Calabi-Yau metrics on C n for n ≥
3, with maximal volume growth, and singular tangent cones at infinity.In addition we construct Calabi-Yau metrics in neighborhoods of certain iso-lated singularities whose tangent cones have singular cross section, generalizingwork of Hein-Naber [18]. Introduction
Since the seminal work of Yau [34], Calabi-Yau metrics have been studied ex-tensively in K¨ahler geometry. Beyond the case of compact K¨ahler manifolds, therehave been many constructions of non-compact Calabi-Yau metrics with various be-haviors at infinity, by Cheng-Yau [4], Tian-Yau [31, 30] and others. In this paperwe are concerned with constructing new non-compact Calabi-Yau manifolds thathave Euclidean volume growth at infinity. In this case there exists a tangent cone atinfinity [3], which is expected to be unique [6, 13], and a natural problem is to tryconstructing complete Calabi-Yau metrics with prescribed tangent cones. Thereare many such constructions in the literature, with tangent cones that have smoothlinks [32, 16, 8, 9], as well as singular links [1, 21, 7].Our work pushes these methods further, obtaining a large class of new exampleson C n , for n ≥
3. In particular these give counterexamples to a conjecture ofTian [29, Remark 5.3], stating that the flat metric is the unique Calabi-Yau metricon C n with maximal volume growth. Some of these examples have very recentlybeen independently obtained by Li [22] and Conlon-Rochon [10], using somewhatdifferent techniques. See Section 1.2 for a comparison with our work. In addition wealso construct Calabi-Yau metrics in neighborhoods of certain isolated singularities,with singular tangent cones, extending unpublished work of Hein-Naber [18].Consider the hypersurface X ⊂ C n +1 given by the equation z + f ( x , . . . , x n ) = 0 , for a polynomial f , so that X is biholomorphic to C n . Suppose in addition thatif we let the x i have weights w i >
0, then f has degree d >
1. If we write F t ( z, x , . . . , x n ) = ( tz, t w x , . . . , t w n x n ), then F − t X has the equation t − d z + f ( x , . . . , x n ) = 0 , and so if d >
1, then F − t X → X as t → ∞ , where X = C × f − (0) . Suppose that X admits a (singular) Ricci flat cone metric ω whose homothetictransformations are the maps F t . It is then natural to expect that we can use ω to define an asymptotically Ricci flat metric on X whose tangent cone at infinityis ( X , ω ), that we can perturb to a complete Calabi-Yau metric on X with the same tangent cone. This almost fits into the class of problems studied by Conlon-Hein [8], except for the fact that X has more than just an isolated singularity atthe origin.We restrict ourselves to the simplest situation, when V = f − (0) ⊂ C n has anisolated normal singularity at the origin, so that X is only singular along C × { } .In particular n ≥
3. If we focus on the slice { } × V ⊂ X near the singular ray,then the corresponding slice of F − t X is a smoothing t − d + f ( x , . . . , x n ) = 0of the cone V . Our strategy is to write down a metric on X by combining aperturbation of the singular Calabi-Yau metric on X with a Calabi-Yau metricon this smoothing of V near the singular rays. The Calabi-Yau metric on thesmoothing is obtained using the results of Conlon-Hein [8]. Our main result thenis as follows. Theorem 1.
Under the assumption that the hypersurface V admits a Calabi-Yaucone metric, there is a complete Calabi-Yau metric on C n whose tangent cone atinfinity is C × V . As an example we can consider the case when n = 3, and f ( x , x , x ) = x + x + x k for k ≥
2. The hypersurfaces f − (0) are the A k − singularities and they all ad-mit flat cone metrics, being cyclic quotients of C . Therefore we obtain infinitelymany complete Calabi-Yau metrics on C with tangent cones C × A k − at infinity.Simple higher dimensional examples can be obtained by taking products with C .The corresponding metric when k = 2 has recently been constructed by Li [22]and Conlon-Rochon [10] independently. Another example in higher dimensions isobtained with f ( x , . . . , x n ) = x + . . . + x n , for n ≥
3, i.e. the A singularity, which admits the Stenzel cone metric. Wetherefore obtain complete Calabi-Yau metrics on C n with tangent cone C × A atinfinity.Consider now the hypersurface X ⊂ C n +1 with an isolated singularity 0 ∈ X ,given by z p + f ( x , . . . , x n ) = 0 , where p >
1, and f is as above. With F t as before, the equation of F − t X is now t p − d z p + f ( x , . . . , x n ) = 0 , and so if p > d , then we have F − t X → X as t →
0, with X = C × V . It istherefore natural to expect that X admits a metric which is asymptotically Calabi-Yau as we approach the singular point, and whose tangent cone at 0 is C × V . Usingan argument that is essentially identical to part of the proof of Theorem 1, we provethat this is the case, and in fact this metric can be perturbed to be Calabi-Yau ina neighborhood of the singular point. This result generalizes unpublished work ofHein-Naber [18], which applies to the case when f = x + . . . + x n , and p > n − n − ,using different techniques. EGENERATIONS OF C n AND CALABI-YAU METRICS 3
Theorem 2. If p > d , then X admits a Calabi-Yau metric on a neighborhood ofthe singular point , whose tangent cone at is C × V . It is likely that such constructions can be applied in a much more general setting,and we focus on these relatively explicit examples for simplicity. In general supposethat X ⊂ C N is a subvariety which admits a possibly singular Calabi-Yau conemetric whose homothetic (Reeb) vector field is ξ = P w i z i ∂ z i for weights w i > F t of C N . Let X ⊂ C N be another subvariety, and suppose that one of the followingtwo conditions holds:(1) lim t →∞ F − t X = X ,(2) or lim t → F − t X = X .Case (1) is the setting of Theorem 1, and here one expects to be able to constructa Calabi-Yau metric on X near infinity, with tangent cone X at infinity. If X is smooth, then one can hope to deform this into a global Calabi-Yau metric on X using techniques of Tian-Yau [31]. In case (2), by contrast, we necessarily have0 ∈ X , and one expects to be able to contruct a Calabi-Yau metric on X ona neighborhood of 0, whose tangent cone at 0 is X , as in Theorem 2. It seemslikely that our methods can be generalized to prove these expectations wheneverthe singular behavior of X has enough structure, such as the iterated edge spacesof Degeratu-Mazzeo [11].Note that for a given X , one typically expects infinitely many possible X tofit in case (1) above, as in Theorem 1, but there should be at most one X fittinginto case (2). This is because the tangent cone of a Calabi-Yau metric on X atthe singularity is expected to be independent of the metric (see Hein-Sun [20] for aspecial case of this).This discussion fits into the framework developed by Donaldson-Sun [13] foranalyzing the metric tangent cones of singular Calabi-Yau metrics under certainassumptions. They show that (under additional assumptions) the metric tangentcone C ( Y ) at 0 of a Calabi-Yau metric on X is an affine algebraic variety whichcan be obtained from X by a “two step” degeneration. In a first step we associateto X its weighted tangent cone W at 0 for a suitable canonical valuation at 0. Thisvaluation is obtained, roughly speaking, from the rate of growth of germs of holo-morphic functions on X with respect to the Calabi-Yau metric. The metric tangentcone C ( Y ) is then obtained by a further degeneration (or test-configuration) of W .One can think of these two steps as being analogous to the Harder-Narasimhanand Jordan-H¨older filtrations of an unstable, respectively semistable, vector bun-dle. Note that in the setting of Theorem 2 both W and C ( Y ) equal X and so itwould be interesting to construct examples where X = W = C ( Y ).1.1. Outline.
We now give an outline of the proof of Theorem 1. First, in Section 2we construct a complete Calabi-Yau metric on the smoothing V = f − (1) of theCalabi-Yau cone V . This is already contained in work of Conlon-Hein [8], but wegive some of the main points since they are a simpler version of what we need later.In Section 3 we write down a metric ω on X , which is approximately Calabi-Yau near infinity, modeled on the product metric on X = C × V . This productis singular along the rays C × { } , and so near these singular rays we glue insuitable scaled copies of the Calabi-Yau metric on V using cutoff functions. The G´ABOR SZ´EKELYHIDI key technical result is the estimate in Proposition 5 of the Ricci potential of ω , insuitable weighted H¨older spaces.The next step, in Section 7, is to modify the metric ω to improve the decay of itsRicci potential. This is analogous to Lemma 2.12 in Conlon-Hein [8] and is basedon inverting the Laplacian in suitable weighted spaces. After this we can furtherdeform ω to a Calabi-Yau metric using a non-compact version of Yau’s Theoremdeveloped by Tian-Yau [31]. We use the version of this due to Hein [17].The technical heart of the paper is in Sections 5 and 6, inverting the Laplacianin suitable weighted spaces. On asymptotically conical manifolds with smoothlink at infinity there is a well developed theory for this, going back to Lockhart-McOwen [23], however the tangent cone X of ( X , ω ) has a singular cross section- it is the double suspension of the link of V , which has a circle of singularitiesmodeled on V . There have been several works dealing with the Laplacian on similarspaces, most notably the theory of QALE spaces due to Joyce [21], and the moregeneral QAC spaces studied by Degeratu-Mazzeo [11]. On the one hand ( X , ω )does not quite fit into the QAC framework, but more crucially, when solving theequation ∆ u = f , these works require at least quadratic decay of f (at least awayfrom the singular rays), since they essentially rely on taking a convolution withthe Green’s function. In our application, however, we also deal with f which haveslower decay (this is the case when the degree d ≤ u = f for f which do not decay sufficiently fast, insteadof analysing the Green’s function, we construct an approximate inverse for ∆ onsuitable local patches, which we can then glue together using cutoff functions. Thisis analogous to the method employed in many other geometric gluing problems(see e.g. Donaldson-Kronheimer [14, Chapter 7] or [28]). One of the local modelsthat we have to analyze is the space X \ ( C × { } ) which the part of X awayfrom the singular rays of X is modeled on. The other model space is C × V , onwhich neighborhoods of those points in X are modeled on that are close to thesingular rays of X . The basic strategy for solving ∆ u = f approximately is to firstdecompose f into pieces that are supported inside the model neighborhoods, theninvert the Laplacian on the model spaces, and finally patch the results back togetherusing cutoff functions. We then need to control the errors that are introduced bythe cutoff functions on the one hand, and by replacing the metric ω on the modelneighborhoods by the corrsponding model metrics.1.2. Relation to other recent works.
As this project was nearing completion,two other works appeared that have significant overlap with our results. One isthe paper of Yang Li [22] mentioned above. It deals with the particular case ofconstructing a metric on C with tangent cone C × A at infinity. It relies on fairlyexplicit calculations, exploiting the symmetries of the situation. It is likely that hisapproach gives better asymptotic understanding of this metric, and this may beimportant in applications to gluing problems.The other, even more recent, paper is Conlon-Rochon [10], constructing Calabi-Yau metrics on C n with tangent cones C × V for suitable Calabi-Yau cones V .It relies on extending the work of Degeratu-Mazzeo [11] to a class of “warpedQAC” metrics, and as such it still requires faster than quadratic decay of the Riccipotential of the approximate solution. In terms of our Theorem 1 this is the casewhen the degree d >
3. One important case that this does not cover is when f ( x ) = x + . . . + x n , and n >
3, since here d = 2 n − n − ≤
3, although in a subsequent
EGENERATIONS OF C n AND CALABI-YAU METRICS 5 version of [11] the authors have overcome this issue by constructing a metric whoseRicci potential decays more rapidly. Nevertheless, our approach to inverting theLaplacian also has the advantage that the same method applies to Theorem 2.
Acknowledgements.
I am grateful to Hajo Hein for multiple helpful discussions.This work was supported in part by NSF grant DMS-1350696. In addition I thankthe anonymous referees for their careful reading of the paper and helpful sugges-tions. 2.
The smoothing model
Suppose that V ⊂ C n is a hypersurface given by the equation f ( x , . . . , x n ) = 0,with an isolated singularity at the origin. Suppose that we have a weight vector ξ = ( w , . . . , w n ) with positive, possibly non-integral entries, giving rise to theaction t · ( x , . . . , x n ) = ( t w x , . . . t w n x n )on C n for t >
0. We denote the degree of f under this action by d , i.e. f ( t · x ) = t d f ( x ). This action generates the action of a complex torus T c on C n , which fixes V . We write T ⊂ T c for the maximal compact torus. We will also write z · x for z ∈ C ∗ , which means that we choose a branch of log z to define the non-integerpowers of z . The choice of branch will not matter.There is a nowhere vanishing holomorphic ( n − V \ { } given byΩ = dx ∧ dx ∧ . . . ∧ dx n ∂ x f where ∂ x f = 0, and by similar expressions where ∂ x i f = 0 for i >
1. We assumethat Ω has degree n − n X i =1 w i = d + n − . Finally we suppose that V admits a Ricci flat K¨ahler cone metric ω V , whosehomothetic transformations are given by the action above. Equivalently we have ω n − V = ( √− ( n − Ω ∧ Ω. This setup has been studied extensively in the literature,in particular in relation to Sasakian geometry (see [15]).
Lemma 3.
Under these assumptions, unless V ∼ = C n − , we have d > .Proof. Let us denote by w min the smallest weight. The Lichnerowicz obstructionof Gauntlett-Martelli-Sparks-Yau [15] implies that since V admits a Calabi-Yaucone metric, we have w min > w min = 1 then necessarily V ∼ = C n − ). It thenfollows that d >
2, since d ≥ w min , unless f is linear. (cid:3) Because of this Lemma we can assume throughout that d >
2. Otherwise V ∼ = C n − in which case Theorems 1 and 2 are clear. Here we are interested in theexistence of complete Ricci flat metrics on the smoothing V of V , given by theequation 1 + f ( x ) = 0. This question was addressed by Conlon-Hein [8], and herewe review the main points since our result can be thought of as a generalization oftheir work.The Ricci flat metric on V is given by ω V = √− ∂∂r , where r is the radialdistance from the vertex of the cone. The idea is to use r to write down a K¨ahler G´ABOR SZ´EKELYHIDI potential on V near infinity, which is approximately Ricci flat. Since V is asymp-totic to V near infinity, a natural approach, followed by Conlon-Hein, is to simplyuse an orthogonal projection from V to V (outside a bounded set) to pull backthe potential r . We follow a slight variant of this method. First let us define thefunction R : C n → R , by letting R = 1 on the Euclidean unit sphere, and extend-ing R to have degree 1 under the action of F t . As shown in He-Sun [19, Lemma2.2], then the form √− ∂∂R defines a cone metric on C n . By the homogeneity,its restriction to V is uniformly equivalent to ω V . In particular the function R restricted to V is also uniformly equivalent to the distance function r . We nowtake any smooth extension of r from V to C n \ { } , which has degree 1 under theaction ξ . This can be defined by first extending r on the sphere R = 1, and thenextending it further by homogeneity. We will continue to write r for this extendedfunction.We claim the following. Proposition 4.
There exists a constant
A > such that (a) The form ω = √− ∂∂r is positive definite on V ∩ { R > A } . (b) The Ricci potential h = log ( √− ∂∂r ) n − ( √− ( n − Ω ∧ Ω of ( V , ω ) satisfies (2.1) ∇ i h = O ( R − d − i ) , as R → ∞ , for all i ≥ , measured with respect to ω .Proof. Given large K , let us analyze the region { K/ < R < K } by scaling themetric by a factor of K − . At the same time let us introduce rescaled coordinates˜ x = K − · x , and ˜ r = K − r, ˜ R = K − R . The rescaled metric is given by K − ω = √− ∂∂ ˜ r , and in the rescaled coordinates the equation of V is K − d + f (˜ x ) = 0 . Let us write this as V K − d . Restricted to the annular region { / < ˜ R < } , thesubmanifolds V K − d converge in C ∞ to V uniformly. In addition ˜ r as a functionof ˜ x is independent of K by the homogeneity of r .Using the implicit function theorem we can cover V K − d ∩ { / < ˜ R < } by afinite number of coordinate balls B i in each of which V K − d is a hyperplane, andmoreover we can find holomorphic functions G i : B i → C n such that G i ( V K − d ) ⊂ V , and G i ( y ) = y + O ( K − d ). Since we also control the derivatives of the G i and˜ r is a fixed smooth function, we obtain˜ r − G ∗ i ˜ r = O ( K − d )in C ∞ , on each of our coordinate balls. We measure derivatives here with respectto a fixed metric √− ∂∂ ˜ R . In particular once K is sufficiently large, √− ∂∂ ˜ r defines a positive definite form on V K − d , and in fact it is uniformly equivalent to √− ∂∂ ˜ R .To control the Ricci potential, we simply need to compare Ω with G ∗ i Ω, sinceby assumption we have ω nV = ( √− ( n − Ω ∧ Ω. By the same reasoning we have
EGENERATIONS OF C n AND CALABI-YAU METRICS 7 Ω − G ∗ i Ω = O ( K − d ). Since the Ricci potential is invariant under scaling we findthat on the annulus { K/ < R < K } we have |∇ i h | K − √− ∂∂R = O ( K − d ) , where we indicate that we measure derivatives with respect to K − √− ∂∂R . Since √− ∂∂R is uniformly equivalent to √− ∂∂r (once K is sufficiently large), thisimplies the result. (cid:3) We can now write down a metric ω V on V , which agrees with √− ∂∂r onthe set where R is sufficiently large. One way to do this is to consider the K¨ahlerpotential C ′ (1 + | x | ) α on C n for large C ′ . For sufficiently small α > r , but if C ′ is sufficiently large, then we can ensure that C ′ (1 + | x | ) α > r on the set where R < K/
2, say. We can now define a regularized maximum (seeDemailly [12, § g max n C ′ (1 + | x | ) α , r o , and let ω V = √− ∂∂ Φ | V . This defines a smooth metric on V , which equals √− ∂∂r where R is sufficiently large.The estimate (2.1) says that the Ricci potential h of ω V satisfies h ∈ C ∞− d interms of the weighted spaces used by Conlon-Hein [8]. Their Theorem 2.1 thenimplies that we can perturb ω V to a Calabi-Yau metric η V = √− ∂∂φ on V ,where the decay of φ − r can be controlled. Since by Lemma 3 we have d >
2, from[8, Theorem 2.1] it follows that we can ensure φ − r ∈ C ∞− c ( V ) for some c >
0. If d > c >
1. We can write this estimate in the followingform: there are constants C i , such that if x lies in the region where 1 / < R < λ · x ∈ V for some λ >
1, then(2.2) (cid:12)(cid:12)(cid:12) ∇ i h r − λ − φ ( λ · x ) i(cid:12)(cid:12)(cid:12) √− ∂∂R < C i λ − − c . Moreover by the uniqueness statment of [8, Theorem 2.1], φ is invariant under theaction of T . 3. The approximate solutions on C n We now consider X ⊂ C n +1 given by z + f ( x , . . . , x n ) = 0 , where f is the polynomial from Section 2. Recall that we have a Ricci flat conemetric on V = f − (0) ⊂ C n , whose distance function is r , and we have smoothlyextended r so that it is defined on C n \ { } and has degree 1 for the action ξ .The hypersurface X = C × V ⊂ C n +1 then has a Ricci flat cone metric √− ∂∂ ( | z | + r ). The corresponding homothetic scalings are given by the ac-tion with weights (1 , w , . . . , w n ). This metric is uniformly equivalent to √− ∂∂ρ ,where ρ = | z | + R , in terms of R from above.We would like to define a metric on X using the potential | z | + r just as we didfor V above, but now X and | z | + r are singular along C × { } . Our approach is G´ABOR SZ´EKELYHIDI to use | z | + r as the potential away from the singular rays, and a suitable scalingof the potential φ on V near the singular rays.Let us denote by γ ( s ) a cutoff function satisfying γ ( s ) = ( s >
20 if s < , and write γ = 1 − γ . We then define the approximate solution on X , at least onthe set where ρ > P for sufficiently large P , by(3.1) ω = √− ∂∂ (cid:16) | z | + γ ( Rρ − α ) r + γ ( Rρ − α ) | z | /d φ ( z − /d · x ) (cid:17) , where α ∈ (1 /d,
1) is to be chosen. As before we must choose a branch of log z todefine z − /d · x , however the value of φ ( z − /d · x ) is independent of this choice since φ is T -invariant. Note moreover that if z + f ( x ) = 0 , then 1 + f ( z − /d · x ) = 0 , and so z − /d · x ∈ V , where φ is defined. Writing φ ( x ) = r + φ − c ( x ) we have ω = √− ∂∂ (cid:16) | z | + r + γ ( Rρ − α ) | z | /d φ − c ( z − /d · x ) (cid:17) , and the estimate (2.2) implies that the term involving φ − c is of lower order than | z | + r . It follows from this that ω = √− ∂∂ Φ where the potential Φ has thesame growth rate as | z | + r = ρ . In particular if ω is positive definite on theset where ρ > P , we can argue as in the construction of ω V above, to construct ametric on X that agrees with ω where ρ > P .The holomorphic n -form Ω = dz ∧ dx ∧ . . . ∧ dx n ∂ x f restricts to a nowhere vanishing n -form on X as well as on X , and as in theprevious section we wish to estimate the Ricci potential of ω with respect to Ω. Wehave the following in analogy with Proposition 4. Proposition 5.
Fix α ∈ (1 /d, . The form ω defines a metric on the subset of X where ρ > P , for sufficiently large P . For suitable constants κ, C i > and weight δ < /d , the Ricci potential h of ω satisfies, for large ρ , |∇ i h | ω < C i ρ δ − − i if R > κρC i ρ δ R − − i if R ∈ ( κ − ρ /d , κρ ) C i ρ δ − /d − i/d if R < κ − ρ /d . If in addition d > and α is chosen sufficiently close to , then we can even choose δ < , i.e. in this case h decays faster than quadratically away from the singularrays in this case.Proof. The proof is similar to that of Proposition 4, but we scale our metric in adifferent way near the singular rays of X , and so we have several different regionsto study separately. EGENERATIONS OF C n AND CALABI-YAU METRICS 9
Region I : Consider the region where
R > κρ , and ρ ∈ ( D/ , D ) for some large D . Here we are uniformly far away from the singular rays. We study the scaledform D − ω , in terms of rescaled coordinates˜ z = D − z ˜ x = D − · x, and we let ˜ r = D − r (recall that here D − · x is defined using the action withweights w i ). On this region, once D is large enough, we have γ = 1 , γ = 0, andso D − ω = √− ∂∂ (cid:16) | ˜ z | + ˜ r (cid:17) . In addition in terms of these coordinates X has equation D − d ˜ z + f (˜ x ) = 0 , using that f has degree d , and so f ( D · ˜ x ) = D d f (˜ x ). Note that R ∈ ( κρ, ρ ), and r is uniformly equivalent to R , and in addition | ˜ z | <
2. It follows that we are inessentially the same situation as in the proof of Proposition 4. The same argumentsshow that |∇ i h | D − ω ≤ C i D − d , and so on this region |∇ i h | ω ≤ C i D − d − i . We can therefore choose any δ such that δ − > − d , i.e. δ > − d . If d > δ <
0, while if d >
2, then 3 − d < /d and so we can choose δ < /d .This implies the required estimates on this region. Region II : Suppose now that R ∈ ( K/ , K ) for some K < κρ , but K/ > ρ α .In addition let ρ ∈ ( D/ , D ). In this case ρ is comparable to | z | , and here we stillhave γ = 1 , γ = 0. We assume that for some fixed z we have | z − z | < K (sothat z will be in a unit ball centered at z after scaling). We now scale our formby K , and define(3.2) ˜ z = K − ( z − z ) , ˜ x = K − · x, ˜ r = K − r. In terms of these we have K − ω = √− ∂∂ (cid:16) | ˜ z | + ˜ r (cid:17) , and the equation of X is K − d ( K ˜ z + z ) + f (˜ x ) = 0 . In addition | ˜ z | <
1. We can still argue essentially like in Proposition 4, and now theerrors we obtain are of order K − d D since | ˜ z | < | z | ∼ D . The Ricci potentialtherefore satisfies |∇ i h | ω ≤ C i DK − d − i . Since d >
K > ρ α , we have DK − d K − − i < CD α (2 − d ) K − − i < Cρ α (2 − d ) R − − i , for suitable C . We need to choose δ so that 1 + α (2 − d ) < δ . If α is sufficientlyclose to 1, we can choose any δ > − d , which is the same constraint as in RegionI. If we merely have α > /d , then we still have 1 + α (2 − d ) < /d , as required. Region III : We now suppose that we are in the gluing region, so R ∈ ( K/ , K ),but K ∈ ( ρ α , ρ α ). Here the cutoff functions γ , γ are not constant. Suppose that ρ ∈ ( D/ , D ), and so | z | is comparable to D . We use the same scaling and changeof variables (3.2) as in Region II. We then have K − ω = √− ∂∂ (cid:16) | ˜ z | + γ ˜ r + γ K − | K ˜ z + z | /d φ (( K ˜ z + z ) − /d K · ˜ x ) (cid:17) . and the cutoff functions γ , γ have bounded derivatives in terms of these rescaledcoordinates. The equation of X is K − d ( K ˜ z + z ) + f (˜ x ) = 0 , as above. We still want to compare the metric to √− ∂∂ ( | ˜ z | + ˜ r )on X which has equation f (˜ x ) = 0. We use the estimate (2.2) to get that in thesecoordinates(3.3) ∇ i h K − | K ˜ z + z | /d φ (( K ˜ z + z ) − /d K · ˜ x ) − ˜ r i = O (cid:16) ( K − D /d ) c (cid:17) . In other words, in the Ricci potential we will have the same error K − d D as inRegion II from comparing the two equations, as well as a new error ( K − D /d ) c from the error in the K¨ahler potential. Since now K ∼ D α , this new term can beestimated as follows: ( K − D /d ) c < CD cd − cα K − . We need δ so that cd − cα < δ . Choosing α is sufficiently close to 1, we can chooseany δ > (2 + c ) /d − c . If in addition d >
3, then we have seen that we can choose c >
1, and so (2 + c ) /d − c <
0, and δ can be chosen negative. In general if weonly have α > /d , then we still have (2 + c ) /d − αc < /d , and so we can choose δ < /d . Region IV : We now consider R ∈ ( K/ , K ), but K ∈ ( κ − ρ /d , ρ α / γ = 0 , γ = 1. We suppose that ρ ∈ ( D/ , D ) and so | z | is comparable to D .We scale in the same way as in Regions II, III, so we make the change of variables(3.2). We have K − ω = √− ∂∂ (cid:16) | ˜ z | + K − | K ˜ z + z | /d φ (( K ˜ z + z ) − /d K · ˜ x ) (cid:17) , and the equation of X is still K − d ( K ˜ z + z ) + f (˜ x ) = 0 . Instead of comparing X to X , this time we want to compare X to the product C × V , scaled suitably. Note that K − d ( K ˜ z + z ) to leading order is K − d z , andso we compare X to the variety C × V K − d z with equation(3.4) K − d z + f (˜ x ) = 0 . The error introduced in passing from X to this variety is of order K − d (since | ˜ z | < EGENERATIONS OF C n AND CALABI-YAU METRICS 11
It remains to study the difference in the K¨ahler potentials. On the variety withequation (3.4) we have the metric √− ∂∂ ( | ˜ z | + K − | z | /d φ ( K | z | − /d · ˜ x )) , and so we need to estimate the difference E = K − | K ˜ z + z | /d φ (( K ˜ z + z ) − /d K · ˜ x ) − K − | z | /d φ ( K | z | − /d · ˜ x ) . Let us write φ = r + φ − c . By the homogeneity of r , we have(3.5) E = K − | K ˜ z + z | /d φ − c (( K ˜ z + z ) − /d K · ˜ x ) − K − | z | /d φ − c ( K | z | − /d · ˜ x ) . In addition since | ˜ z | < , | z | ∼ D and K ≪ D , we have(3.6) K ( K ˜ z + z ) − /d = z − /d K (cid:0) O ( KD − ) (cid:1) . Using (2.2) for φ − c , we therefore find that in the rescaled coordinates(3.7) |∇ i E | < C i ( | z | − /d K ) − − c D − K = O ( K − − c D cd − ) . Combining the error K − d from changing the equation of the variety with thiserror in the K¨ahler potentials, we find that the Ricci potential is of order K − d + K − − c D cd − . We therefore need to ensure that the choice of δ satisfies K − d + K − − c D cd − < CD δ K − . Suppose first that d >
3, so that also c >
1. Then K − d = K − d K − < CD /d − K − , and also K − − c D cd − = ( KD − /d ) − c D d − K − . Since 3 /d − <
0, we can choose δ < d >
2, so c >
0, then K − − c D cd − = ( KD − /d ) − c ( KD − ) D /d K − . Since for some C we have C − D /d < K < CD , it follows that we can choose δ < /d . Region V : Finally we suppose that
R < κ − ρ /d . Again, let z be very closeto z , with ρ , and therefore | z | comparable to D . We now scale by | z | /d , and sowe change variables to˜ z = z − /d ( z − z ) , ˜ x = z − /d · x, ˜ r = | z | − /d r, so that | ˜ z | , ˜ r < C . We have | z | − /d ω = √− ∂∂ h | ˜ z | + | z | − /d (cid:12)(cid:12) z /d ˜ z + z (cid:12)(cid:12) /d φ (cid:0) z /d ( z /d ˜ z + z ) − /d · ˜ x (cid:1)i , and the equation of X is z /d − ˜ z + 1 + f (˜ x ) = 0 . We want to compare this to C × V , with equation1 + f (˜ x ) = 0 , and metric √− ∂∂ (cid:2) | ˜ z | + φ (˜ x ) (cid:3) . The difference in the equations results in an error of order D /d − . For the K¨ahlerpotential note that z /d ( z /d ˜ z + z ) − /d = 1 + O ( D /d − ) , and so | z | − /d (cid:12)(cid:12) z /d ˜ z + z (cid:12)(cid:12) /d φ (cid:0) z /d ( z /d ˜ z + z ) − /d · ˜ x (cid:1) − φ (˜ x ) = O ( D /d − ) . In sum the Ricci potential is of order D /d − , and we need δ that satisfies D /d − < CD δ − /d , i.e. δ > /d −
1. In particular if d > δ < /d , while if d > δ < (cid:3) Weighted spaces on X As in the discussion before Proposition 5, we define a metric ω on all of X ,which agrees with the form defined in (3.1) for sufficiently large ρ . Our eventualgoal is to perturb the metric ω to a Calabi-Yau metric on the set { ρ > A } forsufficiently large A , at which point we will be able to apply Hein [17, Proposition4.1] to construct a global Calabi-Yau metric on X . The main difficulty is to invertthe linearized operator, which is the Laplacian on ( X , ω ).The analysis of the Laplacian on asymptotically conical spaces has been studiedextensively (see e.g. Lockhart-McOwen [23]), and was used in the work of Conlon-Hein [8] discussed in Section 2 above. The difference is that we now need to invertthe Laplacian in a more complicated weighted space that accounts for the singularrays in the tangent cone at infinity. This almost fits into the framework of QACspaces studied by Degeratu-Mazzeo [11], which in turn generalizes the work ofJoyce [21] on QALE manifolds, but unfortunately those results cannot be applieddirectly in our setting. The issue is that at distance D along the singular raysour metric ω is modeled on a product of C with a scaling of the metric ω V by afactor of D /d . In the QAC or QALE geometries there is no such additional scalefactor on the geometry “transverse” to the singular rays. Very recently, Conlon-Rochon [10] have extended the techniques of Degeratu-Mazzeo to such a moregeneral setup, however an additional difficulty in our setting is that we need toinvert the Laplacian on functions that decay more slowly than what they consider.Because of this, we follow a different approach, which has the added advantage thatour method applies with essentially no changes to constructing certain Calabi-Yaumetrics in a neighborhood of an isolated singularity, as we will see in Section 8.We first define weighted spaces C k,αδ,τ ( X , ω ), and then define C k,αδ,τ ( ρ − [ A, ∞ ) , ω )by restricting functions to the set where ρ ≥ A . The norm of a function on ρ − [ A, ∞ ) is defined to be the infimum of the corresponding norms of its extensionsto X .On the set where ρ < P for some fixed large P (we will have P < A ) we usethe usual H¨older norms. When ρ > P then we define the weighted spaces in termsof the functions ρ (which is essentially the radial distance), and R (which controls EGENERATIONS OF C n AND CALABI-YAU METRICS 13 the distance from the singular rays). To make the analogy with Degeratu-Mazzeo’sweighted spaces we define a smooth function w satisfying w = R > κρ,R/ ( κρ ) if R ∈ ( κ − ρ /d , κρ ) ,κ − ρ /d − if R < κ − ρ /d , for the same κ as in Proposition 5.Similarly to Degeratu-Mazzeo, we define the H¨older seminorm[ T ] ,γ = sup ρ ( z ) >K ρ ( z ) γ w ( z ) γ sup z ′ = z,z ′ ∈ B ( z,c ) | T ( z ) − T ( z ′ ) | d ( z, z ′ ) γ . Here c > X has bounded geometry on balls of radius c > T could be a tensor, in which case we compare T ( z ) with T ( z ′ ) using parallel transport along a geodesic.The weighted norm of f is then defined by(4.1) k f k C k,αδ,τ = k f k C k,α ( ρ< P ) + k X j =0 sup ρ ( z ) >P ρ − δ + j w − τ + j |∇ j f | + [ ρ − δ + k w − τ + k ∇ k f ] ,α . A more concise way to express these weighted norms is in terms of the conformalscaling ρ − w − ω . More precisely for this we should replace ρ by a smoothing ofmax { , ρ } . In terms of these weight functions our weighted norms could be definedequivalently as k f k C k,αδ,τ = k ρ − δ w − τ f k C k,αρ − w − ω . The estimate in Proposition 5 can now be stated as saying that the Ricci potential h is in the weighted space C k,αδ − , − , with δ as in the Proposition. Moreover since w ≤
1, we also have h ∈ C k,αδ − ,τ − for any τ < X , ω ) to suitablemodel spaces in different regions. This will be used to study the Laplacian on( X , ω ) in Section 6 below. We will write g, g X for the Riemannian metrics givenby ω, ω X respectively.Let us first consider the region U = { ρ > A, R > Λ ρ /d } ∩ X , for large A, Λ, and let G : U → X be the nearest point projection inside C n +1 , for the cone metric √− ∂∂ ( | z | + R ).We have G ( z, x ) = ( z, x ′ ) , where x ′ ∈ V is the nearest point projection of x ∈ C n under the metric √− ∂∂R . Proposition 6.
Given any ǫ > we can choose Λ > Λ( ǫ ) , and A > A ( ǫ ) sufficientlylarge so that on U we have |∇ i ( G ∗ g X − g ) | g < ǫw − i ρ − i , for i ≤ k + 1 . In particular in terms of our weighted spaces we have k G ∗ g X − g k C k,α , < ǫ. Proof.
The proof is very similar to the analysis of regions I,II, III, IV in the proofof Proposition 5. Let us first consider region I. Suppose that ρ ∈ ( D/ , D ), with D > A , and
R > κρ . Here we have w ∼
1, the notation a ∼ b meaning that C − < a/b < C for a constant C . The estimate that we need to show is equivalentto | D − ∇ i ( G ∗ g X − g ) | D − g < ǫ. We can work in the rescaled coordinates ˜ z, ˜ x as in Proposition 5. In these coor-dinates X is given by the equation D − d ˜ z + f (˜ x ) = 0, and X by the equation f (˜ x ) = 0. In Proposition 5 we were only interested in estimating the Ricci poten-tial, and so we compared the two hypersurfaces in local holomorphic charts (see theproof of Proposition 4). Now, however, we want a more global comparison, usingthe projection map G which is not holomorphic. This is closer to the approachtaken by Conlon-Hein [8]. The end result is the same, since in local charts thedifference between G and the identity map is of order D − d . Since d > ǫ by taking A large.Regions II, III follow similar calculations to the proof of Proposition 5. Let ustherefore consider region IV, which is a little different, since previously we comparedthis region to C × V , whereas now we are comparing it to X = C × V . Supposethat ρ ∈ ( D/ , D ), and R ∈ ( K/ , K ) with Λ ρ /d < K < ρ α /
2. In this region ρ iscomparable to | z | , and we suppose that z is close to some z , with | z | ∈ ( D/ , D ).Introducing coordinates ˜ z, ˜ x as before, the equation for X is K − d ( K ˜ z + z ) + f (˜ x ) = 0 , while the equation for X is f (˜ x ) = 0. As long as | ˜ z | <
1, the error introducedby orthogonal projection is of order K − d , which can be made arbitrarily small bychoosing D large.The two K¨ahler potentials that we need to compare are √− ∂∂ (cid:16) | ˜ z | + K − | K ˜ z + z | /d φ (cid:0) ( K ˜ z + z ) − /d K · ˜ x (cid:1)(cid:17) , and √− ∂∂ (cid:0) | ˜ z | + ˜ r (cid:1) . If | ˜ z | < | z | ∼ D , by the estimate (2.2) the difference between the K¨ahlerpotentials is of order ( KD − /d ) − c − < C Λ − c − . By choosing sufficiently large Λ, we can ensure that this is less than ǫ . (cid:3) Next, we focus on the region where ρ > A , but
R < Λ ρ /d . Fix z ∈ C with | z | > A , and a large constant B . Consider the region V ⊂ X given by points ( z, x )satisfying | z − z | < B | z | /d and where R < Λ ρ /d . Let us define new coordinatesˆ x = z − /d · x, ˆ z = z − /d ( z − z ) , EGENERATIONS OF C n AND CALABI-YAU METRICS 15 and let ˆ R = | z | − /d R . In addition we let ˆ ζ = max { , ˆ R } . Note that (ˆ z, ˆ x ) satisfythe equation(4.2) ˆ z /d − ˆ z + 1 + f (ˆ x ) = 0 , and | ˆ z | < B , | ˆ R | < C Λ for some fixed constant C (since ρ is comparable to | z | ).We define the map H : V → C × V by letting H (ˆ z, ˆ x ) = (ˆ z, ˆ x ′ ), where ˆ x ′ is the nearest point projection from thesolutions of (4.2) to solutions of 1 + f (ˆ x ) = 0. Proposition 7.
Given ǫ, Λ > if A > A ( ǫ, Λ , B ) , then we have |∇ i ( H ∗ g C × V − | z | − /d g ) | | z | − /d g < ǫ ˆ ζ − i , for i ≤ k + 1 . In terms of weighted spaces we then have k| z | /d H ∗ g C × V − g k C k,α , < ǫ. Proof.
This follows the analysis of our metric in regions IV, V in Proposition 5.Let us focus on region IV, which is the more complicated one. As before, we have ρ ∈ ( D/ , D ), R ∈ ( K/ , K ), such that κ − D /d < K < Λ D /d . Here | z | iscomparable to ρ , and ˆ ζ is comparable to ˆ R = | z | − /d R , so ˆ ζ ∼ | z | − /d K . Theestimate we need to show is therefore |∇ i ( | z | /d K − H ∗ g C × V − K − g ) | K − g < ǫ. We introduce the coordinates ˜ z, ˜ x as in Proposition 5, which in terms of ˆ x, ˆ z are˜ x = K − z /d · ˆ x, ˜ z = K − z /d ˆ z. In these coordinates X is given by K − d ( K ˜ z + z ) + f (˜ x ) = 0 . We need to compare the metric ω on X with the product metric C × V K − d z onthe hypersurface with equation K − d z + f (˜ x ) = 0 , under the closest point projection map. By the same calculations as in the analysisof region IV in Proposition 5 we find that the error between the two metrics is oforder BD /d − , and this can be made arbitrarily small by taking D large, since d > (cid:3) We will need an extension operator C ,αδ,τ ( ρ − [ A, ∞ ) , ω ) → C ,αδ,τ ( X , ω ). Moresophisticated methods could be used to deal with the C k,α spaces for k > k = 0 suffices. Proposition 8.
For sufficiently large A , there is a linear extension operator E : C ,αδ,τ ( ρ − [ A, ∞ ) , ω ) → C ,αδ,τ ( X , ω ) , whose norm is bounded independently of the choice of A . Proof.
The basic observation is that a C ,α function f on a half space R n + = { x n ≥ } ⊂ R n can be extended to a function on R n by reflection, while pre-serving the C ,α -norm. In addition, multiplying by a cutoff function, we can de-fine an extension E ( f ) supported in the set { x n > − } , such that k E ( f ) k C ,α ≤ C k f k C ,α ( R n + ) . The same applies if instead of a half space, f is defined on the set { x n ≥ F ( x , . . . , x n − ) } , where F is a C ,α function on R n − , and the norm of theextension operator will then also depend on the C ,α -norm of F .We can globalize this construction to extending functions from ρ − [ A, ∞ ) to X ,using that near each point in ρ − ( A ) we control the geometry of ω at suitable scales.To do this, let x ∈ ρ − ( A ) and consider the ball B ( x, r x ), where r x is defined asfollows: r x = κA/ , if R > κρ,R/ , if κ − ρ /d < R < κρ,A /d , if R < κ − ρ /d . These radii are chosen so that on B ( x, r x ) we have ρw ∼ r x . Let us analyze theseballs at the scale r x . • If at x we have R > κρ , then from the analysis of Region I in Proposition 5we know that the scaled down metric A − ω on B ( x, r x ) converges to thecone metric on X as A → ∞ on a ball of radius A − r x = κ/
10 in X .Moreover this ball is centered at a point ˜ x satisfying ρ (˜ x ) = 1, R (˜ x ) > κ .We therefore control the geometry of the boundary ρ − ( A ) uniformly, andwe can extend functions from ρ − ([ A, ∞ )) ∩ B ( x, r x ) to B ( x, r x ) as above.Note in addition that by the definition of the weighted norms we have k f k C ,αδ,τ ( B ( x,r x ) ,ω ) ∼ A δ k f k C ,α ( B ( x,r x ) ,A − ω ) , and so we can control the weighted H¨older norm of the extension. • If at x we have κ − ρ /d < R < κρ , and R ∈ ( K/ , K ), then from theanalysis of Regions II,III, IV in Proposition 5 we know that the scaleddown metric K − ω approaches the model metric on C × V s for some s with | s | ≤
1. In each of these spaces we control the geometry of ρ − ( A ), andso we can define an extension map. The weighted H¨older norms here arerelated to the norms with respect to the rescaled metric by k f k C ,αδ,τ ( B ( x,r x ) ,ω ) ∼ A δ − τ K τ k f k C ,α ( B ( x,r x ) ,K − ω ) . • In a similar way, if x is in the third region R < κ − ρ /d , then the rescaledmetric A − /d ω approaches the product metric C × V , and again we candefine an extension map. The relation between the weighted H¨older norm,and the H¨older norm with respect to the rescaled metric is k f k C ,αδ,τ ( B ( x,r x ) ,ω ) ∼ A δ − τ + τ/d k f k C ,α ( B ( x,r x ) ,A − /d ω ) . Finally, to globalize these local extensions we can use cutoff functions. (cid:3)
The next result shows that the tangent cone of ( X , ω ) at infinity is X = C × V . Proposition 9.
Let ǫ > . If D is sufficiently large, then the Gromov-Hausdorffdistance between the regions defined by ρ ∈ ( D/ , D ) in ( X , ω ) , and in X = C × V with the product metric ω , is less than Dǫ . EGENERATIONS OF C n AND CALABI-YAU METRICS 17
Proof.
It will be useful to introduce the notation S Λ = { x ∈ X : R < Λ ρ /d } , which we should think of as a region near the singular rays in X . Let us denote by X D , X D the two annular regions that we are considering, with their metrics scaleddown by a factor of D . Our goal is to define a map G : X D → X D , such that forsufficiently large D we have(4.3) | d ( G ( x ) , G ( x )) − d ( x , x ) | < ǫ, and the image G ( X D ) is ǫ -dense in X D .Let Λ be a large constant. On the set X D \ S Λ we define G by the nearest pointprojection, as in Proposition 6, while on the set X D ∩ S Λ we define G by projectiononto the z -axis. From Proposition 6 we know that on X D \ S Λ the rescaled metrics D − ω and D − G ∗ ω can be made arbitrarily close by choosing Λ , D large. We cantherefore assume that for any curve γ in this region(4.4) | length D − ω ( γ ) − length D − G ∗ ω ( γ ) | < ǫ, and in particular if x , x are in this region, then d X D ( x , x ) < d X D ( G ( x ) , G ( x )) + ǫ. Note that the reverse inequality is not yet clear since the shortest curve between x , x in X D may pass through S Λ .At the same time, on X D ∩ S , Proposition 7 shows that after suitable scalingswe can approximate X D by the product metric C × V . In particular, choosing firstΛ and then D sufficiently large, we can assume that the projection π : X D ∩ S → C onto the C factor satisfies | dπ | < ǫ . It follows from this that for any curve γ inthis region(4.5) length X D ( γ ) ≥ (1 − ǫ )length C ( π ◦ γ ) . Note that if x , x ∈ S Λ , then the shortest curve between them in X D will remainin the region S since on the “annular” region S \ S Λ the metric can be madearbitrarily close to the cone X D (if Λ and D are sufficiently large). It follows using(4.5) that d X D ( x , x ) > d C ( π ( x ) , π ( x )) − ǫ. In addition if we fix a reference point o ∈ V , then for any ( p, z ) ∈ X D ∩ S Λ wehave(4.6) d X D (( p, z ) , ( o, z )) < C Λ D /d − < ǫ, for some constant C (recall that we choose D after choosing Λ). It then followsthat if ( p , z ) , ( p , z ) are two points in this region, we have d X D (( p , z ) , ( p , z )) < | z − z | + ǫ, and similarly d X D (( p , z ) , ( p , z )) < | z − z | + ǫ. We have therefore shown that if x , x ∈ S Λ , then | d X D ( x , x ) − d X D ( G ( x ) , G ( x )) | < ǫ. It remains to bound the distance from below between points x , x ∈ X D \ S Λ .Let γ be the shortest curve from x to x . We only have to deal with the possibility that this shortest curve enters the closure of the region S Λ . Let x ′ be the first, and x ′ be the last point along the curve in this region. Then by the observations above,the segment of γ joining x ′ , x ′ must lie in S , and therefore we have d X D ( x ′ , x ′ ) > d C ( π ( x ′ ) , π ( x ′ )) − ǫ > d X D ( G ( x ′ ) , G ( x ′ )) − ǫ. Then by the triangle inequality and the estimate (4.4) we get d X D ( x , x ) > d X D ( G ( x ) , ( x )) − ǫ for a larger value of ǫ . This shows (4.3). The fact that G ( X D ) is ǫ -dense in X D follows from the inequality analogous to (4.6) for X D . (cid:3) It follows from this result in particular that if o ∈ X is a fixed basepoint, thenthe distance function d ( o, · ) is uniformly equivalent to ρ . Later on we will alsoneed that ( X , ω ) satisfies the following “relatively connected annuli”, or RCA,condition. This is an easy consequence of the fact that the tangent cone at infinityof ( X , ω ) is a metric cone over a compact connected length space. Proposition 10.
For sufficiently large D , any two points x , x ∈ X with d ( o, x i ) = D can be joined by a curve of length at most CD , lying in the annulus B ( o, CD ) \ B ( o, C − D ) , for a uniform constant C . The Laplacian on the model spaces
Here we consider two model problems, namely the Laplacians on the cone X = C × V , and on the product C × V . These will be the building blocks for invertingthe Laplacian on X in Section 6. As a preliminary step we consider the Laplacianon the product C × V , but using weighted spaces which involve a weight functiononly in the V factor.5.1. The model space C × V . Consider the product metric g = g C + g V , andlet r be the distance function on V as before. We define weighted spaces on C × V in the usual way, analogously to (4.1) with the weight function given by r . A moreconcise definition is obtained by conformally scaling g to r − g . In terms of thisscaled metric our weighted spaces can be written as C k,ατ = r τ C k,αr − g , with corresponding norm given by k f k C k,ατ = k r − τ f k C k,αr − g . Our goal is to show that the kernel of the Laplacian on ( C × V , g ) is trivial inthe weighted H¨older space C k,ατ , for τ ∈ (4 − n, V is m = 2 n −
2, so τ ∈ (2 − m,
0) is the usual “good” range of weights for theLaplacian on V . We use the Fourier transform in the C direction in a similar wayto what was done by Walpuski [33] which in turn is based on Brendle [2] (see alsoMazzeo-Pacard [25]), although the details will be slightly different.For any function χ on C × V with sufficient decay in the C direction we definethe Fourier transform ˆ χ ( ξ, x ) = Z C χ ( z, x ) e −√− ξ · z dz, where we think of C as R . EGENERATIONS OF C n AND CALABI-YAU METRICS 19
Proposition 11.
Let χ be a smooth function on C × V , such that ˆ χ has compactsupport away from { } × V . In particular χ is supported in C × K for a compact K ⊂ V . We can then find f solving ∆ f = χ , and moreover (1) f ∈ C k,ατ for all τ ∈ (2 − m, , (2) In addition f decays exponentially in the C direction. More precisely forany a > there is a constant C such that k f k C k,ατ ( | z | >A ) < C (1 + A ) − a forany A > and τ ∈ (2 − m, .Proof. Taking the Fourier transform, the equation that we need to solve is(5.1) ∆ V ˆ f ( ξ, x ) − | ξ | ˆ f ( ξ, x ) = ˆ χ ( ξ, x ) . For each fixed ξ = 0 we can solve this in polar coordinates on V , using the spectraldecomposition for the link of V . It reduces to analyzing ODEs of the form(5.2) ∂ r ˆ f + m − r ∂ r ˆ f + 1 r λ ˆ f − | ξ | ˆ f = ˆ χ λ , where λ ≤ V . Note that theˆ χ λ have compact support in r . For ξ = 0 the ODE has a fundamental solutionwhich decays exponentially as r → ∞ , and is bounded near r = 0. Using this weobtain solutions ˆ f ( ξ, x ) of (5.1) decaying exponentially as r → ∞ , and in particularsatisfying bounds k ˆ f ( ξ, · ) k C τ ≤ C k ˆ χ ( ξ, · ) k C , where C is uniform as long as ξ isbounded.Our results follow by taking the inverse Fourier transform of ˆ f . Item (1) abovefollows from the properties of the ODE solutions. Item (2) follows from the factthat if ˆ χ has compact support away from { } × V , then ˆ f is smooth when viewedas a function R → C τ ( V ). This can be seen by differentiating (5.1) with respectto ξ . We find that any derivative ∂ lξ ˆ f satisfies an equation of the form∆ V ∂ lξ ˆ f ( ξ, x ) − | ξ | ∂ lξ = ∂ lξ ˆ χ ( ξ, x ) + X | i | < | l | a i ( ξ ) ∂ iξ ˆ f ( ξ, x )in terms of lower order derivatives. Inductively we find that each partial derivative ∂ lξ ˆ f ( ξ, x ) is bounded near r = 0 and decays exponentially as r → ∞ , using thatˆ f ( ξ, x ) is identically zero for ξ in a neighborhood of 0.Taking the invese Fourier transform we obtain exponential decay in | z | of the C τ norms of f in the vertical slices { z } × V . The decay of the derivatives of f isobtained from this using Schauder estimates since χ is compactly supported in the V direction. (cid:3) Corollary 12.
Suppose that ∆ f = 0 , for some f ∈ C k,ατ ( C × V ) with τ ∈ (2 − m, .Then f = 0 .Proof. We will argue by taking the distributional Fourier transform of f , and show-ing that it is supported at the origin. It will be more convenient to use weighted L -spaces than the H¨older spaces. For this let τ , τ be such that τ < τ < τ , anddefine a weight function σ on V such that σ = r τ for large r , and σ = r τ forsmall r . We define the weighted L σ space on V using the norm k F k L σ = Z V | F | σ − r − m dV. Our choice of τ , τ ensures that we can view f as a bounded map f : C → L σ ( V ) . In addition the dual space under the L pairing is L σ ′ , where σ ′ = σ − r − m . Let ˆ f be the Fourier transform of f in the C -direction as above, so now ˆ f is a distributionvalued in L σ . If g is a smooth map C → L σ ′ of compact support, then the pairingˆ f ( g ) is defined as ˆ f ( g ) = Z C h f, ˆ g i dz. We first claim that ˆ f is a distribution of finite order (in fact order at most 4).For this suppose that g : C → L σ ′ has compact support K , and satisfies |∇ i g | ≤ A for i ≤
4, where the derivatives are in the C -direction. Then the usual integrationby parts argument shows that k ˆ g ( ξ ) k ≤ C K,A (1 + | ξ | ) − , for a constant dependingon K, A , and so | ˆ f ( g ) | ≤ Z C k f ( z, · ) k L σ k ˆ g ( z ) k L σ ′ dz ≤ C K,A , since f is bounded.Next we claim that ˆ f is supported at the origin, i.e. if g is smooth (in the C -directions) and has compact support away from the origin, then ˆ f ( g ) = 0. We canapproximate g in the space C ( C , L σ ′ ) with smooth functions that have compactsupport away from { } × V , and so we assume that g is of this type. We can thenapply Proposition 11 to the function χ = ˆ g , and so we find h on C × V satisfying∆ h = ˆ g , such that h ∈ C k,ατ , and h decays exponentially in the C -direction. Itfollows that ˆ f ( g ) = Z C × V f ˆ g dV = Z C × V f ∆ h dV = Z C × V h ∆ f dV = 0 , where the integration by parts is justified by the decay properties of h .Since ˆ f is supported at the origin, it follows that ˆ f is a linear combination ofderivatives of delta functions at the origin, with coefficients in L σ . This follows theusual argument from the scalar case (see Rudin [26, Theorem 6.25]). Therefore wehave f ( z ) = l X i,j =1 z i ¯ z j f ij , for f ij ∈ L σ . Since f is bounded, we have f ij = 0 unless i = j = 0, and so f isindependent of the z -variable. The result now follows since the ODEs (5.2) with ξ = 0 , ˆ χ λ = 0 have no solutions with growth rates in the range (2 − m, (cid:3) EGENERATIONS OF C n AND CALABI-YAU METRICS 21
The model space X . We now consider the cone X , or rather X \ C × { } .In polar coordinates the corresponding metric is g X = dr + r h L , where h L is the metric on the link L . The link ( L, h L ) is incomplete (since weremove the singular rays from X ), and its metric completion L is the doublesuspension of the link of the cone V , since X = C × V . In particular L hasa circle of singularities modeled on the cone V . In the language of [11], L is asmoothly stratified space of depth 1, and h L is an iterated edge metric.We define weighted H¨older norms on ( L, h L ) in terms of the weight function w given by smoothing out the distance from the singular stratum. As above, theweighted norms can be expressed concisely in terms of the conformally scaled metric w − h L : k f k C k,ατ = k w − τ f k C k,αw − hL . The norm depends on the precise smoothing chosen, but we end up with equivalentnorms. The main result about the Laplacian on L that we need is the following. Proposition 13.
Let a ∈ C , and consider the map ∆ h L + a : C k,ατ ( L ) → C k − ,ατ − ( L ) , for τ ∈ (2 − m, (recall m = 2 n − is the real dimension of V ). If Im a = 0 , orif a ∈ R avoids a discrete set of values, then ∆ + a is invertible.Proof. This follows from the Fredholm theory for edge operators of Mazzeo [24].It implies that for our range of weights the image of ∆ + a is L -orthogonal to itskernel, and moreover ∆ has discrete real spectrum. (cid:3) We now consider the cone X over L . It is more convenient to conformally scalethe cone metric g X to a cylinder, using the radial distance function r :˜ g X = r − g X = dt + h L , where t = ln r . The equation ∆ g X f = u is then equivalent to∆ h L f + ∂ t f + (2 n − ∂ t f = e t u. The relevant weighted H¨older norms, compatible with the definition (4.1) can beformulated in terms of the further conformal scaling w − ˜ g X , using the weightfunction w on L above. We then define k f k C k,αδ,τ = k e − δt w − τ f k C k,αw − gX . Our goal is to show that the map L = ∆ h L + ∂ t + (2 n − ∂ t : C k,αδ,τ ( R × L ) → C k − ,αδ,τ − ( R × L )is invertible. It is convenient to conjugate the operator by e δt , to reduce to the casewhen δ = 0. Writing L δ ( f ) = e δt L ( e − δt f ), we have L δ = ∆ h L + ∂ t + (2 n − − δ ) ∂ t + ( δ − (2 n − δ ) , and our main result is the following. Proposition 14.
For δ avoiding a discrete set of indicial roots, and τ ∈ (2 − m, ,the operator L δ : C k,ατ ( R × L ) → C k − ,ατ − ( R × L ) is invertible. Here we write C k,ατ = C k,α ,τ for simplicity. We first have the following result, analogous to Proposition 11, except we do notneed the support to avoid { } × L . Proposition 15.
Suppose that δ avoids a discrete set of indicial roots. Let χ besmooth on R × L , such that ˆ χ has compact support. We can then find f solving L δ f = χ , such that (1) f ∈ C k,ατ for any τ < , (2) For any a > there is a constant C such that k f ( t, x ) k C k,ατ ( | t | >A ) < C (1 + A ) − a for any A > .The same applies to the operator L ∗ δ = ∆ h L + ∂ t − (2 n − − δ ) ∂ t + ( δ − (2 n − δ ) , which is the adjoint of L δ .Proof. The proof is similar to that of Proposition 11. Taking the Fourier transformin the t variable we need to solve the equations∆ h L ˆ f − (cid:2) ξ ˆ f − √− ξ (2 n − − δ ) − δ + (2 n − δ (cid:3) ˆ f = ˆ χ. This equation is of the form (∆ h L + a ) ˆ f = ˆ χ . Suppose that δ = n −
1. If ξ = 0then we have Im a = 0, while if ξ = 0, then a = δ − (2 n − δ . Therefore if δ isgeneric, we can apply Proposition 13 to find ˆ f no matter what ξ is. In addition,since C k,ατ ⊂ C k,ατ ′ if τ > τ ′ , we have that f ∈ C τ ′ for any τ ′ <
0. The Schauderestimates then imply that f ∈ C kτ ′ for any k > f is a smooth function of ξ . This can be seen bydifferentiating the equation as we did in Proposition 11. Inductively we find thateach derivative ∂ lξ ˆ f satisfies an equation of the form∆ h L ∂ lξ ˆ f + a ( ξ ) ∂ lξ ˆ f = g ( ξ ) , where g ( ξ ) has compact support in ξ , and g ( ξ ) ∈ C k,ατ for all τ <
0. In addition a ( ξ ) is such that Proposition 13 applies. In this way we can bound the ξ -derivativesof ˆ f , and so the inverse Fourier transform f will have the required decay. (cid:3) This has the following corollary.
Corollary 16.
Suppose that f ∈ C k,ατ satisfies L δ f = 0 for some generic δ , and τ ∈ (2 − m, . Then f = 0 .Proof. Suppose that f is nonzero. We can then find χ with ˆ χ having compactsupport such that R f χ = 0. We apply Proposition 15 to solve L ∗ δ h = χ , with h ∈ C k,ατ ′ , where τ ′ is chosen so that 2 − m − τ < τ ′ <
0. We then have Z R × L f χ dV = Z R × L f L ∗ δ h dV = Z R × L ( L δ f ) h dV = 0 , EGENERATIONS OF C n AND CALABI-YAU METRICS 23 where the integration by parts is justified by the decay properties of h and thechoice of τ ′ . This contradicts our choice of χ . (cid:3) We next use a standard blowup argument to obtain the following.
Proposition 17.
Let τ ∈ (2 − m, , and δ generic. There exists a constant C such that for any f ∈ C k,ατ ( R × L ) we have k f k C k,ατ ≤ C kL δ f k C k − ,ατ − . In particular L δ has closed range.Proof. Note first that since the metric w − ˜ g X has bounded geometry, we can usethe Schauder estimates to obtain(5.3) k f k C k,ατ ≤ C ( kL δ f k C k − ,ατ − + k f k C τ ) . If follows that it is enough to prove that k f k C τ ≤ C kL δ f k C k − ,ατ − for a uniform constant C .We argue by contradiction. Suppose that we have a sequence of functions f i ∈ C k,ατ with k f i k C τ = 1, but kL δ f i k C k − ,ατ − < /i . We can find points ( t i , x i ) ∈ R × L such that | f i ( t i , x i ) | > w ( x i ) τ . By translating in the t direction we can assume that t i = 0 for each i .There are two possibilities. If w ( x i ) is bounded away from zero, then by choosinga subsequence we can assume that x i → x for some x ∈ L . The Schauder estimate(5.3) implies that choosing a further subsequence we can assume that the f i convergelocally in C k,α ′ to a limit f ∈ C k,ατ , which then must satisfy L δ f = 0. Corollary 16implies that f = 0, which contradicts that | f ( x ) | ≥ w ( x ) τ .The other possibility is that w ( x i ) →
0. Consider the rescaled metrics w ( x i ) − ˜ g X .If we take the pointed limit of R × L with these rescaled metrics, based at the points(0 , x i ), then (up to choosing a subsequence) we obtain the limit space R × V , withthe product metric ω Euc + ω V , with basepoint (0 , x ) for some x ∈ V at distance1 from the vertex. This is just the statement that h L is modeled on S × V nearthe circle of singularities.Under taking this pointed limit the rescaled functions w ( x i ) − τ f k converge locallyin C k,α ′ to a limit f ∈ C k,ατ ( R × V ), satisfying ∆ f = 0. Corollary 12 implies that f = 0, contradicting | f (0 , x ) | ≥ / (cid:3) We can finally prove Proposition 14.
Proof of Proposition 14.
It is enough to show that L δ : C k,ατ ( R × L ) → C k − ,ατ − ( R × L )is surjective, since Corollary 16 implies that it is injective. While we already knowthat the image is closed, and moreover Proposition 15 provides us with many ele-ments in the image, these functions do not form a dense set in C k − ,ατ − , so we cannotimmediately conclude. Instead let u ∈ C k − ,ατ − ( R × L ). We can find a sequence of smooth χ i such thatˆ χ i has compact support, χ i → u locally uniformly and moreover k χ i k C k − ,ατ − < C for a constant C depending on u . By Proposition 15 we can find f i ∈ C k,ατ ( R × L )such that L δ f i = χ i , and by Proposition 17 we have k f i k C k,ατ < C, for C independent of i . Up to choosing a subsequence we can take a limit f i → f ,with the convergence holding locally in C k,α ′ , and such that k f k C k,ατ ≤ C . Thelimit satisfies L δ f = u , and so L δ is surjective. (cid:3) The model space C × V . Let us now move on to the Laplacian on C × V .Here the relevant weighted spaces are defined in terms of the weight function ζ on V , which is a smoothed out version of max { , d ( · , o ) } for a point o ∈ V . As above,the weighted H¨older spaces can be defined in terms of a conformally scaled versionof the product metric g = g Euc + g V : k f k C k,ατ = k ζ − τ f k C k,αζ − g . The weighted norms on V are defined analogously.We first have the following result, analogous to Proposition 13. Proposition 18.
Let τ ∈ (2 − m, , λ ≥ , and let u be smooth with compactsupport on V . We can find a smooth function f on V such that (5.4) ∆ V f − λf = u, and in addition we have an estimate | f | ≤ Cζ τ for a constant that depends on k u k C τ − and τ , but not on λ . Recall that here m = 2 n − is the dimension of V .Proof. When λ = 0, this follows from the standard theory for the Laplacian actingin weighted spaces on the asymptotically conical manifold V (see e.g. Lockhart-McOwen [23]): ∆ V : C k,ατ ( V ) → C k − ,ατ − ( V ) . For our choice of weight the Laplacian is self-adjoint, and moreover any decayingelement in f ∈ ker ∆ V decays at the rate of at least d ( · , o ) − m . It follows byintegration by parts that f is constant, but since it decays, we have in fact f = 0.Hence ∆ V is invertible.When λ >
0, then we can also solve Equation (5.4) using that ∆ V − λ is anessentially self-adjoint operator on L , whose kernel is trivial. Moreover using theSchauder estimates it follows that the solution f decays faster than any inversepower of ζ . What remains is to obtain a uniform estimate for this decay, indepen-dent of λ (in particular as λ → b ∈ C k,ατ be a solution of∆ V b = − ζ τ − . By the maximum principle we have that b >
0. We claim that for sufficiently large C , depending on k u k C τ − , we have an estimate | f | < Cb , independent of λ . Let usshow that f < Cb : if this estimate were to fail, then the function f − Cb would EGENERATIONS OF C n AND CALABI-YAU METRICS 25 achieve a maximum at some point x max , using the fast decay of f . In particular f ( x max ) >
0, and at the same time by the maximum principle0 ≥ ∆( f − Cb )( x max )= u ( x max ) + λf ( x max ) + Cζ τ − ( x max ) > u ( x max ) + Cζ τ − ( x max ) . If C is chosen large depending on k u k C τ − , this is a contradiction. In a similar wayone can prove that f > − Cb for the same C . (cid:3) The next result is analogous to Proposition 15.
Proposition 19.
Suppose that χ is a smooth function on C × V such that ˆ χ hascompact support. We can then solve the equation ∆ f = χ such that (1) f ∈ C k,ατ for any τ > − m , (2) If ˆ χ is supported away from { } × V , then in addition for any a > thereis a constant C such that k f k C k,ατ ( | z | >A ) < C (1 + A ) − a for any A > .Proof. The proof is similar to the proofs of Propositions 11, 15. After Fouriertransforming, the relevant equations are(5.5) ∆ V ˆ f − | ξ | ˆ f = ˆ χ. Proposition 18 implies that we can solve these equations with uniform estimates on k ˆ f k C τ . Taking the inverse Fourier transform we obtain a solution of ∆ f = χ , with f ∈ C τ . The Schauder estimates then imply that f ∈ C k,ατ .In order to get the decay property (2), we can argue as in Propositions 11, 15,by differentiating Equation (5.5). For ξ = 0, the solutions of (5.5) decay faster thanany inverse power of ζ in the V -direction, and so inductively we find that eachderivative ∂ lξ ˆ f has the same decay. Therefore ˆ f is a smooth function C → C τ ( V )with compact support, and from this we obtain the required decay for f in the C direction. (cid:3) We can next follow the proofs of Corollary 12 and Proposition 17 closely to prove
Proposition 20.
Let τ ∈ (2 − m, . There exists a constant C such that for any f ∈ C k,ατ ( C × V ) we have k f k C k,ατ ≤ C k ∆ f k C k − ,ατ − . Finally we can prove the following.
Proposition 21.
The Laplacian ∆ C × V : C k,ατ ( C × V ) → C k − ,ατ − ( C × V ) is invertible for τ ∈ (2 − m, .Proof. The proof of this, based on Corollary 12 and Propositions 19 and 20 isessentially identical to the proof of Proposition 14. (cid:3) Inverting the Laplacian
In this section we study the mapping properties of the Laplacian in the weightedspaces that we have defined. The main result is the following.
Proposition 22.
Suppose that we choose τ ∈ (4 − n, and δ avoids a discreteset of indicial roots. For sufficiently large A > the Laplacian ∆ : C ,αδ,τ ( ρ − [ A, ∞ ) , ω ) → C ,αδ − ,τ − ( ρ − [ A, ∞ ) , ω ) is surjective with inverse bounded independently of A . The proof of this result will take up the remainder of this section. The strategy isto construct an approximate inverse for the Laplacian by localizing the problem onthe different regions of ρ − [ A, ∞ ) for sufficiently large A , studied in Propositions 6and 7, and then using the inverses constructed on corresponding model spaces inSection 5.Suppose that we have a function u ∈ C ,αδ − ,τ − ( ρ − [ A, ∞ ) , ω ), with norm k u k C ,αδ − ,τ − <
1. The goal is to construct, once A is sufficiently large, a function f = P u on X with k f k C ,αδ,τ ( X ,ω ) < C for a uniform C such that k ∆ f − u k C ,αδ − ,τ − ( ρ − [ A, ∞ ) ,ω ) < . Then the operator P is an approximate inverse for ∆, so ∆ P is invertible, and asa consequence ∆ has a bounded right inverse.Using the extension map defined in Proposition 8 we can assume that u is actuallydefined on all of X and k u k C ,αδ − ,τ − ( X ,ω ) < C, for a constant C independent of A . We will decompose u using cutoff functionsinto various different pieces. Let us recall the cutoff functions γ , γ from before,so that γ ( s ) is supported where s >
1, and γ + γ = 1. Let us choose a largenumber Λ, and write u = u + u , where u i = γ i ( R Λ − ρ − /d ) u. So u is supported on the set where R > Λ ρ /d . Let us use the notation U = { R > Λ ρ /d } ∩ { ρ > A } from Proposition 6. We then have a map G : U → X such that(6.1) k G ∗ ω X − ω k C k,α , < ǫ. We can use the map G to view u (at least on the set where ρ > A ) as a functionon X , supported away from C ×
0. Proposition 14 implies
Proposition 23.
As long as τ ∈ (4 − n, , and δ avoids a discrete set of indicialroots, the Laplacian ∆ X : C k,αδ,τ ( X ) → C k − ,αδ − ,τ − ( X ) is invertible. We can therefore define
P u on X satisfying ∆ X P u = u , and satisfying theestimate k P u k C ,αδ,τ ( X ) < C, EGENERATIONS OF C n AND CALABI-YAU METRICS 27 for a uniform C . In order to transfer this function back to X , we use anothercutoff function β = γ (cid:18) ln( R Λ − / ρ − /d )ln Λ / (cid:19) . This has the property that β = 0 on the region where R < Λ / ρ /d , while β = 1on the support of γ ( R Λ − ρ − /d ), which is where u is supported. Moreover in ourweighted spaces we have the estimate(6.2) k∇ β k C k,α − , − ( ρ − (1 , ∞ ) ∩ X ) < C ln Λ . We have ∆ X ( β P u ) = u + 2 ∇ β · ∇ ( P u ) + (∆ X β ) P u , and so using the multiplication properties k f g k C k,αa + b,c + d ≤ C k f k C k,αa,c k g k C k,αb,d together with (6.2) we obtain(6.3) k ∆ X ( β P u ) − u k C ,αδ − ,τ − ( ρ − (1 , ∞ ) ∩ X ) < C ln Λ . Using the map G again to view β P u as a function on X , and using (6.1) tocompare the Laplacians on X and X we find that(6.4) k ∆ ω ( β P u ) − u k C ,αδ − ,τ − ( ρ − [ A, ∞ )) < ǫ, once Λ and A are sufficiently large.We next need to examine the piece u , which is supported in the region where R < ρ /d , and we are assuming that in addition ρ > A . Note that here we have ρ ∼ | z | . Geometrically this region can be thought of roughly as a fibration over theset {| z | > A } ⊂ C , whose fiber over z is the region { r < } ⊂ V , scaled downby a factor of | z | /d . We decompose u into pieces whose supports are localized inthe z -plane. Proposition 7 tells us that suitably scaled, on these regions we canapproximate our space with a corresponding region in the product C × V . ByProposition 21, we can invert the Laplacian there.Let us choose a large B >
0. We construct cutoff functions χ i on C suchthat P χ i = 1 on the set where | z | > A as follows. Consider ( C , ˜ g ), where ˜ g = B − | z | − /d g Euc is a conformal scaling of the Euclidean metric. Since d >
1, on theset where | z | > A for sufficiently large A , this metric is close to being Euclideanon larger and larger scales. We can then cover this region with disks of radius 2(in the metric ˜ g ) centered at points z i , such that the corresponding disks of radius1 are disjoint, and define the cutoff funtions χ i supported in the disks of radius 2,and equal to 1 on the disks of radius 1. Scaling back the metric on C we have that χ i = 1 on the ball of radius B | z i | /d around z i , and χ i = 0 outside of the ball ofradius 2 B | z i | /d around z i . In addition |∇ l χ i | g Euc = O ( B − l | z i | − l/d ) for all l ≥ χ i in a similar way, which we will use totransfer our local solutions back to X (analogous to β above). The ˜ χ i equal 1 onthe support of χ i , and are supported in the balls of radius 3 B | z i | /d . Furthermore |∇ l ˜ χ i | = O ( B − l | z i | − l/d ) as well. An additional important property of these cutofffunctions, which can be seen more clearly in terms of the conformally scaled metric ˜ g , is that any z with | z | > A is in the support of only a fixed bounded number N of the ˜ χ i ( N is independent of the choices of large B, A ).We now decompose u into the pieces u = P i χ i u , at least on the region where | z | > A . By construction the function χ i u is supported on a region where z ∈ B ( z i , B | z i | /d ) for a point z i ∈ C , and in addition R < | z | /d . By Proposition 7,on this region the scaled metric | z i | − /d ω can be approximated by the productmetric ω C × ω V on C × V , using the map H . Here we have the following resultfrom Section 5. Proposition 24.
The Laplacian ∆ C × V : C k,ατ ( C × V ) → C k − ,ατ − ( C × V ) is invertible for τ ∈ (4 − n, . In order to apply this, we need to view χ i u as a function on C × V , using themap H , and relate its weighted norm in C ,ατ − on C × V to the norm in C ,αδ − ,τ − on X . For this, note that on our region we have ρ ∼ | z i | , and so by the definitionof the weight function w we have ρ − w − ∼ max {| z i | /d , R } − ρ δ − w τ − ∼ | z i | δ − τ max {| z i | /d , R } τ − . At the same time the weight function ˆ ζ used to define the weighted spaces on C × V (using the notation from Proposition 7), is comparable to max { , | z i | − /d R } . Itfollows that the estimate k u k C ,αδ − ,τ − < C on X translates to(6.5) k u k C ,ατ − ( C × V ) < C | z i | δ − τ + τ − d on the support of χ i . Note also that by construction, on the support of u we have(6.6) |∇ l χ i | < CB − l < CB − l Λ l ˆ ζ − l , thinking of χ i as a function on C × V and using the Euclidean metric on C (sinceˆ ζ < C Λ). It follows that once B is sufficiently large, depending on Λ, the estimate(6.5) implies k χ i u k C ,ατ − ( C × V ) < C | z i | δ − τ + τ − d . We now apply Proposition 24, but note that ∆ | z i | − /d ω = | z i | /d ∆ ω . We thereforeuse the Proposition to define P ( χ i u ) by∆ C × V P ( χ i u ) = | z i | /d χ i u , satisfying the bound k P ( χ i u ) k C ,ατ < C | z i | δ − τ + τd . We need to transfer this function back to the manifold X . Note that in terms ofthe coordinate ˆ z i from Proposition 7 on the C factor and the distance function ˆ ζ on V , the function χ i u is supported in the region where | ˆ z i | < B and ˆ ζ < χ i from above, which equal 1 on the supports of χ i ,and are supported, in these coordinates, where | ˆ z i | < B . In addition we use thecutoff function β = γ ln(ˆ ζ − Λ − )ln Λ ! , EGENERATIONS OF C n AND CALABI-YAU METRICS 29 which equals 1 where ˆ ζ < ζ > and has the property thatin our weighted spaces(6.7) k∇ β k C k,α − < C ln Λ . We now need an estimate analogous to (6.3) for the difference∆ C × V ( ˜ χ i β P ( χ i u )) − | z i | /d χ i u = 2 ∇ ( β ˜ χ i ) · ∇ P ( χ i u ) + ∆ C × V ( β ˜ χ i ) P ( χ i u ) . Using the estimate (6.7) for β and an estimate analogous to (6.6) for ˜ χ i (with Λreplaced by Λ and B chosen correspondingly larger), we find that k ∆ C × V ( ˜ χ i β P ( χ i u )) − | z i | /d χ i u k C ,ατ − ( C × V ) < Cǫ | z i | δ − τ + τd . Proposition 7 allows us to estimate the difference between | z | − /d ω and the productmetric on C × V under the identification using the map H , and this leads to k ∆ | z i | − /d ω ( ˜ χ i β P ( χ i u )) − | z i | /d χ i u k C ,ατ − ( C × V ) < Cǫ | z i | δ − τ + τd . Dividing through by | z i | /d , and translating the estimate back to our weightedspaces on X , we get(6.8) k ∆ ω ( ˜ χ i β P ( χ i u )) − χ i u k C ,αδ − ,τ − < Cǫ. Finally we define
P u = β P u + X i β ˜ χ i P ( χ i u ) . We use the estimates (6.4) and (6.8), together with the observation that any givenpoint is contained in at most a fixed number of our regions, to deduce that k ∆( P u ) − u k C ,αδ − ,τ − < Cǫ < / , for sufficiently small ǫ , which by the above discussion we can achieve by first choos-ing Λ, then B , and finally A sufficiently large. In this case ∆ P is invertible, with k (∆ P ) − k <
2. Since by construction the operator P has bounded norm indepen-dent of A , we have constructed a right inverse P (∆ P ) − for ∆ for sufficiently large A , with norm independent of A . This completes the proof of Proposition 22.7. Calabi-Yau metrics on C n In Section 3 we wrote down a form ω on the hypersurface X , whose Riccipotential decays in a suitable weighted space by Proposition 5. Our goal is to usethe linear theory developed in Section 6 to improve the decay of the Ricci potentialenough to be able to apply Hein [17, Proposition 4.1] to construct a global Calabi-Yau metric on X . Since we will use a similar method in Section 8 below, we willactually perturb ω to a metric ˜ ω which is Calabi-Yau on the set ρ − [ A, ∞ ) forsufficiently large A . Proposition 25.
Suppose that A is sufficiently large, τ < is sufficiently close to0, and δ < /d is as in Proposition 5. Then there exists a small u ∈ C ,αδ,τ such that ( ω + √− ∂∂u ) n = ( √− n Ω ∧ Ω on the set ρ − [ A, ∞ ) . Proof.
Note first that if k u k C ,αδ,τ < ǫ , then by the definition of the weighted normswe have |∇ u | ω < Cǫρ δ − w τ − . Since w > C − ρ /d − this implies |∇ u | ω < Cǫρ δ − /d − τ (1 − /d ) . If τ is close to zero, and δ < /d , then for sufficiently small ǫ , the form ω + √− ∂∂u defines a metric uniformly equivalent to ω . This is the reason for our requirementthat δ < /d in Proposition 5.Let us define B = { u ∈ C ,αδ,τ : k u k C ,αδ,τ ≤ ǫ } , where ǫ is sufficiently small so that ω + √− ∂∂u is uniformly equivalent to ω . Letus define the operator F : B → C ,αδ − ,τ − ( ρ − [ A, ∞ )) u log ( ω + √− ∂∂u ) n ( √− n Ω ∧ Ω (cid:12)(cid:12)(cid:12)(cid:12) ρ − [ A, ∞ ) . Our goal is to find u ∈ B such that F ( u ) = 0.Let us write(7.1) F ( u ) = F (0) + ∆ ω u + Q ( u )for a suitable nonlinear operator Q . Denoting by P the right inverse for ∆ foundin Proposition 22, it is enough to solve u = P ( − F (0) − Q ( u )) , i.e. we are looking for a fixed point of the map N ( u ) = P ( − F (0) − Q ( u )). Note thatwe have a uniform bound for P independent of A (for sufficiently large A ), since theright inverse for one choice of A also provides a right inverse for all larger choicesof A . In addition either from an explicit formula for Q , or from differentiatingEquation (7.1) and estimating the difference ∆ ω + √− ∂∂u − ∆ ω + √− ∂∂v , we see thatas long as u, v ∈ B we have the estimate k Q ( u ) − Q ( v ) k C ,αδ − ,τ − ≤ C ( k u k C ,α , + k v k C ,α , ) k u − v k C ,αδ,τ . For this note that the C ,α , norms of u controls √− ∂∂u in C ,α , , which in turn isthe difference between the metrics ω and ω + √− ∂∂u .It follows that there is a constant ǫ such that N is a contraction on B as longas k u k C ,α , < ǫ for all u ∈ B . This holds if ǫ in the definition of B is sufficiently small, since asabove we have ρ δ w τ ≤ Cρ δ − τ − /d − ρ w , which implies k u k C ,α , ≤ C k u k C ,αδ,τ . We can therefore assume that k N ( u ) − N ( v ) k < k u − v k for u, v ∈ B . EGENERATIONS OF C n AND CALABI-YAU METRICS 31
Finally we just have to ensure that N maps B to itself. For this first note thatby the estimates of Proposition 5 we have F (0) ∈ C ,αδ ′ − ,τ − for some δ ′ < δ that issufficiently close to δ . It follows that k F (0) k C ,αδ − ,τ − ( ρ − [ A, ∞ )) < CA δ ′ − δ , which can be made arbitrarily small by choosing A large. Next, we have that if u ∈ B , then k N ( u ) k C ,αδ,τ ≤ k N (0) k C ,αδ,τ + k N ( u ) − N (0) k C ,αδ,τ ≤ C k F (0) k C ,αδ − ,τ − ( ρ − [ A, ∞ )) + 12 k u k C ,αδ,τ ≤ CA δ ′ − δ + ǫ . For sufficiently large A we therefore have N ( u ) ∈ B . Therefore we can find a fixedpoint of N , as required. (cid:3) We can now complete the proof of Theorem 1, by applying Hein [17, Proposition4.1] to the metric ˜ ω = ω + √− ∂∂u constructed in the previous proposition. Notethat by elliptic regularity ˜ ω is actually smooth. Since ˜ ω is asymptotically a smallperturbation of ω , Proposition 9 shows that the tangent cone at infinity of ( X , ˜ ω )is the cone X . To apply Hein’s result we need to check that ( X , ˜ ω ) satisfiesthe condition SOB (2 n ) (see [17, Definition 3.1]), and it has a C ,α quasi-atlas.The condition SOB (2 n ) includes the connectedness of certain annuli on X , butfor the proof it is actually enough to check the RCA condition used in Degeratu-Mazzeo [11] for instance. Since ˜ ω is uniformly equivalent to ω , this condition holdsby Proposition 10. To check that ( X , ˜ ω ) satisfies SOB (2 n ) it is then enough toshow that for a constant C > s > C the volume of the ball B ( x, s )satisfies C − s n < Vol( B ( x, s )) < Cs n , for any x ∈ X . This can be seen for instance by using Colding’s volume convergencetheorem [5] under Gromov-Hausdorff limits, and by noting that ˜ ω is Ricci flatoutside of a compact set, and has a tangent cone at inifinity that is non-collapsed(i.e. has Euclidean volume growth).The existence of a C ,α quasi-atlas, i.e. charts of a uniform size around each pointin which the metric is controlled in C ,α , can also be seen using Propositions 6, 7.These results show that for the metric ω we actually have charts of size ρw aroundany point, in which the metric is controlled in C k,α . Note that since w > κ − ρ /d − we have ρw > κ − ρ /d , which goes to infinity as ρ → ∞ . When we perturb themetric to ˜ ω , then by our construction we a priori only control ˜ ω in C ,α in thesecharts, but elliptic regularity allows us to improve this to C k,α .We can therefore apply Proposition 4.1 from Hein [17] to further perturb ˜ ω toa global Calabi-Yau metric on X . This perturbation does not change the tangentcone at infinity, and so we obtain the required Calabi-Yau metric on C n ∼ = X withtangent cone X = C × V at infinity. Calabi-Yau metrics in a neighborhood of an isolated singularity
In this section we show that the ideas developed in the previous sections can alsobe used to construct Calabi-Yau metrics on neighborhoods of certain isolated sin-gularities ( X , X is the hypersurface(8.1) z p + x + . . . + x n = 0 , with p > n − n − .We consider the situation where X ⊂ C n +1 is the hypersurface z p + f ( x , . . . , x n ) = 0 , where as before, V = f − (0) ⊂ C n admits a Calabi-Yau cone metric. As before,we let the degree of f be d for the homothetic action with weight w = ( w , . . . , w n )on x , and this time we require that p > d . For the hypersurface (8.1) the condition p > d coincides with Hein-Naber’s condition p > n − n − . Our goal is to construct aCalabi-Yau metric in a neighborhood of the singular point 0 ∈ X , whose tangentcone at 0 is C × V .As before, we have the nowhere vanishing holomorphic n -formΩ = dz ∧ dx ∧ . . . ∧ dx n ∂ x f on X , and the first step is to write down a metric ω on X whose Ricci potential h = log ω n ( √− n Ω ∧ Ωdecays in a suitable weighted space near the origin. The definition of ω is completelyanalogous to (3.1), given by ω = √− ∂∂ (cid:16) | z | + γ ( Rρ − α ) r + γ ( Rρ − α ) | z | p/d φ ( z − p/d · x ) (cid:17) , where this time we choose α ∈ (1 , p/d ), and γ i , ρ, R are just as before. Note thatthe potential is asymptotic to | z | + r as ρ → Proposition 26.
The form ω defines a metric on X \{ } on the set where ρ < P − for sufficiently large P . In addition we can find a weight δ > p/d for which theRicci potential h of ω satisfies |∇ i h | ω < C i ρ δ − − i if R > κρ,C i ρ δ R − − i if R ∈ ( κ − ρ p/d , κρ ) ,C i ρ δ − p/d − ip/d if R < κ − ρ p/d , for suitable κ, C i > . Recall that by Lemma 3 we can assume d > .Proof. The proof of these estimates is very similar to the proof of Proposition 5,estimating the Ricci potential by comparing ω to various model metrics in differentregions, by scaling. We will keep the notation the same as in the proof of Propo-sition 5 to make the similarities apparent. The main difference is that now we areinterested in estimating the errors as ρ → ρ → ∞ . EGENERATIONS OF C n AND CALABI-YAU METRICS 33
Region I : Consider the region where
R > κρ , and ρ ∈ ( D/ , D ) for small D .We scale the metric to D − ω and use coordinates˜ z = D − z, ˜ x = D − · x, ˜ r = D − r. We have D − ω = √− ∂∂ ( | ˜ z | + ˜ r ) , and the equation of X is D p − d ˜ z p + f (˜ x ) = 0 . As before, we obtain |∇ i h | D − ω ≤ C i D p − d . If p > d >
2, then p − d > p/d −
2, and so we can choose δ > p/d satisfying D p − d < D δ − . Region II : Here R ∈ ( K/ , K ) for some K ∈ (4 ρ α , κρ ), and ρ ∈ ( D/ , D ).Let z be close to z such that | z | ∼ D . In terms of coordinates˜ z = K − ( z − z ) , ˜ x = K − · x, ˜ r = K − r, we have K − ω = √− ∂∂ ( | ˜ z | + ˜ r ) , and the equation of X is K − d ( K ˜ z + z ) p + f (˜ x ) = 0 , where | ˜ z | ≤
1. Arguing as before, when we compare this to the equation f (˜ x ) = 0we obtain an error of order K − d D p . This implies |∇ i h | K − ω ≤ C i K − d D p = C i K − d D p K − . On this region
K > ρ α (and d > K − d D p < D p +(2 − d ) α . Since α < p/d we have p + (2 − d ) α > p/d , so we can choose δ > p/d . Region III : Here R ∈ ( K/ , K ) for K ∈ ( ρ α , ρ α ), and ρ ∈ ( D/ , D ). Weconsider z close to z , and since in this region ρ is comparable to | z | , we have | z | ∼ D . With the same scaling as in Region II, we have K − ω = √− ∂∂ (cid:16) | ˜ z | + γ ˜ r + γ K − | K ˜ z + z | p/d φ (cid:0) ( K ˜ z + z ) − p/d K · ˜ x (cid:1)(cid:17) . The equation of X is K − d ( K ˜ z + z ) p + f (˜ x ) = 0 . As in Proposition 5 we compare this to the metric √− ∂∂ ( | ˜ z | + ˜ r )on X with equation f (˜ x ) = 0. To compare the potentials, similarly to (3.3) wehave ∇ i h K − | K ˜ z + z | p/d φ (( K ˜ z + z ) − p/d K · ˜ x ) − ˜ r i = O (cid:16) ( K − D p/d ) c (cid:17) . Arguing as before we have an error of order K − d D p from comparing the twoequations, which is bounded in the same way as in Region II. Since in this region K ∼ D α , the new error from comparing the K¨ahler potentials satisfies( K − D p/d ) c = K − c D pd (2+ c ) K − = D pd + c ( pd − α ) K − . We need δ so that this is bounded by D δ K − (as D → α < p/d , we canchoose δ > p/d . Region IV : Here R ∈ ( K/ , K ) with K ∈ ( κ − ρ p/d , ρ α / ρ ∈ ( D/ , D ).We choose z close to z , with | z | ∼ D . We scale the same way as in Regions II,III. As before, we have K − ω = √− ∂∂ (cid:16) | ˜ z | + K − | K ˜ z + z | p/d φ (cid:0) ( K ˜ z + z ) − p/d K · ˜ x (cid:1)(cid:17) , and the equation of X is K − d ( K ˜ z + z ) p + f (˜ x ) = 0 . Similarly to the proof of Proposition 5 we now compare K − ω to the product metricon C × V K − d z p with equation K − d z p + f (˜ x ) = 0 . The error given by the difference of the equations is of order K − d z p − = O ( K − d D p − ).Let us denote by E the difference in K¨ahler potentials, E = K − | K ˜ z + z | p/d φ (cid:0) ( K ˜ z + z ) − p/d K · ˜ x (cid:1) − K − | z | p/d φ ( Kz − p/d · ˜ x ) . Since K ( K ˜ z + z ) − p/d = z − p/d K (1 + O ( KD − )) , similarly to (3.5), (3.6), (3.7) we get |∇ i E | < C i ( | z | − p/d K ) − − c D − K = O ( K − − c D (2+ c ) pd − ) . Combining this with the error of order K − d D p − , we need δ > p/d such that K − d D p − + K − − c D (2+ c ) pd − < CD δ K − . This is equivalent to K − d D p − (cid:0) KD − p/d ) d − − c (cid:1) < CD δ . We can assume that c > d − − c > c > d − KD − p/d is bounded away from zero, it is enough to choose δ so that K − d D p − ( KD − p/d ) d − − c < CD δ , i.e. K − c D (2+ c ) pd − < CD δ . Since
K < D , in order to be able to choose δ > p/d , we need1 − c + (2 + c ) pd − > pd . This is equivalent to p > d , which holds in our setting.
EGENERATIONS OF C n AND CALABI-YAU METRICS 35
Region V : Here
R < κ − ρ p/d , ρ ∈ ( D/ , D ), and z is close to z satisfying | z | ∼ D . We rescale the metric by | z | p/d . We introduce new coordinates˜ z = z − p/d ( z − z ) , ˜ x = z − p/d · x, ˜ r = | z | − p/d r, so that | ˜ z | <
1. We have | z | − p/d ω = √− ∂∂ h | ˜ z | + | z p/d ˜ z + z | p/d | z | − p/d φ (cid:0) ( z p/d ˜ z + z ) − p/d z p/d · ˜ x (cid:1)i , and the equation of X is z − p ( z p/d ˜ z + z ) p + f (˜ x ) = 0 . Note that z − p ( z p/d ˜ z + z ) p = 1 + O ( z p/d − ) = 1 + O ( D p/d − ) , and so comparing to C × V with equation 1 + f (˜ x ) = 0 we introduce an error oforder D p/d − .The difference in the corresponding K¨ahler potentials is E = | z p/d ˜ z + z | p/d | z | − p/d φ (cid:0) ( z p/d ˜ z + z ) − p/d z p/d · ˜ x (cid:1) − φ (˜ x ) . Since ( z p/d ˜ z + z ) − p/d z p/d = 1 + O ( z p/d − ) = 1 + O ( D p/d − ) , we have |∇ i E | | z | − p/d ω = O ( D p/d − ). Therefore we need to be able to choose δ > p/d for which D p/d − < CD δ − p/d . This is possible, since we assumed p/d > (cid:3) Abusing notation, let us now denote by ω a metric on X \ { } which agreeswith the ω constructed above on a neighborhood of 0. The rest of the proof ofTheorem 2 is essentially the same as the proof of Proposition 25, in order to find a u such that ( ω + √− ∂∂u ) n = ( √− n Ω ∧ Ωon the set where ρ < A − for sufficiently large A . The key ingredient is to find aright inverse for the Laplacian(8.2) ∆ ω : C k,αδ,τ ( ρ − (0 , A − ]) → C k − ,αδ − ,τ − ( ρ − (0 , A − ]) , for sufficiently large A , where the weighted spaces are defined analogously to before.Here we use smooth weight functions ρ, w satisfying that ρ agrees with | z | + R near the origin (this is essentially the radial distance from the singular point), and w = R > κρ,R/ ( κρ ) if R ∈ ( κ − ρ p/d , κρ ) ,κ − ρ p/d − if R < κ − ρ p/d . The reason for the definition of w is that near the singular rays C × { } our metric ω is now modeled on D p/d ω V , transverse to the C -factor. We have results analogous to Propositions 7 and 6. Here we consider two typesof regions: U = { ρ < A − , R > Λ ρ p/d } , for large A, Λ, and V = {| z − z | < B | z | p/d , ρ < A − , R < Λ ρ p/d } , where z , B are viewed as fixed. In an identical way to Propositions 7 and 6 wecan define maps G : U → X and H : V → C × V . The Riemannian metrics g, g X , g C × V defined by ω, ω and the product metrics on C × V then satisfy k G ∗ g X − g k C k,α , < ǫ if Λ > Λ( ǫ ) and A > A ( ǫ ), and k| z | p/d H ∗ g C × V − g k C k,α , < ǫ, for any z , B, Λ, once
A > A ( ǫ, Λ , B ).Just as in Proposition 9, these estimates imply that the tangent cone of ( X , ω )at 0 is given by the cone X = C × V . Moreover the existence of a right inversefor the Laplacian in (8.2) once A is sufficiently large follows exactly the argumentfrom Section 6.Proposition 26 implies that the Ricci potential h of ω satisfies h ∈ C ,αδ ′ − ,τ − forsome δ ′ > p/d , and any τ <
0. If follows that for slightly smaller δ > p/d wehave k h k C ,αδ − ,τ − ( ρ − (0 ,A − ]) < CA δ − δ ′ , which can be made arbitrarily small by choosing A large. We can now follow theproof of Proposition 25 to solve Equation 8 with u ∈ B , where B = { u ∈ C ,αδ,τ : k u k C ,αδ,τ < ǫ } , for sufficiently small ǫ , with τ < k u k C ,α , ( ρ − (0 ,A − ]) ≤ C k u k C ,αδ,τ ( ρ − (0 ,A − ]) . This follows since the inequality w > C − ρ p/d − , together with δ > p/d implies ρ δ w τ < Cρ w . In particular if u ∈ B with sufficiently small ǫ , then ω + √− ∂∂u is uniformlyequivalent to ω . The rest of the argument is identical to the proof of Proposition 25.Finally, the tangent cone of ω + √− ∂∂u at 0 agrees with the tangent cone of ω ,since if u ∈ C ,αδ,τ for our choices of δ, τ , then |√− ∂∂u | ω → ρ → References [1]
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