Extremal metrics on blowups along submanifolds
Abstract.
We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of ArezzoPacardSinger, who considered blowups in points.
1. Introduction
A basic question in Kähler geometry is the existence of extremal metrics on Kähler manifolds, in the sense of Calabi [4]. A Kähler metric on is an extremal metric if the gradient of its scalar curvature is a holomorphic vector field on . The YauTianDonaldson conjecture [26, 22, 7, 18] relates the existence of an extremal metric on a compact Kähler manifold to an algebrogeometric stability condition, but so far there are only a few existence results beyond the KählerEinstein case [25, 22, 5].
In this paper, following the works of ArezzoPacard [1, 2], ArezzoPacardSinger [3] and the second author [19, 21] we investigate the existence of an extremal metric on a blowup of along a smooth submanifold, assuming that admits an extremal metric . The main new feature in our work is that we allow , while previous works focused on blowups in points with the exception of Hashimoto [8] who considered blowups of projective spaces in lines.
In order to state our result we set up some notation. We suppose that is a codimension submanifold of , and we write for the group of Hamiltonian isometries of . There is an associated moment map
normalized so that has zero integral for each . Denoting by the space of codimension complex submanifolds of , the group acts on , preserving a natural symplectic form, and we have a moment map
We identify using the product on Hamiltonian functions, and so we can naturally think of as a vector field on . In analogy with the result in [19], in this paper we prove the following.
Theorem 1.
Suppose that is a submanifold such that and the vector field are tangent to . Assume also that the codimension of is . Then admits an extremal metric in the class for sufficiently small .
The strategy of the proof is very similar to that employed in [19]. Because of technical difficulties we have not obtained the same result when , although it is very likely that it is true in that case as well.
Our result can be used to obtain many new examples of extremal metrics. The simplest situation is when has trivial isometry group, and so in particular has constant scalar curvature. In this case the moment map is trivial, and so for any submanifold of codimension greater than two the blowup admits a constant scalar curvature metric in for small . A more general result, allowing for a nontrivial automorphism group, analogous to [3, Theorem 2.4], is the following.
Corollary 2.
Suppose that is an extremal metric on , and let be a maximal torus in the isometry group of . Suppose that has codimension greater than 2, and the action of preserves . Then admits an extremal metric in for sufficiently small .
Proof.
The vector field is invariant under the adjoint action of the isometry group of , and so it is in the center of the Lie algebra of the isometry group . In particular , where is the Lie algebra of . Similarly because the moment map is equivariant, is in the center of the stabilizer of under the infinitesimal action of . By our assumption this stabilizer contains , so any element in its center must belong to . In particular (here we are identifying as before), and so fixes . Theorem 1 then applies. ∎
This corollary applies for example to subspaces as long as . In this way we obtain some extensions of the work of Hashimoto [8], who showed that admits an extremal metric for all . More generally we can let be any toric manifold which admits an extremal metric, for instance a KählerEinstein metric obtained using the existence result of WangZhu [24]. We can then choose to be a toric submanifold of codimension greater than 2.
There are also more general submanifolds satisfying the assumption in Theorem 1 that is tangent to . The condition means that is a balanced embedding, and Donaldson [6] showed that if is trivial, and admits a constant scalar curvature metric, then there are balanced embeddings for sufficiently large . This result was generalized by the first named author [16] to the case when has nontrivial automorphisms, and admits an extremal metric (see also Mabuchi [11, 14], Hashimoto [9] for other work in this direction). As a consequence we have the following.
Corollary 3.
Let be an extremal Kähler manifold, with for a line bundle . Fix an integer , and an embedding using a basis of sections of . If is sufficiently large, then admits an extremal metric in the class for small .
Proof.
Theorem 1.1 in [16] implies that under the assumptions there exist relatively balanced embeddings using a basis of sections of for sufficiently large . This means precisely that under these embeddings , identified with a vector field on , is tangent to . Our main result, Theorem 1, then implies the required result. ∎
The structure of the paper is the following. In Section 2 we will write down a metric on giving a first approximation to the metric that we are looking for, and we will present the main gluing result that we need to prove. In Section 3 we will show that the linearized operator of our problem is invertible. We will complete the proof of our main result in Section 4 by constructing a better approximate solution , in an analogous way to what was done in [3, 19], and then controlling the relevant nonlinear terms.
2. The gluing problem
2.1. A first approximate solution
Suppose as in the introduction that is a compact Kähler manifold such that is an extremal metric. Let be a codimension submanifold, where . Our goal in this section is to construct a Kähler metric on the blowup in the class for sufficiently small , which will be a first approximation to the extremal metric that we seek. In previous work by ArezzoPacard [1], ArezzoPacardSinger [3] and the second author [19], was a point, and the approximate solution on was constructed by identifying an annulus around by an annulus inside the blowup . When is a submanifold, there is no longer a standard form of a neighborhood of , and so we will instead view as a completion of under a suitable metric. In other words we will identify the complement of the exceptional divisor in with , and our constructions will primarily take place on . We then simply need to ensure that our metric extends to a smooth metric on , which we will achieve by using the usual coordinate charts covering the blowup.
The basic building block for constructing extremal metrics on blowups is the BurnsSimanca metric [17] on . This is a scalar flat, asymptotically flat Kähler metric
where is smooth up to the boundary, and is in the weighted space , i.e. for all , as . In addition is a cutoff function such that for and for . There are also more refined expansions of . We will need to use that (see e.g. [19, Lemma 26])
where .
We will define by using the Kähler potential of the BurnsSimanca metric, but replacing by the distance function to the submanifold , with respect to the metric . Note that is a smooth function in a tubular neighborhood of . For small let us define for
In addition let be a cutoff function as above, and define by , and . So is supported away from , while is supported near .
Finally we define
on .
Proposition 4.
For sufficiently small , the form defines a Kähler metric on , extending to a smooth metric on .
In the proof we will need the following.
Lemma 5.
At any point , we can choose coordinates , defined for , such that and
where , and all derivatives of are bounded. In addition
where for , while higher order derivatives are bounded. All of these bounds can be chosen to be uniform in the point .
Proof of Proposition 4.
We will work on four separate regions:

On the set where , we have , so it is a smooth metric.

On the set , we have contributions from the derivatives of , but the term involving is not present. The asymptotics of imply that, measured with respect to the metric , we have
for all . It follows that for sufficiently small , the form is also positive.

On the set we have and . We change coordinates, using Lemma 5. In terms of above, we set
By shifting the center of the coordinate system we can assume that . We can compare with the product metric on . In our coordinates a Kähle potential for is given by
where is given by
At the same time the product metric has Kähler potential
and this product metric is uniformly equivalent to the Euclidean metric in the coordinates. We have
The estimates we have for the derivatives of imply, that in the coordinates
At the same time, we have
and so the decay estimates for imply that
Since and , we have
(1) using also that is comparable to . For small the form will then be a small perturbation of the product metric, since on this region .

Finally, to examine the set where we perform a different change of coordinates. In terms of above, we set
We use this chart at points where , say, and permute the coordinates appropriately at other points. On our region, once is sufficiently small, we have . It follows that , , and by changing the basepoint for the coordinate system, we can assume that . In these coordinates we have
Once again we will see that in these coordinates is well approximated by the product metric on . Indeed, in these coordinates a Kähler potential for is given by
while a Kähler potential for the product metric is
where
On our region we have , and the derivatives of are all bounded. It follows that
(2) for all , where we are taking derivatives in the coordinates. In particular, once is sufficiently small, will define a smooth metric uniformly equivalent to the product metric on this set.
∎
2.2. The gluing result
The overall strategy to proving Theorem 1 is the same as in [19]. We first choose a maximal torus in the stabilizer of the submanifold , and work equivariantly throughout. Let denote the centralizer of , and the Hamiltonian functions corresponding to the Lie algebra of (including the constants). So . Write for the subspace corresponding to . The elements in lift to the blowup in a natural way, giving Hamiltonians of holomorphic vector fields with respect to the metric . For this note that the function is invariant under the action of , and so is invariant.
In [19] we defined a lifting of the rest of the functions in using cutoff functions, but here we proceed in a slightly different way, simply pulling back the functions under the blowdown map .
Definition 6.
We define a map
depending on , in the following way. We fix a decomposition . We lift elements to in the natural way: if we write , then on we have
where is the holomorphic vector field corresponding to . This function extends to give a smooth function on , and it is the Hamiltonian, with respect to , of the vector field .
For we simply define , and note that this also defines a smooth function on , since the blowdown map is smooth.
Given this definition, the gluing result that we need to show is the following.
Proposition 7.
Suppose that is such that is tangent to . There are constants such that for all we can find and satisfying
(3) 
In addition we have an expansion
where are constants, and for some .
Based on this proposition, the proof of Theorem 1 is identical to the argument in [19, p. 1426]. For the reader’s convenience we give the main points here.
Proof of Theorem 1.
We are assuming that and are tangent to . We choose our maximal torus so that . Note that we also have because is in the center of . The complexification of the group acts on the space , and we want to show that for sufficiently small we can find an element near the identity, so that Proposition 7 applied to the perturbed submanifold yields an extremal metric on .
The key point for this is that , as a map from submanifolds to can be viewed as a perturbation of a moment map, and so [19, Proposition 8] can be applied. We obtain a small perturbation of , such that when Proposition 7 is applied at , then the vector field induced by is tangent to . In particular the metric constructed in Proposition 7 will then be an extremal metric on . At the same time, , and so we obtain the required extremal metric on . ∎
3. The linearized problem
In this section we study the linearized problem corresponding to Equation (3). Let us denote by the invariant functions on satisfying for all , where the inner product is computed using . We will consider the linear operator
Here denotes the linearization of the scalar curvature operator at , i.e.
for a suitable nonlinear operator , and . Recall (see e.g. [20, Section 4.1]) that we have
in terms of the Ricci curvature of . In addition we will need to relate this to the Lichnerowicz operator , where . We have
We will show that the operator is invertible, and that we can control the norm of its inverse in suitable weighted spaces.
3.1. Weighted spaces
We will next define the weighted Hölder spaces that we will use. Let us define the weight function by
and extend it to by continuity. We define the weighted space on , depending on , as follows. The estimate means that for any the norm of on an ball of radius around is bounded by , measured with respect to the scaled up metric .
In practice we can control these weighted norms as follows. On the region where , the metric is uniformly equivalent to , and so on this region means that
for , with the derivatives measured using . On the region where , as we have seen in the proof of Proposition 4, the scaled up metric is uniformly equivalent to the product metric on . This in turn is uniformly equivalent, on suitable charts, with the Euclidean metric in terms of our coordinates , or from the proof of Proposition 4. It follows that in these coordinates means either
(4) 
in the charts (where is comparable to ), or
in the charts.
We also define analogous weighted spaces on using the weight function given by , where is distance from the exceptional divisor in , and weighted spaces on using the weight function in terms of the coordinate on the factor.
We have the following estimate of our liftings in the weighted spaces.
Lemma 8.
If we have then
Here denotes any fixed norm on the finite dimensional vector space .
Proof.
Using the splitting from Definition 6, if , then . On the region where , we certainly have
for all , since in fact all the derivatives are bounded uniformly, and .
On the region where we change coordinates to the from the proof of Proposition 4, and note that in terms of the local coordinates we have
Using that , we obtain the required estimate.
If , then is defined by
where , and
At the same time, in the coordinates from Lemma 5, the vector field in this case is of the form
where since is tangent to . Consider the region where , which is the only place where the problematic term appears. Here , and note that , while has bounded derivatives. It follows that on this region
which, combined with the estimate above for implies the result we need.
Note that if the vector field were not parallel to , then would blow up near . This is our reason for lifting elements in in a different way. ∎
3.2. Controlling the inverse
We will now think of our operator as a map between suitable weighted spaces:
and our goal is the following result.
Proposition 9.
For sufficiently small and , the operator is invertible, with a bound on its inverse independent of .
We prove this result using a blowup argument, following the exposition in [20, Theorem 8.14]. We will need the following three lemmas.
Lemma 10.
Define the linear operator
If and is in the weighted space with , then .
Proof.
The restriction on the weight ensures that is a distributional solution of on all of , and in particular extends smoothly to . On we have the equation and for all . This implies that both and vanish identically. ∎
Lemma 11.
If is in the weighted space with , and , then . Here denotes the Lichnerowicz operator on the product space.
Proof.
We use an argument with the Fourier transform similar to that in MazzeoPacard [15] (see also Walpuski [23, Lemma A.1]). Let us write , where denotes the coordinate on . We have
where are the Laplacians on the two factors, and denotes the Ricci form of , with the indices raised. In particular the terms only involve derivatives on the factor.
We take the Fourier transform of in the variable. This way we obtain a distribution on , satisfying the equation
In terms of the Lichnerowicz operator on this can be written as
We claim that this implies that the distribution is supported on the set . To show this, let be a smooth function with compact support away from . We need to show that .
For this we claim that there is a solution of the equation
with decaying faster than any negative power of in the direction, and with if is sufficiently large (outside the support of ). This follows from the fact that for any fixed the operator
on is essentially selfadjoint, and has trivial kernel in . Indeed any solution function in which is in the kernel would have to be rapidly decaying by applying Schauder estimates (on balls of radius at distance from the exceptional divisor), and then an integration by parts shows that .
We now have that
where the integration by parts is justified since is rapidly decaying in the direction, and has bounded support in the direction.
Now we know that is supported on , and as a result it is a linear combination of derivatives of the delta function at the origin in (with coefficients given by functions of ). In other words we can write
where each denotes an derivative of the delta function at the origin. It follows that
where each is an degree homogeneous polynomial in . Since is bounded, only a constant polynomial can appear, and so we find that is purely a function of . Since , and we are assuming that decays at infinity in the coordinate, it follows (see e.g. ArezzoPacard [1], KovalevSinger [10]) that . ∎
Lemma 12.
If is in the weighted space with and , then .
Proof.
Again a local argument shows that actually extends to a smooth function on , satisfying . In addition we know that the function decays at infinity in the factor. An argument identical to that in the previous lemma shows that . ∎
We will now use these results to prove Proposition 9
Proof of Proposition 9.
We use an argument by contradiction. We follow the exposition in Székelyhidi [20, Theorem 8.14] closely.
To emphasize the presence of the parameter , we will denote by our operator with respect to the metric . Our weighted spaces were defined in terms of local coordinates in which is uniformly equivalent to the Euclidean metric. Using the Schauder estimates in these local charts we obtain a uniform constant (independent of ), such that
(5) 
for all (recall that ). We want to show that with a possibly larger constant, the same inequality holds without the terms on the right hand side, for sufficiently small . Arguing by contradiction, let us suppose that for a sequence we have corresponding functions and satisfying
where note that the norms of are computed using .
Using the equation (5) we can choose a subsequence of the , converging to a limit , in the space with , and . In particular we find that
where denotes the operator in Lemma 10. From Lemma 10 we have that . This implies that , and so up to choosing a further subsequence we can assume that
i.e. we can drop the term involving in the definition of .
We now need to examine the points , where achieves its maximum. Recall that is the weight function on (or on ) we defined before, with respect to . By our assumption on , we have . We already know that on compact sets away from , which implies that we must have . We have two cases depending on whether is bounded or not.
Suppose first that for some we have for all . For sufficiently large , the points will be in charts of the form considered in Lemma 5, and by changing to or coordinates as in the proof of Proposition 4 we can view as a point in , at distance at most from , where is the exceptional divisor. Moveover the pullbacks of in this chart will converge to the product metric on on compact sets. Choosing a further subsequence we can the extract a limit of the functions , locally in , where is in the weighted space , satisfying . Lemma 11 implies that , contradicting that .
Finally we suppose that , but . It follows that in our charts from Lemma 5 we have , up to a bounded factor. Arguing as above, by taking a subsequence we can extract a limit of the , giving a function , in the weighted space satisfying . Lemma 12 implies that , which is a contradiction again.
In sum we find that there is a constant , such that for sufficiently small the estimate
holds. This shows in particular that has trivial kernel, and since it has index zero it must be invertible. In addition we obtain the required uniform bound on its inverse. ∎