aa r X i v : . [ m a t h . DG ] M a r BLOWING UP EXTREMAL K ¨AHLER MANIFOLDS II
G ´ABOR SZ´EKELYHIDI
Abstract.
This is a continuation of the work of Arezzo-Pacard-Singer andthe author on blowups of extremal K¨ahler manifolds. We prove the conjecturestated in [32], and we relate this result to the K-stability of blown up manifolds.As an application we prove that if a K¨ahler manifold M of dimension greaterthan 2 admits a cscK metric, then the blowup of M at a point admits acscK metric if and only if it is K-stable, as long as the exceptional divisor issufficiently small. Introduction
We continue our study [32] of extremal metrics on blown-up manifolds, followingthe work of Arezzo-Pacard [1, 2] and Arezzo-Pacard-Singer [3]. See Pacard [21] fora survey and see also LeBrun-Singer [18], Rollin-Singer [23], Tipler [35], Biquard-Rollin [4] for related work. The starting point is a compact K¨ahler manifold M withan extremal metric ω . The notion of extremal metric was introduced by Calabi [6],and it means that the gradient of the scalar curvature ∇ s ( ω ) is a holomorphicvector field. The basic question that we study is whether the blowup Bl p ,...,p n M of M in a finite number of points admits an extremal metric in the K¨ahler class(1) π ∗ [ ω ] − ε ( a [ E ] + a [ E ] + . . . a n [ E n ]) , where π is the blowdown map, a , . . . , a n > ε > E i are the exceptional divisors. Our methods, following [1, 2, 3] are perturbative,restricting the results to sufficiently small ε >
0. In addition our results will berestricted to blowing up only one point, and dimension m >
2. We expect that withsome more work our method can deal with the case m = 2, but blowing up morethan one point introduces more serious difficulties as we will explain in Section 3.2.To state the main result, let us write G for the group of Hamiltonian isometriesof ( M, ω ) and g for its Lie algebra. Let(2) µ : M → g ∗ be the equivariant moment map for the action of G normalized in such a way thatthe Hamiltonian functions h µ, ξ i have zero mean on M for all ξ ∈ g . From now onwe will identify g with its dual, using the inner product given by the L product onHamiltonian functions. Let ∆ µ be the Laplacian of µ taken componentwise afteridentifying g ∗ with R l for some l . A central role is played by the perturbed momentmap µ ( p ) + δ ∆ µ ( p )for small δ . Note that this is simply the moment map for the action of G on M with respect to the K¨ahler form ω − δρ , where ρ is the Ricci form of ω . Let us write G c for the complexification of G , acting on M by biholomorphisms. With this themain result is as follows, confirming Conjecture 6 in [32] in the case when m > Theorem 1.
Assume that the dimension m > , and suppose that ∇ s ( ω ) vanishesat p ∈ M . There is a δ > depending on ( M, ω ) with the following property.Suppose that for some δ ∈ (0 , δ ) there is a point q in the G c -orbit of p such thatthe vector field µ ( q ) + δ ∆ µ ( q ) vanishes at q . Then the blowup Bl p M admits anextremal metric in the K¨ahler class π ∗ [ ω ] − ε [ E ] , for all sufficiently small ε > . Suppose that M is a projective variety and ω ∈ c ( L ) for a line bundle L over M . The condition in the theorem can be interpreted as relative stability of thepoint p with respect to the natural linearization of the G c -action on the Q -linebundle L + δK M for small rational δ , where K M is the canonical bundle. In thisterminology, our earlier result in [32] only dealt with the case when p is relativelystable with respect to the linearization on L . This in turn refined earlier results ofArezzo-Pacard-Singer [3], where some extra conditions were required. Allowing asmall perturbation of the line bundle L gives more precise information about casewhen the point p is strictly semistable.In the case when ( M, ω ) is a constant scalar curvature K¨ahler (or cscK) manifold,then we can show that Theorem 1 actually gives a complete characterization ofthe possible blowup points. Suppose again that M is projective and ω ∈ c ( L ).For small rational ε let us write L ε = π ∗ L − ε [ E ] for an ample Q -line bundleon the blowup Bl p M . The Yau-Tian-Donaldson conjecture [37, 34, 11] predictsthat the existence of a cscK metric on the blowup Bl p M in the first Chern class c ( L ε ) is related to the K-stability of the pair (Bl p M, L ε ). In Section 4 we definea simple version of K-stability for K¨ahler manifolds, restricting attention to test-configurations with smooth central fibers. Using this and Theorem 1 we obtain thefollowing. Theorem 2.
Let ( M, ω ) be a cscK manifold of dimension m > , and let p ∈ M .Then for sufficiently small ε > the following are equivalent, and are independentof ε : (1) The blowup Bl p M admits a cscK metric in the class π ∗ [ ω ] − ε [ E ] , (2) The pair (Bl p M, π ∗ [ ω ] − ε [ E ]) is K-stable with respect to smooth test-configurations, (3) There is a point q in the G c -orbit of p such that µ ( q ) + ε ∆ µ ( q ) = 0 . This extends our earlier result in [32], where (
M, ω ) was assumed to be a K¨ahler-Einstein manifold. A natural problem is to generalize this result to extremal met-rics, and we will discuss the difficulty that arises in Section 3.2.The contents of the paper are as follows. In Section 2 we prove a finite dimen-sional perturbation result, which is sharper than the result we used in [32]. Theheart of the paper is Section 3 where the main analytic gluing theorem, Theorem 12is proved, and the proof of Theorem 1 is given at the end of Section 3.5. The mainingredient is a refined expansion of the solution to the equation introduced in [32].This is similar to other obstructed perturbation problems in the literature such asPacard-Xu [22], but an important difference is that computing one more term inour case is enough to get a sharp existence result for cscK metrics when n >
2. Thereason for this is the algebro-geometric structure of the problem. We discuss thisaspect in the final Section 4, where we extend [32, Theorem 5] to K¨ahler manifoldswhich are not necessarily algebraic, and we give the proof of Theorem 2.
LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 3
Acknowledgements.
I would like to thank Frank Pacard and Michael Singer forseveral useful discussions. 2.
Relative stability
In this section we will consider the action of a compact Lie group G on a compactK¨ahler manifold M , which is Hamiltonian with respect to a K¨ahler form ω . Theaction of G extends to a holomorphic action of the complexification G c . Let uswrite µ : M → g for the moment map, where we have identified g ∼ = g ∗ using an invariant innerproduct. Since we want to work on K¨ahler manifolds which may not be algebraic, wewill review the basic ideas from relative stability [31] in this setting. The usual GITtheory has been extended to this setting by several authors, for instance Mundet iRiera [20], and Teleman [33]. Our definitions will not necessarily match with theirssince we just want to cover the bare minimum of the theory that we will use.We will need to work with maximal compact subgroups of G c other than G . If K ⊂ G c is maximal compact, then K = Ad g ( G ) for some g ∈ G c . The metric( g − ) ∗ ω is K -invariant, and a corresponding moment map is given by(3) µ K ( g · p ) = ad g µ ( p ) . This way of assigning “compatible” moment maps for the actions of all maximalcompact subgroups of G c is analogous to a linearization of the action in GIT, andit is called a “symplectization” of the action in [33]. Definition 3. (1)
A point p ∈ M is stable for the action of G c (and for ourchoice of symplectization), if the stabilizer G cp is trivial, and there exists apoint q ∈ G c · p such that µ ( q ) = 0 . (2) A point p ∈ M is semistable for the action of G c if there is a point in theorbit closure q ∈ G c · p such that µ ( q ) = 0 . (3) A point p ∈ M is relatively stable for the action of G c , if there exists apoint q ∈ G c · p such that µ ( q ) ∈ g q . It follows from (3) that the definition is independent of the choice of maximalcompact subgroup. The main observation in [31] (see also Kirwan [15]) is thatrelative stability of a point p is equivalent to stability of p for the action of asubgroup of G c . The relevant subgroup can be defined for any complex torus T c ⊂ G c . For this, let K ⊂ G c be a maximal compact subgroup containing thecompact torus T . Let us write k T ⊥ = { ξ ∈ k : ad ξ ( η ) = 0 , h ξ, η i = 0 for all η ∈ t } , and let K T ⊥ be the corresponding subgroup of K (this is a closed subgroup by [30,Lemma 1.3.2]). We then let G cT ⊥ be the complexification of K T ⊥ . We will need thefollowing result. Proposition 4.
Suppose that p ∈ M is relatively stable, and T ⊂ G p is a maximaltorus, such that T c ⊂ G cp is also maximal. Then there exists a point q ∈ G cT ⊥ · p such that µ ( q ) ∈ g q . G´ABOR SZ´EKELYHIDI
Proof.
Using (3), we can choose a maximal compact subgroup K ⊂ G c such that µ K ( p ) ∈ k p . Let T ′ ⊂ K p be a maximal torus. It follows then that p has trivialstabilizer in the group K T ′⊥ , and(4) proj k T ′⊥ µ K ( p ) = 0 , so from [32, Lemma 16] we find that the stabilizer of p in G cT ′⊥ is trivial. Thisimplies that T ′ c is a maximal torus in G cp . We can therefore conjugate T ′ c into T c using an element of G cp , and by taking the corresponding conjugate of the maximalcompact K , we can assume that T ′ = T .From (4) we then have that p is stable for the action of G cT ⊥ , and in particularusing the maximal compact subgroup G T ⊥ we find that there is a point q ∈ G cT ⊥ · p such that(5) proj g T ⊥ µ ( q ) = 0 . Since elements in G cT ⊥ commute with T , we have that T ⊂ G q . Since µ ( q ) is G q invariant, we have that µ ( q ) commutes with t . It then follows from (5) that µ ( q ) ∈ t ,and in particular µ ( q ) ∈ g q . This is what we set out to prove. (cid:3) We will also need a K¨ahler version of the Hilbert-Mumford criterion, developedin [20] and [33]. For any ξ ∈ g and p ∈ M we define the weight(6) W ( p, ξ ) = lim t →−∞ h µ ( e itξ · p ) , ξ i . We then have the following result from Teleman [33].
Proposition 5.
Suppose that the stabilizer G cp is trivial. (1) The point p ∈ M is stable, i.e. there is q ∈ G c · p with µ ( q ) = 0 , if and onlyif there is a constant δ > , such that W ( p, ξ ) > δ for all ξ with k ξ k = 1 . (2) The point p ∈ M is semistable, i.e. there is q ∈ G c · p with µ ( q ) = 0 , if andonly if W ( p, ξ ) > for all ξ ∈ g . Note that the strict lower bound δ > W ( p, ξ ) as afunction on the unit sphere in g is lower semicontinuous, being the supremum of afamily of continuous functions.2.1. A perturbation problem.
We will now assume that we have two K¨ahlerforms ω, η on M , and the action of G is assumed to be Hamiltonian with respectto both K¨ahler forms. We have equivariant moment maps µ, ν : M → g . We will therefore have to specify which moment map (or symplectization) we usewhen we speak of stable or semistable points. In our application we will have ν = µ + δ ∆ µ for small δ , corresponding to the K¨ahler form ω − δ ρ . We will againassume that p ∈ M has trivial stabilizer. Our goal is the following result. Theorem 6.
Let p have trivial stabilizer. Suppose that µ ε : M → g is a family ofcontinuous functions such that µ ε = µ + εν + O ( ε κ ) , for some κ > . Suppose that for all sufficiently small ε > we can find q ∈ G c · p such that µ ( q ) + εν ( q ) = 0 . Then for sufficiently small ε > we can find q ∈ G c · p such that µ ε ( q ) = 0 . LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 5
Proof.
The assumption implies that for each small ε > q ε ∈ G c · p such that µ ( q ε ) + εν ( q ε ) = 0 . By compactness of M , we can assume up to choosing a subsequence, that q ε → q as ε →
0, for some q ∈ M . It follows that µ ( q ) = 0 . Since q ∈ G c · p , the point p is semistable with respect to µ . If p were actuallystable with respect to µ , then we could apply Proposition 8 from [32] to find q ∈ G c · p for sufficiently small ε > µ ε ( q ) = 0. The difficulty now is that the point q may be on the boundary of the G c -orbit of p .By the assumption, we can choose a small δ > p is stable with respectto µ + δν . For ε ≪ δ we have δδ − ε µ ε = µ + εδ − ε ( µ + δν ) + O ( ε κ ) , and so by replacing ν by µ + δν , we can assume that p is stable with respect to ν .Let us define the weights W µ ( p, ξ ) and W ν ( p, ξ ) as in (6). We then know fromProposition 5 that W µ ( p, ξ ) > ξ , and there is a c > W ν ( p, ξ ) > c , for all ξ with k ξ k = 1 . Fix an ε >
0. By linearity we have W µ + εν ( p, ξ ) > ξ = 0, and in fact W µ + εν ( p, ξ ) > εc , for all ξ with k ξ k = 1.Using the compactness of the unit sphere, it follows that there is a large radius R > k µ ( e iξ · p ) + εν ( e iξ · p ) k > εc , for all ξ with k ξ k = R. Since p is stable with respect to µ + εν , it follows that there is a unique ξ ε suchthat µ ( e iξ ε · p ) + εν ( e iξ ε · p ) = 0 . Consider the maps
F, F ε : g → g given by F ( ξ ) = µ ( e iξ · p ) + εν ( e iξ · p ) F ε ( ξ ) = µ ε ( e iξ · p ) . From (7) we know that F induces a map F : ∂B ( R ) → g \ { } , where ∂B ( R ) is the R -sphere in g . By a homotopy argument we can see that thedegree of this map is ±
1, since F also induces a map F : g \ { ξ ε } → g \ { } , while at ξ ε the derivative of F is an isomorphism. If ε is chosen sufficiently small,then by our assumption | F ε − F | < εc , so F ε also defines a map with nonzero degree from ∂B ( R ) to g \ { } . But thenthere must be a ξ ∈ g such that F ε ( ξ ) = 0. (cid:3) G´ABOR SZ´EKELYHIDI
We also need the following, which is analogous to Proposition 12 in [32], butapplies in the non-algebraic case as well.
Proposition 7.
Assume again that the stabilizer of p is trivial. There is a δ > ,depending on µ, ν , such that the following are equivalent: (1) For some δ ∈ (0 , δ ) we can find q ∈ G c · p such that µ ( q ) + δν ( q ) = 0 . (2) We have W µ ( p, ξ ) > for all ξ , and W ν ( p, ξ ) > for all ξ = 0 for which W µ ( p, ξ ) = 0 . (3) For all δ ∈ (0 , δ ) we can find q ∈ G c · p such that µ ( q ) + δν ( q ) = 0 .In particular whether or not we can find q ∈ G c · p such that µ ( q ) + δν ( q ) = 0 isindependent of δ ∈ (0 , δ ) .Proof. Let us write S ⊂ g for the unit sphere. To prove (1) ⇒ (2), note that(1) implies that W µ + δν ( p, ξ ) > ξ = 0. Suppose first that W µ ( p, ξ ) < ξ ∈ S . This would imply that for all sufficiently small δ > W µ + δν ( p, ξ ) <
0, which is a contradiction if δ is chosen to be sufficiently small. So(1) implies that W µ ( p, ξ ) > ξ . In addition if ξ = 0, but W µ ( p, ξ ) = 0, thenclearly we must have W ν ( p, ξ ) > ⇒ (3) note that the set of ξ ∈ S for which W µ ( p, ξ ) = 0 is a closedsubset in S by lower semicontinuity of W . It follows that there is a c > W ν ( p, ξ ) > c , for all ξ ∈ S such that W µ ( p, ξ ) = 0 . The set of ξ ∈ S for which W ν ( p, ξ ) > c is an open set U , whose complement in S is compact. Again by lower semicontinuity, W µ ( p, ξ ) > c for some c > ξ ∈ S \ U . The boundedness of W ν ( p, ξ ) for ξ ∈ S then easily implies that forsufficiently small δ we have W µ + δν ( p, ξ ) > , for all ξ = 0 . The implication (3) ⇒ (1) is immediate. (cid:3) The gluing theorem
The goal of this section is to state and prove the main gluing theorem that wewill use, namely Theorem 12.3.1.
Preliminary discussion.
We will use a technique very similar to that em-ployed in our earlier work [32]. The first step is to use cutoff functions to gluethe extremal metric on M to a model metric (the Burns-Simanca metric) on theblowup of C m at the origin, scaled down by a factor of ε . The gluing is performedon a small annulus around the point p . This results in a metric ω ε on Bl p M , whosescalar curvature is controlled, since the Burns-Simanca metric is scalar flat. Theproblem of perturbing ω ε to an extremal metric can then be written as finding azero of a map(8) F : C ∞ (Bl p M ) × h → C ∞ (Bl p M ) , where h is the space of Hamiltonian holomorphic vector fields on Bl p M . For theexact form of F , see Section 3.5 and note that in practice we must work with variousweighted H¨older spaces instead of C ∞ (Bl p M ). The main technical difficulty inconstructing an extremal metric on the blowup Bl p M is that in general the space h has lower dimension than the space g of Hamiltonian holomorphic vector fieldson M . The way this manifests itself in the analysis is that it is more difficult to LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 7 find a well-controlled right-inverse for the linearization of F as ε →
0. In [32] weovercome this problem by introducing a more general operator of the form(9) e F : C ∞ (Bl p M ) × g → C ∞ (Bl p M ) , such that if f ∈ h , then e F ( ϕ, f ) = F ( ϕ, f ). So if we find a zero e F ( ϕ, f ) = 0 with f ∈ h , then we have an extremal metric. A right-inverse is not hard to constructfor the linearization of e F , so we find a solution ( ϕ p , f p ). If we blow up at a differentpoint q , then we obtain a different pair ( ϕ q , f q ). The crucial point is to computethe leading terms in f p as ε →
0. This can be done by finding better approximatesolutions than ω ε . In [32] we found that the first non-trivial term is µ ( p ), using atechnique similar to [3] to improve the approximate solution. A finite dimensionalperturbation argument then shows that if the vector field µ ( p ) vanishes at p , thenfor small ε there is a point q ∈ G c · p such that f q vanishes at q . This gives us anextremal metric on Bl q M , but Bl q M is biholomorphic to Bl p M .To prove Theorem 1 we need to find more terms in the expansion of f p as ε → Possible generalizations.
We will now briefly discuss the new difficultiesthat arise in trying to generalize our results.3.2.1.
The case m = 2 . The main issue with the case when m = 2 is that we needto compute more terms in the expansion of f in Theorem 12. This can most easilybe seen in the formula for the Futaki invariant on a blowup in Corollary 36, sincethe term involving ∆ h v vanishes for m = 2. The algebro-geometric formula hasbeen computed to more terms in [32], and the problem is to construct sufficientlygood approximate solutions to obtain a corresponding expansion of f . We believethat this should be possible, but it needs a deeper analysis of the linearized problemthan what we have performed in the case m > More blowup points.
A more significant issue arises when we try to blow upmore than one point, in contrast to previous works [1, 2, 3, 32], where the numberof points made little difference. The new complication in Theorem 1 is that we needto perform a gluing construction at the points q δ in the G c -orbit of p , for which µ ( q δ )+ δ ∆ µ ( q δ ) vanishes at q δ , for arbitrarily small δ . In the borderline case when p is not relatively stable with respect to µ , the corresponding points q δ will approachthe boundary of the G c -orbit of p as δ →
0. When there is only one blowup point,then this is not a problem, since the geometry of the manifolds Bl q δ M is controlledas δ →
0. When we blow up an n -tuple ( p , . . . , p n ), however, then as we approachthe boundary of the orbit, some of the points may approach each other. In thiscase the geometry of the blowup is only controlled as long as the n -tuple staysaway from the “large diagonal” in the n -fold product M × M × . . . × M , where atleast two points coincide. This means that in order to use the same strategy aswhat we used in the proof of Theorem 1, we would need to obtain results analogousto Propositions 19 and 20, where the norm of the inverse will now depend on thedistance of our n -tuple from the large diagonal in a suitable sense. Alternativelyone may try to make contact with the results on constructing cscK or extremalmetrics on iterated blowups, developed for K¨ahler surfaces in LeBrun-Singer [18],Rollin-Singer [23] and Tipler [35] for instance. G´ABOR SZ´EKELYHIDI
Extremal metrics.
When (
M, ω ) is cscK, we were able to obtain a sharpexistence result for cscK metrics on the blowups Bl p M . Many of the argumentscan be adapted with little difficulty to extremal metrics, however we were not ableto show that when the hypothesis of Theorem 1 fails, then the blowups (Bl p M, [ ω ε ])are relatively K-unstable for sufficiently small ε . The basic reason is that the innerproduct of vector fields lifted to a blowup is not the same as their inner producton M . The inner product enters in the definition of relative stability, and in orderto obtain a sharp existence result, we would need a version of Theorem 1 wherethe inner product on g is also perturbed in order to match with the inner productof lifted vector fields. In practice this means that we would need to compute moreterms in the expansion of f , similarly to the m = 2 case. An alternative approachwould be to show directly that the map p f p above can itself be thought of as amoment map for a suitable perturbed K¨ahler form on M together with a perturbedinner product on g , without necessarily knowing the expansion explicitly.3.3. Burns-Simanca metric.
To obtain the first approximate solution ω ε , wewant to glue the extremal metric ω on M to a rescaling of a suitable model metricon Bl C m , ie. on the blowup of C m at the origin. This model metric is a scalarflat metric found by Burns (see LeBrun [16]) for m = 2 and by Simanca [24] for m >
3. Away from the exceptional divisor it can be written in the form η = i∂∂ (cid:18) | w | + ψ ( w ) (cid:19) , where w = ( w , . . . , w m ) are standard coordinates on C m . The function ψ can befound by solving an ODE. We will need the following result from Gauduchon [14]about the asymptotics of ψ . Note that we use i∂∂ as opposed to dd c which intro-duces a factor of 2 in our formula, and also our normalization of the volume of theexceptional divisor is different. Lemma 8. If m > then the K¨ahler potential for a suitable scaling of the Burns-Simanca metric η = i∂∂ ( | w | / ψ ( w )) satisfies (10) ψ ( w ) = − π m − ( m − | w | − m + d | w | − m + d | w | − m + O ( | w | − m ) , where d > . The scaling of η is such that the exceptional divisor has volume m − . The important aspect of this result for us is the formula for the coefficient of thefirst term, and the sign of the second. Note that the scaling is chosen in such a waythat if we construct ω ε as in Section 3.4, then we end up with a metric in the class π ∗ [ ω ] − ε [ E ].3.4. The metric ω ε on Bl p M . Suppose as before that ω is an extremal K¨ahlermetric on M . Let X s be the Hamiltonian vector field corresponding to the scalarcurvature s ( ω ). Write G for the Hamiltonian isometry group of ( M, ω ), so the Liealgebra g of G consists of holomorphic Killing fields with zeros.Choose a point p ∈ M where the vector field X s vanishes, and let T ⊂ G p be atorus fixing p whose Lie algebra contains X s . Let H ⊂ G consist of the elementscommuting with T and let us write h ⊂ C ∞ ( M ) for the space of Hamiltonian LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 9 functions of vector fields in the Lie algebra of H . Note that h contains the constantsas well. Let us also write t ⊂ h for the Hamiltonian functions corresponding to thesubgroup T ⊂ H .Given a small parameter ε >
0, we will construct an approximate solution toour problem on Bl p M in the K¨ahler class π ∗ [ ω ] − ε [ E ]. For simplicity assume thatthe exponential map is defined on the unit ball in the tangent space T p M (if not,we can scale up the metric ω ). Choose local normal coordinates z near p such thatthe group T acts by unitary transformations on the unit ball B around p (thisis possible by linearizing the action, see Bochner-Martin [5] Theorem 8). In thesecoordinates we can write ω = i∂∂ (cid:0) | z | / ϕ ( z ) (cid:1) , where ϕ = O ( | z | ). At the same time the Burns-Simanca metric from Section 3.3has the form η = i∂∂ (cid:0) | w | / ψ ( w ) (cid:1) . We glue ε η to ω using a cutoff function in the annulus B r ε \ B r ε in M , where r ε = ε α for some α < z = εw .To do this, let γ : R → [0 ,
1] be smooth such that γ ( x ) = 0 for x < γ ( x ) = 1for x >
2. Define γ ( r ) = γ ( r/r ε ) , and write γ = 1 − γ . Then for small ε we can define a K¨ahler metric ω ε on Bl p M which on the annulus B \ B ε is given by ω ε = i∂∂ (cid:18) | z | γ ( | z | ) ϕ ( z ) + γ ( | z | ) ε ψ ( ε − z ) (cid:19) . Moreover outside B r ε the metric ω ε = ω while inside the ball B r ε we have ω ε = ε η .Note that the action of T lifts to Bl p M giving biholomorphisms, and that ω ε is T -invariant.In order to define the operator e F from (9), we need to lift elements in h to Bl p M . Definition 9.
We define a linear map l : h → C ∞ (Bl p M ) as follows. Decompose h into a direct sum h = t ⊕ h ′ in such a way that eachfunction in h ′ vanishes at p . Each f ∈ t corresponds to a holomorphic Hamiltonianvector field X f on M vanishing at p . We then define l ( f ) to be the Hamiltonianfunction with respect to ω ε of the holomorphic lift of X f to Bl p M , normalized sothat f = l ( f ) outside B . For f ∈ h ′ we define l ( f ) = γ f near p using the cutofffunction γ from above. We can then think of this l ( f ) as a function on Bl p M .Finally we can extend l to all of h by linearity. Note that in contrast to [32] we are not assuming that T ⊂ G p is a maximaltorus. This is necessary for technical reasons, namely we will want to be able towork with all T -invariant points at the same time. On the other hand it impliesthat even if f ∈ h corresponds to a holomorphic vector field vanishing at p , its lift l ( f ) will not give rise to a holomorphic vector field, unless f ∈ t .We will also need lifts corresponding to metrics other than ω ε . If Ω = ω ε + i∂∂ Φand Φ is T -invariant, then we define l Ω ( f ) = l ( f ) + 12 ∇ Φ · ∇ f. If f ∈ t , then l Ω ( f ) is a Hamiltonian function for the vector field X f , with respectto Ω. In particular this has the following consequence. Lemma 10. If f ∈ t , then Z Bl p M l Ω ( f ) Ω m = Z Bl p M l ( f ) ω mε . Proof.
This can be checked by using Ω t = ω ε + ti∂∂ Φ, and differentiating withrespect to t . Alternatively, in the algebraic case, note that the integral of the Hamil-tonian function of a holomorphic vector field can be computed algebro-geometrically,so in particular it is independent of the metric. (cid:3) The extremal metric equation.
We will now write down what the opera-tors F and e F in (8) and (9) look like. We have a torus T ⊂ G fixing p , and we seeka T -invariant function ϕ on Bl p M such that ω ε + i∂∂ϕ is an extremal metric.We need the following which can also be found in [3], [32]. Lemma 11.
Suppose that ϕ ∈ C ∞ (Bl p M ) T and f ∈ t such that (11) s ( ω ε + i∂∂ϕ ) − ∇ l ( f ) · ∇ ϕ = l ( f ) , where the gradient and inner product are computed with respect to the metric ω ε .Then ω ε + i∂∂ϕ is an extremal metric. In order to solve Equation (11) as a perturbation problem, we will write it inthe form(12) s ( ω ε + i∂∂ϕ ) − ∇ l ( s + f ) · ∇ ϕ = l ( s + f ) , where s ∈ t is the scalar curvature of the extremal metric ω . The advantage of thisis that we now seek ϕ and f which are small, or in other words, setting ϕ = 0 and f = 0 we get an approximate solution to the equation.For any K¨ahler metric Ω let us define the operators L Ω and Q Ω by(13) s (Ω + i∂∂ϕ ) = s (Ω) + L Ω ( ϕ ) + Q Ω ( ϕ ) , where L is the linearized operator. A simple computation shows that L Ω ( ϕ ) = − ∆ ϕ − Ric(Ω) i ¯ j ϕ i ¯ j , and analysing this operator will be crucial later on. Note that we are using thecomplex Laplacian here which is half of the usual Riemannian one. The linearoperator appearing in the linearization of Equation (12) is then(14) ( ϕ, f ) L Ω ε ( ϕ ) − ∇ l ( s ) · ∇ ϕ − l ( f ) , which is closely related to the Lichnerowicz operator that we will discuss in Sec-tion 3.6.We can now state the main gluing result that we will prove, which correspondsto finding a zero of the operator e F in (9). The proof of this theorem will appear atthe end of Section 3.12. LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 11
Theorem 12.
Fix a torus T ⊂ G such that s ∈ t . There are constants ε > and κ > m with the following property. Suppose that p ∈ M is a fixed point of T . Forevery ε ∈ (0 , ε ) we can find u ∈ C ,α (Bl p M ) T and f ∈ h satisfying the equation (15) s ( ω ε + i∂∂u ) = l ω ε + i∂∂u ( f ) = l ( f ) + 12 ∇ l ( f ) · ∇ u. In addition the element f ∈ h has an expansion f = s + C − ε m − (cid:18) c − ε m ! s ( p ) (cid:19) µ ( p ) − ε m c ∆ µ ( p ) + O ( ε κ ) , for some κ > m , where C is a constant depending on ε , and c , c > . Assuming this result, we can prove Theorem 1.
Proof of Theorem 1.
By replacing p with a different point in its G c -orbit, we canassume that the stabilizer G p is a maximal compact subgroup of the complex sta-bilizer G cp . Since the scalar curvature s is G -invariant, the new point will also be acritical point of ∇ s .Let T ⊂ G p be a maximal torus. Then T c ⊂ G cp is also a maximal torus,and writing t for the Lie algebra, we have s ∈ t . Let H ⊂ G consist of theelements of G commuting with T . As above, we write h for functions on M whoseHamiltonian vector fields are in h . We apply Theorem 12 to the set M T of T -invariant points in M . For every q ∈ M T and ε ∈ (0 , ε ) we obtain a T -invariantfunction u q,ε ∈ C ,α (Bl q M ) and f q,ε ∈ h such that if f q,ε ∈ t , then we have anextremal metric on Bl q M . From Theorem 12 we know that f q,ε = s + C − ε m − (cid:18) c − ε m ! s ( q ) (cid:19) µ ( q ) − ε m c ∆ µ ( q ) + O ( ε κ ) , where κ > m , C is a constant depending on ε , and c , c >
0. Let us write µ ε ( q ) = − ε − m (cid:18) c − ε m ! s ( q ) (cid:19) − ( f q,ε − s − C )= µ ( q ) + ε c c ∆ µ ( q ) + O ( ε κ ′ ) , where κ ′ >
2. Then f q,ε ∈ t if and only if µ ε ( q ) ∈ t .We will now apply Theorem 6 to the action of G cT ⊥ on M , where G T ⊥ ⊂ H is the group introduced in Section 2. The corresponding moment maps µ T ⊥ and∆ µ T ⊥ are µ , ∆ µ projected to g T ⊥ . From the assumption of Theorem 1, togetherwith Proposition 4, we know that for some sufficiently small δ > q in the G cT ⊥ -orbit of p such that µ ( q ) + δ ∆ µ ( q ) ∈ g q . Moreover, since T c is a maximal torus in G cp , and G T ⊥ commutes with T , we have g q = t . It follows that µ T ⊥ ( q ) + δ ∆ µ T ⊥ ( q ) = 0 . Proposition 7 and Theorem 6 now imply that for all sufficiently small ε > q ∈ G cT ⊥ · p such that(16) pr g T ⊥ µ ε ( q ) = 0 . By construction, µ ε ( q ) ∈ h , i.e. µ ε ( q ) commutes with T , so (16) implies that µ ε ( q ) ∈ t . This implies that f q,ε ∈ t , and so we have obtained an extremal metric on Bl q M , in the class π ∗ [ ω ] − ε [ E ]. Since q ∈ G c · p , the manifold Bl q M isbiholomorphic to Bl p M . (cid:3) The Lichnerowicz operator.
For any K¨ahler metric Ω on a manifold X wehave the operator D Ω : C ∞ ( X ) → Ω , ( T , X ) , given by D ( ϕ ) = ∂ ∇ , ϕ where ∂ is the natural ∂ -operator on the holomorphictangent bundle. The Lichnerowicz operator is then the fourth order operator D ∗ Ω D Ω : C ∞ ( X ) → C ∞ ( X ) , whose significance is that the kernel consists of precisely those functions whose gra-dients are holomorphic vector fields. The relation to the operator in Equation (14)is that a computation (see eg. LeBrun-Simanca [17]) shows that(17) D ∗ Ω D Ω ( ϕ ) = − L Ω ( ϕ ) + 12 ∇ s (Ω) · ∇ ϕ. When comparing this to Equation (14), note that in general s (Ω ε ) is not equal to l ( s ). The difference will be sufficiently small though.3.7. The Lichnerowicz operator on weighted spaces.
As in Arezzo-Pacard [1,2], Arezzo-Pacard-Singer [3] and also [32], we need to study the invertibility of thelinearized operator between suitable weighted H¨older spaces on the blowup Bl p M .First we need to understand the behaviour the Lichnerowicz operator on weightedspaces on the manifolds M \ { p } and Bl p C m , and then obtain results about theblowup by “gluing” these spaces. This section is parallel to Section 5.1 in [32], butwe need slightly different results.Let us first consider M p = M \ { p } with the metric ω . For functions f : M p → R we define the weighted norm k f k C k,αδ ( M p ) = k f k C k,αω ( M \ B / ) + sup r< / r − δ k f k C k,αr − ω ( B r \ B r ) . Here the subscripts ω and r − ω indicate the metrics used for computing the cor-responding norm. The weighted space C k,αδ ( M p ) consists of functions on M \ { p } which are locally in C k,α and whose k · k C k,αδ norm is finite.We need the following result, which is Proposition 17 in [32]. As before, we havea torus T ⊂ G fixing the point p , and H ⊂ G is the centralizer of T . Proposition 13. If δ < , δ is not an integer, and α ∈ (0 , , then the operator C k,αδ ( M p ) T × t → C k − ,αδ − ( M p ) T ( ϕ, f )
7→ D ∗ ω D ω ϕ − f has a bounded right-inverse. Let us turn now to the manifold Bl C m with the Burns-Simanca metric η . Therelevant weighted H¨older norm is now given by k f k C k,αδ (Bl C m ) = k f k C k,αη ( B ) + sup r> r − δ k f k C k,αr − η ( B r \ B r ) . Here we abused notation slightly by writing B r ⊂ Bl C m for the set where | z | < r (ie. the pullback of the r -ball in C m under the blowdown map).The following is Proposition 18 from [32]. LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 13
Proposition 14. If δ > − m the operator C k,αδ (Bl C m ) → C k − ,αδ − (Bl C m ) ϕ
7→ D ∗ η D η ϕ has a bounded right inverse.If δ ∈ (3 − m, − m ) , let χ be a compactly supported function on Bl C m withnon-zero integral. The operator C k,αδ (Bl C m ) × R → C k − ,αδ − (Bl C m ) , ( ϕ, t )
7→ D ∗ η D η ( ϕ ) + tχ has a bounded right inverse. Weighted spaces on Bl p M . We will need to do analysis on the blown-upmanifold Bl p M endowed with the approximately extremal metric ω ε . For this wedefine the following weighted spaces, which are simply glued versions of the aboveweighted spaces on M \ { p } and Bl p C m .We define the weighted H¨older norms C k,αδ by k f k C k,αδ = k f k C k,αω ( M \ B ) + sup ε r / r − δ k f k C k,αr − ωε ( B r \ B r ) + ε − δ k f k C k,αη ( B ε ) . The subscripts indicate the metrics used to compute the relevant norm. This isa glued version of the two spaces defined in the previous section in the followingsense. If f ∈ C k,α (Bl p M ) and we think of Bl p M as a gluing of M \ { p } and Bl C m then γ f and γ f can naturally be thought of as functions on M \ { p } and Bl C m respectively. Then the norm k f k C k,αδ (Bl p M ) is comparable to k γ f k C k,αδ ( M p ) + ε − δ k γ f k C k,αδ (Bl C m ,η ) . Another way to think about the norm is that if k f k C k,αδ c then f is in C k,α (Bl p M ) and also for i k we have |∇ i f | c for r > |∇ i f | cr δ − i for ε r |∇ i f | cε δ − i for r ε. The norms here are computed with respect to the metric ω ε , and note that on B ε we have ω ε = ε η . We will often use the following to compare the different weightednorms: k f k C k,αδ k f k C k,αδ ′ , if δ ′ > δ,ε δ ′ − δ k f k C k,αδ ′ , if δ ′ < δ. Sometimes we will restrict the norm to subsets such as C k,αδ ( M \ B r ε ) and C k,αδ ( B r ε ). A crucial property of these weighted norms is that(18) k γ i k C ,α c for some constant c independent of ε , where γ i are the cutoff functions from Sec-tion 3.4.In addition we need the following lemma about lifting elements of h ⊂ C ∞ ( M )to C ∞ (Bl p M ) according to Definition 9. Lemma 15.
For any f ∈ h its lifting satisfies k l ( f ) k C ,α c | f | , k∇ l ( f ) k C ,α c | f | , for some constant c independent of ε . Here | · | is any fixed norm on h .Proof. Recall that we defined the lifting using a decomposition h = t ⊕ h ′ , wherethe functions in h ′ vanish at p . Suppose first that f ∈ h ′ . Since f vanishes at p , wehave k f k C ,α ( M p ) c | f | , where c is independent of f . It follows from the multiplication properties of weightedspaces and (18) that k l ( f ) k C ,α c | f | , from which the required inequalities follow.Now suppose that f ∈ t , and write X f for the holomorphic vector field on M corresponding to f . The result is clearly true for constants, so we can assumethat f vanishes at p . On the ball B r ε ⊂ M , the action of X f is given by unitarytransformations, generated by a matrix A , say. Outside B r ε the vector field isunchanged and the metrics ω and ω ε are uniformly equivalent. Inside B r ε themetric ω ε is uniformly equivalent to ε η . It is more convenient to work with η ,since that is a fixed metric, and we can then scale back depending on ε . Let f η be the Hamiltonian function of X f with respect to η , so f = ε f η . In terms of η we are working on the ball B R ε , and outside B the Hamiltonian f η is given by aquadratic function depending on A . It follows that we have pointwise bounds |∇ i f η ( x ) | η C i r ( x ) − i | A | , where r ( x ) = 1 inside B , and r ( x ) is the distance from the exceptional divisoroutside B . We can choose the norm | A | to coincide with the norm | f | chosen onthe finite dimensional vector space h . Rescaling this inequality, together with whatwe already know outside B r ε , we get k l ( f ) k C k,α C | f | , which implies the resultsthat we want. (cid:3) The linearized operator on Bl p M . We now begin studying the linearizedoperator on Bl p M , in terms of the weighted spaces introduced in the previoussection. The constants that appear below will be independent of ε unless thedependence is made explicit.Recall that for any metric Ω we write L Ω ( ϕ ) = − ∆ ϕ − Ric(Ω) i ¯ j ϕ i ¯ j . We want to first study how this varies as we change the metric. For this we havethe following, which is Proposition 20 from [32].
Proposition 16.
Suppose that δ < . There exist constants c , C > such that if k ϕ k C ,α < c then k L ω ϕ ( f ) − L ω ε ( f ) k C ,αδ − C k ϕ k C ,α k f k C ,αδ , where ω ϕ = ω ε + i∂∂ϕ . LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 15
One consequence is an estimate for the nonlinear operator Q ω ε in the formula(19) s ( ω ε + i∂∂ϕ ) = s ( ω ε ) + L ω ε ( ϕ ) + Q ω ε ( ϕ ) . The following is Lemma 21 in [32].
Lemma 17.
Suppose that δ < . There exists a c > such that if k ϕ k C ,α , k ψ k C ,α c , then k Q ω ε ( ϕ ) − Q ω ε ( ψ ) k C ,αδ − C ( k ϕ k C ,α + k ψ k C ,α ) k ϕ − ψ k C ,αδ . We will need one further result, which was not used in [32].
Lemma 18.
Suppose that ω = ω ε + i∂∂ϕ , and k ϕ k C ,α , k ψ k C ,α c , for some sufficiently small c . Then (20) k Q ω ( ψ ) − Q ω ε ( ψ ) k C ,αδ − C k ϕ k C ,α k ψ k C ,α k ψ k C ,αδ . Proof.
Let us write g for a metric, and g + h for a small perturbation, thought ofas matrices in local coordinates. We can write schematically s ( g + h ) = ( g + h ) − ∂ log det( g + h )= g − ( I + g − h ) − ∂ (log det g + log det( I + g − h )) , where I is the identity matrix. Expanding in power series, we find that Q is of theform Q g ( h ) = X i =0 g − (cid:2) ∂ i ( g − h ) (cid:3) F i ( g − h ) , where the F i are power series. In order to estimate Q g ( h ) − Q g ( h ) it is enough toconsider a typical term, for instance g − (cid:2) ∂ ( g − h ) (cid:3) ( g − h ) l − g − (cid:2) ∂ ( g − h ) (cid:3) ( g − h ) l , for some l >
0. In our situation h = i∂∂ψ , and we have k g − − g − k C ,α C k ϕ k C ,α Cc , k g − j h k C ,α C k ψ k C ,α Cc , k g − j h k C ,αδ − C k ψ k C ,αδ , for j = 1 ,
2. From this it is a straightforward calculation to check the estimate(20). (cid:3)
The heart of the matter is to understand the invertibility of the linearized oper-ator of our problem on Bl p M . The following is Proposition 22 from [32]. Proposition 19.
For sufficiently small ε and δ ∈ (4 − m, the operator G : ( C ,αδ ) T × h → ( C ,αδ − ) T ( ϕ, f ) L ω ε ( ϕ ) − ∇ s ( ω ε ) · ∇ ϕ − l ( f ) has a right inverse P , with the operator norm k P k < C for some constant C independent of ε . We will need a slight variation of this result as well, dealing with weights in therange (3 − m, − m ). One can easily obtain a result for δ ∈ (3 − m, − m ) fromthe preivous proposition, but we will only have a bound of the form Cε δ − (4 − m ) forthe inverse. It turns out that if we restrict the range to functions with zero mean,we can obtain an inverse with norm bounded independent of ε . Proposition 20.
Let us write ( C ,αδ − ) T for the elements in ( C ,αδ − ) T which havezero mean on Bl p M , and h for the elements f ∈ h such that l ( f ) has zero meanon Bl p M with respect to ω ε . For sufficiently small ε , and δ ∈ (3 − m, − m ) , theoperator G : ( C ,αδ ) T × h → ( C ,αδ − ) T ( ϕ, f ) L ω ε ( ϕ ) − ∇ s ( ω ε ) · ∇ ϕ − l ( f ) has a right inverse P , with the operator norm k P k < C for some constant C independent of ε .Proof. The proof is very similar to the proof of Proposition 22 in [32], for the m = 2case. The idea is to first work with the operator G : ( C ,αδ ) T × h × R → ( C ,αδ − ) T ( ϕ, f, t )
7→ D ∗ ω ε D ω ε − l ( f ) + tχ, where χ is the function from Proposition 14 and D ∗ ω ε D ω ε = − L ω ε ( ϕ ) + 12 ∇ s ( ω ε ) · ∇ ϕ. One can then use the inverses in Propositions 13 and 14 to construct an approximateright inverse for G , which in turn can be used to show that G has a bounded rightinverse. If ψ ∈ ( C ,αδ − ) T , then we can use this to find ϕ ∈ ( C ,αδ ) T , f ∈ h , t ∈ R such that D ∗ ω ε D ω ε ( ϕ ) − l ( f ) + tχ = ψ. Integrating this over Bl p M we find that t = 0. This shows that we have constructedan inverse for G . (cid:3) Remark . We will need analogous results for operators corresponding to a per-turbation Ω = ω ε + i∂∂ Φ. We have k ( D ∗ ω ε D ω ε − D ∗ Ω D Ω ) ϕ k C ,αδ − C k Φ k C ,α k ϕ k C ,αδ , and k l ( f ) − l Ω ( f ) k C ,αδ − C k∇ l ( f ) · ∇ Φ k C ,αδ − C | f |k Φ k C ,αδ − C | f |k Φ k C ,α , if δ − <
2. So as long as k Φ k C ,α is sufficiently small, we can deduce the invertibilityof the operators corresponding to Ω from the invertibility of those correspondingto ω ε . Note also that(21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Bl p M l Ω ( f ) Ω m − Z Bl p M l ( f ) ω mε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z Bl p M |∇ l ( f ) · ∇ Φ | Ω m + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Bl p M l ( f ) (Ω m − ω mε ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C | f | k Φ k C ,α , LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 17 so if f ∈ h , then we can adjust f while preserving its norm up to a factor, to ensurethat l Ω ( f ) has zero integral with respect to Ω, as long as k Φ k C ,α is sufficiently small.It follows that both propositions can be applied to small perturbations of ω ε .3.10. The approximate solution Ω . We will now work on obtaining a metricΩ on Bl p M , which is closer to being extremal than our previous candidate ω ε . Inthe next section we will use this to find an even better approximate solution Ω ,at which point we will be able to use the contraction mapping theorem to obtain asolution of our equation.There are 3 regions in Bl p M which we need to think about differently, namelythe region Bl p M \ B r ε , the annular region B r ε \ B r ε on which our cutoff fuctions γ and γ live, and B r ε . Here r ε = ε α , and from now on we will work with α = 2 m − m + 1 . Like in Section 3.4, let us write the metric ω in coordinates near the point p . Wewill need a more precise expansion than before, so we write(22) ω = i∂∂ (cid:18) | z | A ( z ) + A ( z ) + ϕ ( z ) (cid:19) , where A ( z ) , A ( z ) are quartic and quintic in z respectively, and ϕ ∈ C k,α ( M p ).Also let us write s = s ( ω ) for the scalar curvature of ω . Lemma 22.
Suppose that ∇ s vanishes at p . Then we have ∆ A = − s ( p ) , ∆ A = 0 , where ∆ is the Laplacian with respect to the Euclidean metric.Proof. This follows from computing the scalar curvature of ω as a perturbation ofthe flat metric near p . s ( ω ) = − ∆ ( A + A + ϕ ) + Q ( A + A + ϕ ) . From Lemma 17 it follows that Q ( A + A + ϕ ) ∈ C k,α ( B \ { p } ) , and so s ( ω ) + ∆ ( A + A ) ∈ C k,α , near p . Since ∆ A is a constant and ∆ A is linear, the result follows. (cid:3) As for the rescaled Burns-Simanca metric, after a change of coordinates we canwrite it as(23) ε η = i∂∂ (cid:16) | z | d ε m − | z | − m + d ε m | z | − m + d ε m − | z | − m + ε ψ − m ( ε − z ) (cid:17) , where ψ − m ∈ C k,α − m (Bl C m ), and d = − π m − ( m − .These are the two metrics that we want to glue across the annular region B r ε \ B r ε . In the construction of ω ε we performed this gluing by multiplying all the termsexcept | z | / we want to only multiply ϕ and ψ − m by cutoff functions. For this we need tomodify ε η so that it contains A ( z ) , A ( z ), and we need to modify ω by d ε m − | z | − m + d ε m | z | − m + d ε m − | z | − m . Let us focus on ω first. For this we have the following. Lemma 23.
We can find T -invariant functions G , G on M \ { p } such that dis-tributionally on M we have (24) D ∗ ω D ω G = f + 4 π m ( m − δ p D ∗ ω D ω G = f − π m ( m − δ p , for some f , f ∈ h , and for any δ > we have (25) G − | z | − m ∈ C k,α − m − δ ( M p ) ,G − | z | − m ∈ C k,α − m − δ ( M p ) . In addition f = − π m ( m − V − + µ ( p )) ,f = 2 π m ( m − µ ( p ) , where V = Vol( M ) .Proof. Let us define e G using a cutoff function to be equal to zero on M \ B , andequal to | z | − m on B / . Comparing ω to the flat metric near p , we find that D ∗ ω D ω e G ∈ C k − ,α − m ( M p ) . We can now use the inverse in Proposition 13. For small δ > − m − δ ∈ (4 − m,
0) (for m > δ = 0), so we obtain a ϕ ∈ C k,α − m − δ and f ∈ h such that D ∗ ω D ω ( e G − ϕ ) = f , on M p , and so we can let G = e G − ϕ . The only contribution to the distributional part of D ∗ ω D ω e G comes from ∆ | z | − m in the flat metric, giving the result.Similarly we define e G using a cutoff function to equal | z | − m on B / and tovanish outside B . Comparing with the flat metric again, we obtain D ∗ ω D ω e G ∈ C k − ,α − m ( M p ) . Once again we can find ϕ ∈ C k,α − m − δ for small δ > − m is an indicialroot), and f ∈ h such that D ∗ ω D ω ( e G − ϕ ) = f , on M p . In this case, there are several contributions to the distributional part at p , but apartfrom the leading contribution of ∆ δ p , the rest is a multiple of δ p . We can thereforefind a constant C such that G = e G − ϕ + CG satisfies our requirements.In order to find f , we take the L -product of (24) with any element g ∈ h , toobtain 0 = h f , g i + 4 π m ( m − g ( p ) , LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 19 since D ∗ ω D ω g = 0. By definition g ( p ) = h µ ( p ) , g i , and so it follows that the projec-tion of f onto h must be pr h f = − π m ( m − µ ( p ) . To obtain f from this, we just need to take the L -product of (24) with the function1. We can obtain the formula for f similarly. (cid:3) Lemma 24.
We can find a function Γ on M \ B r ε such that (26) D ∗ ω D ω Γ = − h on M \ B for some h ∈ h , and satisfying the following properties. On B r ε \ B r ε the function Γ has the form Γ = d ε m − | z | − m + d ε m | z | − m + d ε m − | z | − m + Γ , where k Γ k C k,α − m ( B rε \ B rε ) = O ( ε κ ) , for some κ > m . On M \ B r ε we have (27) (cid:13)(cid:13)(cid:13) s ( ω + i∂∂ Γ) − ∇ s ( ω ) · ∇ Γ − s ( ω ) − h (cid:13)(cid:13)(cid:13) C ,α − − m ( M \ B ε ) = O ( ε κ ) , for some κ > m . In addition (28) h = − πε m − ( m − V − + µ ( p )) + ε m s ( p ) m ! ( V − + µ ( p )) − d ε m π m ( m − µ ( p )= C − ε m − (cid:18) c − ε m ! s ( p ) (cid:19) µ ( p ) − ε m c ∆ µ ( p ) , where V = Vol( M ) , C is a constant, and c , c > .Proof. Let us use a cutoff function to define G to be zero outside B , and equalto | z | − m in B / . We letΓ = d ε m − G + d ε m G − ε m ( m − s ( p )4 π m m ! G + d ε m − G , where G and G are defined in Lemma 23. Then (26) and (28) follow from theproperties of G , G . On the annulus B r ε \ B r ε we haveΓ = d ε m − | z | − m + d ε m | z | − m + d ε m − | z | − m + Γ , where Γ = d ε m − ( G − | z | − m ) + d ε m ( G − | z | − m ) + Cε m | z | − m , for some constant C . From (25) it follows that k Γ k C k,α − m ( B rε \ B rε ) = O ( ε κ )for some κ > m . For instance k G − | z | − m k C k,α − m ( B rε \ B rε ) Cr − δε k G − | z | − m k C k,α − m − δ ( B rε \ B rε ) , and for sufficiently small δ > ε m − r − δε = O ( ε κ ) , for some κ > m , by our choice of r ε , since2 m − m + 1 > m >
3. The other terms are larger and are handled similarly.For the scalar curvature of ω + i∂∂ Γ we have s ( ω + i∂∂ Γ) − ∇ s ( ω ) · ∇ Γ − s ( ω ) − h = Q ω (Γ) − D ∗ ω D ω Γ − h . We will work on the 3 regions M \ B , B \ B / and B / \ B r ε separately.On M \ B we have D ∗ ω D ω Γ + h = 0, and k Q ω (Γ) k C ,α − − m ( M \ B ) C k Γ k C ,α k Γ k C ,α − m . The weight is irrelevant outside B , so we have k Q ω (Γ) k C ,α ( M \ B ) C ( ε m − ) = O ( ε κ )for some κ > m , as long as m > B \ B / we can still ignore the weights, so we still have the same estimatefor Q ω (Γ), but now D ∗ ω D ω Γ + h = Cε m − D ∗ ω D ω ( G ) . It follows from this that kD ∗ ω D ω Γ + h k C ,α ( B \ B / ) = O ( ε m − ) = O ( ε κ )for κ > m , as long as m > B / \ B r ε . It is best to work on the annuli A r = B r \ B r , for r ∈ ( r ε , / D ∗ ω D ω Γ + h = d ε m − D ∗ ω D ω | z | − m , so kD ∗ ω D ω Γ+ h − d ε m − ∆ | z | − m k C ,α − − m = d ε m − k ( D ∗ ω D ω − ∆ ) | z | − m k C ,α − − m . On the annulus A r we have k ( D ∗ ω D ω − ∆ ) | z | − m k C ,α − − m k ϕ k C ,α k| z | − m k C ,α − m Cr · r − m , where ϕ = O ( | z | ). We have r > r ε , and so ε m − r − m = O ( ε κ )for some κ > m , since α < Q ω (Γ) = Q ω ( d ε m − | z | − m ) + h Q ω (Γ) − Q ω ( d ε m − | z | − m ) i . The next highest order term in Γ after d ε m − | z | − m is d ε m | z | − m , so on theannulus A r k Q ω (Γ) − Q ω ( d ε m − | z | − m ) k C ,α − − m Cε m k| z | − m k C ,α k| z | − m k C ,α − m Cε m r − m r − = O ( ε κ ) , for κ > m , since α < m m + 1 . LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 21
What remains is to estimate Q ω ( d ε m − | z | − m ) − ∆ ( d ε m − | z | − m ) . It is not hard to check that both terms are of the same order, and for m > m = 3 we need to work harder. On the annulus A r , using Lemma 18, we have k Q ω ( d ε m − | z | − m ) − Q ( d ε m − | z | − m ) k C ,α − − m Cr ε m − k| z | − m k C ,α k| z | − m k C ,α − m Cε m − r − m = O ( ε κ ) , for κ > m , where Q is given by the flat metric. Using that ε η is scalar flat, wehave0 = s h i∂∂ (cid:16) | z | d ε m − | z | − m + d ε m | z | − m + d ε m − | z | − m + ε ψ − m ( ε − z ) (cid:17)i , so if we write ε η = i∂∂ (cid:16) | z | + ε ψ ( ε − z ) (cid:17) , then we get − ∆ ( d ε m − | z | − m ) + Q ( d ε m − | z | − m ) = ∆ ( ε ψ − m ( ε − z ))++ Q ( d ε m − | z | − m ) − Q ( ε ψ ( ε − z )) , and so on A r we have k − ∆ ( d ε m − | z | − m ) + Q ( d ε m − | z | − m ) k C ,α − − m Cε m − k| z | − m k C ,α − m ++ Cε m k| z | − m k C ,α k| z | − m k C ,α − m Cε m − r − m + Cε m r − − m = O ( ε κ ) , for κ > m , where we used that the largest order term in ε ψ ( ε − z ) after theleading term is ε m | z | − m . Combining all these estimates, we obtain the requiredbound (27). (cid:3) Now to deal with modifying ε η , we have the following. Lemma 25.
We can find a function Ψ on Bl C m of the form Ψ = A ( z ) + A ( z ) + Ψ , where k Ψ k C k,α − m ( B rε \ B rε ) = O ( ε κ ) , and in addition k s ( ε η + i∂∂ Ψ) − s ( p ) k C ,α − − m ( B rε ) = O ( ε κ ) , for some κ > m .Proof. We will work in terms of η with the variable w = ε − z . Write R ε = ε − r ε ,so that in terms of w , we are gluing on the annulus B R ε \ B R ε .Write e A ( w ) , e A ( w ) for the functions A ( w ) and A ( w ) cut off on the annulus B R ε \ B R ε . Since k ε e A + ε e A k C ,α Cε R ε ≪ , we have k Q η ( ε e A + ε e A ) k C ,α C ( ε R ε )( ε ) = Cε R ε . It follows from Proposition 14 that we can find e Ψ such that D ∗ η D η e Ψ = Q η ( ε e A + ε e A ) , and k e Ψ k C ,α Cε R ε = Cε r ε . Setting Ψ ( z ) = ε e Ψ( ε − z ), we then have k Ψ k C ,α − m ( B rε \ B rε ) Cr m +1 ε k Ψ k C ,α ( B rε \ B rε ) Cr m +1 ε r ε = O ( ε κ ) , for κ > m , since α > m m + 3 . In addition, using Lemma 22, s ( η + i∂∂ ( ε e A + ε e A + e Ψ)) = L η ( ε e A + ε e A + e Ψ) + Q η ( ε e A + ε e A + e Ψ)= ε s ( p ) + L η ( e Ψ) + ( L η + ∆ )( ε e A + ε e A )+ Q η ( ε e A + ε e A )+ Q η ( ε e A + ε e A + e Ψ) − Q η ( ε e A + ε e A ) . Using the equality D ∗ η D η = L η we have L η ( e Ψ) + Q η ( ε e A + ε e A ) = 0. Using thefact that η differs from the flat metric by order | w | − m , we get k s ( η + i∂∂ ( ε e A + ε e A + e Ψ)) − ε s ( p ) k C ,α − − m ( B Rε ) C ( ε R ε + ε R mε ) = O ( ε δ ) , for some δ >
0. Since on the ball B R ε in terms of w (and on B r ε in terms of z ) wehave s ( ε η + i∂∂ Ψ) = ε − s ( η + i∂∂ ( ε e A + ε e A + e Ψ)) , it follows that k s ( ε η + i∂∂ Ψ) − s ( p ) k C ,α − − m ( B rε ) = O ( ε κ ) , for some κ > m . (cid:3) We now define our new approximate metric Ω to be equal to ω + i∂∂ Γ on M \ B r ε , equal to ε η + i∂∂ Ψ on B r ε , and on the annular region B r ε \ B r ε we letΩ = i∂∂ (cid:16) | z | A + A + γ ϕ + γ Γ + d ε m − | z | − m + d ε m | z | − m + d ε m − | z | − m + γ ε ψ − m ( ε − z ) + γ Ψ (cid:17) . Lemma 26.
On the annular region B r ε \ B r ε we have k s (Ω ) − s ( p ) k C ,α − − m ( B rε \ B rε ) = O ( ε κ ) , for some κ > m .Proof. We compute the scalar curvature of Ω as a perturbation of the metric ω = i∂∂ (cid:16) | z | d ε m − | z | − m + d ε m | z | − m + d ε m − | z | − m (cid:17) on the annulus B r ε \ B r ε . Since s ( ω + ε i∂∂ψ − m ( ε − z )) = s ( ε η ) = 0 , LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 23 we have s ( ω ) = − ε L ω ( ψ − m ( ε − z )) − Q ω ( ε ψ − m ( ε − z ))= ε ∆ ( ψ − m ( ε − z )) + lower order terms . It follows that k s ( ω ) k C ,α − − m ( B rε \ B rε ) = O ( ε m − r − mε ) = O ( ε κ )for some κ > m . Then s (Ω ) = s ( ω ) + L ω ( A + A + γ ϕ + γ Γ + γ ε ψ − m ( ε − z ) + γ Ψ )+ Q ω ( A + A + γ ϕ + γ Γ + γ ε ψ − m ( ε − z ) + γ Ψ ) . On the annulus B r ε \ B r ε we have A + A + γ ϕ + γ Γ + γ ε ψ − m ( ε − z ) + γ Ψ = A + lower order terms , and also k γ ϕ + γ Γ + γ ε ψ − m ( ε − z ) + γ Ψ k C ,α − m = O ( r m +3 ε + ε m − r − mε + ε κ )= O ( ε κ ) , for some κ > m , and ∆ ( A + A ) = − s ( p ) . It follows that k s (Ω ) − s ( p ) k C ,α − − m k ( L ω + ∆ )( A + A ) k C ,α − − m + O ( ε κ )+ C k A k C ,α k A k C ,α − m = O ( ε κ ) , for some κ > m . (cid:3) Let us write Ω = ω ε + i∂∂u . From the results above, we find Lemma 27.
We have (29) k s (Ω ) − l Ω ( s + h ) k C ,α − − m = O ( ε κ ) , for some κ > m , and (30) k u k C k,α = O ( ε δ ) , for some δ > .Proof. We work on the regions M \ B r ε and B ε separately. On M \ B r ε we have u = Γ, ω ε = ω and l ( h ) = h , so the result follows from Lemma 24, together with k l Ω ( h ) − h k C ,α − − m = (cid:13)(cid:13)(cid:13)(cid:13) ∇ Γ · ∇ h (cid:13)(cid:13)(cid:13)(cid:13) C ,α − − m Cε m − = O ( ε κ ) , for κ > m , since m > B r ε we have k l Ω ( h ) k C ,α − − m Cr m +1 ε k l Ω ( h ) k C ,α Cr m +1 ε ε m − = O ( ε κ )for some κ > m , since α > m +1 . In addition k l Ω ( s − s ( p )) k C ,α − − m Cr m +3 ε k l Ω ( s − s ( p )) k C ,α Cr m +3 ε = O ( ε κ ) , for some κ > m . The bound (29) then follows from Lemmas 25 and 26. As for (30), note that outside B r ε the leading term in u is of order ε m − | z | − m ,while inside B r ε the leading term is of order | z | . The bound (30) is then easy tocheck. (cid:3) The approximate solution Ω . We need to modify Ω once more to obtaina metric Ω = Ω + i∂∂u . For this we want to find ( u , h ) solving(31) − D ∗ Ω D Ω u − l Ω ( h ) = l Ω ( s + h ) − s (Ω ) . We will use the inverse operator from Proposition 20, so in addition to the estimatefrom Lemma 27, we need to bound the integral of l Ω ( s + h ) − s (Ω ). For this wehave the following. Lemma 28.
We have Z Bl p M l Ω ( s + h ) − s (Ω ) Ω m m ! = O ( ε κ ) , for some κ > m .Proof. First note that from Lemma 10 we have Z Bl p M l Ω ( s ) − s (Ω ) Ω m m ! = Z Bl p M l ( s ) − s ( ω ε ) ω mε m ! , since s ∈ t and the total scalar curvature is an invariant of the K¨ahler class.We have(32) Z Bl p M l Ω ( h ) Ω m − Z Bl p M l ( h ) ω mε = Z Bl p M ∇ l ( h ) · ∇ u Ω m ++ Z Bl p M l ( h ) (Ω m − ω mε ) . On M \ B , we can bound this by | h | k u k C ,α , as in Equation (21). We have | h | k u k C ,α ( M \ B ) Cε m − ε m − = O ( ε κ ) , for some κ > m if m >
2. To bound (32) on B \ B r ε , note that k u k C ,α − m ( M \ B rε ) = O ( ε m − ) , and | l ( h ) | + |∇ l ( h ) | Cε m − from Lemma 15. It follows that k l ( h ) (Ω m − ω mε ) k C − m ( B \ B rε ) Cε m − . The integral on B \ B r ε is then bounded by Z r ε ε m − r − m r m − dr Cε m − . Similarly k∇ l ( h ) · ∇ u k C − m Cε m − , so the other term in (32) is also bounded by ε m − . Since m >
2, we have ε m − = O ( ε κ ) for some κ > m .The lift l ( h ) equals h outside B r ε , and the volume of B r ε is of order r mε , sowe have Z Bl p M l ( h ) ω mε m ! = Z M h ω m m ! + O ( ε m − r mε ) = Z M h ω m m ! + O ( ε κ ) , LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 25 for some κ > m .Since s ∈ t , we can relate the integrals on Bl p M and on M using the formulasin Proposition 35. We have Z Bl p M l ( s ) ω mε m ! = Z M s ω m m ! − ε m m ! s ( p ) + O ( ε κ ) , for some κ > m , and also Z Bl p M s ( ω ε ) ω mε m ! = Z M s ω m m ! − πε m − ( m − . Combining these formulas, we have Z Bl p M l ( s + h ) − s ( ω ε ) ω mε m ! = Z M h ω m m ! + 2 πε m − ( m − − ε m m ! s ( p ) + O ( ε κ ) . From the formula for h in Lemma 24, the first 3 terms on the right cancel,leaving only O ( ε κ ). (cid:3) Letting C be the average of l Ω ( s + h ) − s (Ω ) with respect to Ω , we can applyProposition 20 to find u and h ′ ∈ h such that −D ∗ Ω D Ω u − l Ω ( h ′ ) = l Ω ( s + h ) − s (Ω ) − C. Letting h = h ′ − C , this implies that we can solve (31), with(33) k u k C ,α − m + | h | = O ( ε κ ) , with κ > m . We now let Ω = Ω + i∂∂u . This satisfies the following. Lemma 29. k s (Ω ) − l Ω ( s + h + h ) k C ,α − m = O ( ε κ ) , for some κ > m .Proof. First we have s (Ω ) = s (Ω ) + L Ω ( u ) + Q Ω ( u )= s (Ω ) − D ∗ Ω D Ω ( u ) + 12 ∇ s (Ω ) · Ω ∇ u + Q Ω ( u )= l Ω ( s + h + h ) + 12 ∇ s (Ω ) · Ω ∇ u + Q Ω ( u )= l Ω ( s + h + h ) − ∇ l ( s + h + h ) · ∇ u + 12 ∇ s (Ω ) · Ω ∇ u + Q Ω ( u ) , where by · Ω we indicate that the gradients and inner products are taken withrespect to Ω instead of ω ε . We need to estimate ∇ s (Ω ), and for this we have k s (Ω ) − s ( ω ε ) k C ,α − = k L ω ′ ( u ) k C ,α − C k u k C ,α C, where ω ′ = ω ε + ti∂∂u for some t ∈ [0 , k∇ l ( s + h + h ) · ∇ u − ∇ s (Ω ) · Ω ∇ u k C ,α − m C k u k C ,α − m = O ( ε κ ) , by (33). So we only need to estimate Q Ω ( u ), but for this Lemma 17 implies k Q Ω ( u ) k C ,α − m C k u k C ,α k u k C ,α − m Cε − m k u k C ,α − m ε − k u k C ,α − m = O ( ε κ ) , using (33) again. (cid:3) Solving the non-linear equation.
We are finally ready to try solving theequation we need to, i.e. we want u, h such that s (Ω + i∂∂u ) = l Ω + i∂∂u ( s + h + h + h ) . Expanding this in terms of the linearized operator, we have s (Ω ) + L Ω ( u ) + Q Ω ( u ) = l Ω ( s + h + h ) + 12 ∇ u · ∇ l ( s + h + h )+ l Ω ( h ) + 12 ∇ u · ∇ h, which we can write as e G ( u, h ) = l Ω ( s + h + h ) − s (Ω ) + 12 ∇ u · ∇ h − Q Ω ( u ) , where e G is defined by e G ( u, h ) := L Ω ( u ) − ∇ u · ∇ l ( s + h + h ) − l Ω ( h ) . It follows from Remark 21, and Lemma 29 that this operator e G is sufficientlyclose to the operator G in Proposition 19 when ε ≪
1, so that the inverse P fromProposition 19 can be used to obtain an inverse e P for e G with uniformly boundednorm. We are therefore trying to solve the fixed point problem( u, h ) = N ( u, h ) , where N ( u, h ) : C ,αδ × h → C ,αδ × h ( u, h ) e P (cid:16) l Ω ( s + h + h ) − s (Ω ) + 12 ∇ u · ∇ h − Q Ω ( u ) (cid:17) , and δ = 4 − m + τ for sufficiently small τ > Lemma 30.
There is a constant c > such that if k v i k C ,α , | g i | < c , for i = 1 , , then kN ( v , g ) − N ( v , g ) k C ,αδ × h k ( v − v , g − g ) k C ,αδ × h . We can now complete the proof of Theorem 12.
Proof of Theorem 12.
From Lemma 29 we have kN (0 , k C ,αδ × h c ε κ ′ , for some κ ′ > m . Define S = { ( v, g ) : k v k C ,αδ , | g | c ε κ ′ } . LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 27
If ( v, g ) ∈ S , then for sufficiently small ε we have | g | < c with the c from Lemma 30and also k v k C ,α Cε δ − k v k C ,αδ Cc ε κ ′ + δ − < c , for sufficiently small ε , since κ ′ + δ − >
0. It follows that kN ( v, g ) k kN ( v, g ) − N (0 , k + kN (0 , k k ( v, g ) k + c ε κ ′ c ε κ ′ , so N is a contraction mapping S into itself. We can therefore find a fixed point( u, h ) of N in S , and this gives a solution of the equation s (Ω + i∂∂u ) = l Ω + i∂∂u ( s + h + h + h ) . From Lemma 24, Equation (33) and the fact that | h | c ε κ ′ , we have s + h + h + h = s + h + O ( ε κ )= s − ε m − π ( m − V − + µ ( p )) + ε m s ( p ) m ! ( V − + µ ( p )) − d ε m π m ( m − µ ( p ) + O ( ε κ ) , for some κ > m . This is what we wanted to prove. Since the solution is obtainedusing the contraction mapping principle, the solution will depend smoothly on theparameters. In particular when we perform this construction at the set of all T -invariant points in M , then the constant in O ( ε κ ) can be chosen to be uniform. (cid:3) Relative stability of blowups
In this section we will give the proof of Theorem 2. Since we want to dealwith K¨ahler manifolds which are not necessarily algebraic, we will reformulate asimple version of the usual theory of K-stability in the K¨ahler setting, which is moresimilar to Tian’s original definition in [34] than to the more recent algebro-geometricapproach of Donaldson [11].4.1.
Relative K-stability for K¨ahler manifolds.
We will define relative K-stability for K¨ahler manifolds similarly to Tian’s definition [34]. Our definition willactually be simpler since we only consider test-configurations with smooth centralfibers.
Definition 31.
A (smooth) test-configuration for a K¨ahler manifold ( M, ω ) consistsof a holomorphic submersion τ : X → C together with a holomorphic lift v of thevector field ∂∂θ , satisfying the following properties: (1) X admits a K¨ahler metric Ω , for which v is a Hamiltonian Killing field. (2) The pair ( τ − (1) , Ω | τ − (1) ) is isometric to ( M, ω ) . The vector field v gives a Hamiltonian Killing field on the central fiber ( M , ω ),which we will denote by v also. Let us write h v for its Hamiltonian function. TheFutaki invariant of this vector field is defined (see Futaki [12], Calabi [7]) to be(34) Fut( M , [ ω ] , v ) = Z M h v ( s − s ( ω )) ω n , where s is the average of the scalar curvature of ω . The notation indicates that theFutaki invariant does not depend on the particular metric chosen, only its K¨ahlerclass. For the definition of relative stability, we need to recall the extremal vector fielddefined by Futaki-Mabuchi [13]. Let us write Aut ( M ) for the connected componentof the identity in the automorphism group of M , and let g Aut ( M ) be the kernelof the map from Aut ( M ) to the Albanese torus of M . Finally, let G ⊂ g Aut ( M )be a maximal compact subgroup. Suppose that ω is G -invariant. Then the actionof G is Hamiltonian with respect to ω (see e.g. LeBrun-Simanca [17]). This meansthat we can identify elements in the Lie algebra g with their Hamiltonian functions,normalized to have zero mean. Let us write s ext = proj g s ( ω ) , for the L -projection of the scalar curvature of ω onto g . The extremal vector fieldis the vector field corresponding to s ext . The main result in [13] is that this vectorfield is independent of the choice of G -invariant metric ω .Suppose now that T is a maximal torus in G . We say that the test-configuration( X , v ) is compatible with T , if there is a Hamiltonian holomorphic T -action on X preserving the fibers, leaving v invariant, and such that when restricted to τ − (1) itrecovers the T -action on M . We define the modified Futaki invariant of such a test-configuration as follows. As before, the central fiber of the test-configuration hasan induced Hamiltonian Killing field v . Again, let h v be a Hamiltonian functionfor v , normalized to have zero mean. In addition, ( M , ω ) is equipped with aHamiltonian T -action, and in particular the extremal vector field of M inducesa Hamiltonian holomorphic vector field v ext on M , with a Hamiltonian function h v ext . Definition 32.
The modified Futaki invariant of the test-configuration is definedto be
Fut v ext ( M , [ ω ] , v ) = Z M h v ( h v ext − s ( ω )) ω n . Note, in particular, that this coincides with the usual Futaki invariant, if h v and h v ext are orthogonal. Definition 33.
We say that ( M, ω ) is K-semistable (with respect to smooth test-configurations), if Fut( M , [ ω ] , v ) > , for all smooth test-configurations, compatible with a maximal torus T as above. Ifin addition equality only holds if the central fiber M is biholomorphic to M , then ( M, ω ) is K-stable. Relative K-stability is defined analogously with the modifiedFutaki invariant replacing the Futaki invariant. The following proposition follows from the theorem of Chen-Tian [8], that themodified Mabuchi energy is bounded below, if M admits an extremal metric inthe K¨ahler class [ ω ]. For test-configurations with smooth central fibers it is fairlystraight-forward to relate the modified Futaki invariant to the behavior of the mod-ified Mabuchi functional. This is explained carefully in Tosatti [36] and Clarke-Tipler [9]. As a consequence we have the following. Proposition 34. If M admits an extremal metric ω , then ( M, ω ) is relativelyK-semistable (with respect to smooth test-configurations). Using the method of [27] and [29] we can improve the “semistability” to “stabil-ity”, but first we need to study Futaki invariants on blowups.
LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 29
Futaki invariants on blowups.
Suppose that (
M, ω ) is a K¨ahler manifold,and v is a Hamiltonian holomorphic vector field on M , with Hamiltonian function h v . If p ∈ M is such that v vanishes at p , then v lifts to a holomorphic vector fieldˆ v on the blowup Bl p M . Moreover, the lift is Hamiltonian with suitable choices ofK¨ahler metric on the blowup, in the class π ∗ [ ω ] − ε [ E ]. We need to compute theFutaki invariant Fut(Bl p M, π ∗ [ ω ] − ε [ E ] , ˆ v )in terms of the Futaki invariant on M . In the algebraic case this computationwas done by Stoppa [28], and was refined in [32] (see also Della Vedova [10]). OnK¨ahler surfaces the first term of the expansion was calculated by Li-Shi [19] underthe assumption that we blow up a non-degenerate zero of the vector field. We canobtain the general result for K¨ahler manifolds using the simple observation thatthe difference Fut(Bl p M, π ∗ [ ω ] − ε [ E ] , ˆ v ) − Fut( M, [ ω ] , v )can essentially be computed in a neighborhood of p , since we can choose the metricon the blowup to coincide with the metric ω outside a neighborhood of p . Indeed thisis what the metric ω ε in Section 3.4 is like. We can also choose the Hamiltonianfunction h ˆ v to coincide with h v outside a small ball B around p , by choosing h ˆ v = l ( h v ). We then have the following formulas. Proposition 35.
For sufficiently small ε > we have (35) Z M ω m − Z Bl p M ω mε = ε m , Z M h v ω m − Z Bl p M l ( h v ) ω mε = ε m h v ( p ) + ε m +2 m + 1 ∆ h v ( p ) , Z M s ( ω ) ω m − Z Bl p M s ( ω ε ) ω mε = 2 πm ( m − ε m − , Z M h v s ( ω ) ω m − Z Bl p M l ( h v ) s ( ω ε ) ω mε = 2 πm ( m − ε m − h v ( p )+ 2 π ( m − ε m ∆ h v ( p ) . Proof.
Each of these formulas can be reduced to a calculation on the projectivespace P m . The first and third formulas can also be checked easily using the coho-mological interpretations of the integrals. The formulas involving h v could also beapproached using equivariant cohomology, but we will not pursue this.In order to reduce the problem to a calculation on P m , note that each pair ofintegrals coincides outside a ball B (in the notation of Section 3.4 we are taking B = B r ε ). So for instance we have(36) Z M ω m − Z Bl p M ω mε = Z B ω m − Z Bl p B ω mε , with similar formulas for the other 3 integrals. Since v vanishes at p , we can choosecoordinates around p in which v is given by a linear transformation. For small ε >
0, we can therefore choose a metric Ω on P m in the class c ( O (1)), togetherwith a holomorphic Killing field V vanishing at a point P ∈ P m (with Hamiltonian h V ), such that the data ( B r ε ( P ) , Ω , V, h V ) is equivalent in the obvious sense to the corresponding data ( B r ε ( p ) , ω, v, h v ). It follows from (36) together with theanalogous formula on P m , that Z M ω m − Z Bl p M ω mε = Z P m Ω m − Z Bl P P m Ω mε , where Ω ε is a metric on Bl P P m in the class π ∗ [Ω] − ε [ E ] constructed just likewe constructed ω ε . The analogous formula holds for all of the differences that weneed to compute in Equation (35). We have therefore reduced the problem to acalculation on P m .On P m one way to do the calculation would be to perform a computation interms of toric geometry, since we can assume that P is fixed by a maximal torus ofautomorphisms of P m . Alternatively, we can use an algebro-geometric calculation,since by continuity it is enough to deal with the case when ε is rational. It isessentially this that we have already calculated in [32, Lemma 28]. The resultsof that Lemma, together with the calculations in [11, Proposition 2.2.2] imply theresult we want. (cid:3) Using this proposition we can compute the Futaki invariant on a blowup, ex-tending [32, Corollary 29] to the K¨ahler case.
Corollary 36.
Suppose that h v is normalized to have zero mean. For sufficientlysmall ε > we have an expression (37) Fut(Bl p M, [ ω ε ] , ˆ v ) = Fut( M, [ ω ] , v ) + A ε h v ( p ) + B ε ∆ h v ( p ) , where A ε = O ( ε m − ) and B ε = O ( ε m ) are functions of ε depending on ( M, [ ω ]) .One can easily expand A ε , B ε in terms of ε . In general (38) Fut(Bl p M, [ ω ε ] , ˆ v ) = Fut( M, [ ω ] , v ) + 2 πm ( m − ε m − h v ( p ) + O ( ε m ) . Suppose that
Fut( M, [ ω ] , v ) = 0 , and m > . Then (39) Fut(Bl p M, [ ω ε ] , ˆ v ) = 2 πm ( m − ε m − h v ( p )+ ε m (cid:16) π ( m − h v ( p ) − s h v ( p ) (cid:17) + O ( ε m +2 ) , where s is the average scalar curvature of ( M, ω ) . In addition if h v ( p ) = ∆ h v ( p ) =0 , then Fut(Bl p M, [ ω ε ] , ˆ v ) = 0 .Proof. Let us write s ε for the average scalar curvature of ω ε . Then from (35) wehave(40) s ε = R Bl p M s ( ω ε ) ω mε R Bl p M ω mε = R M s ( ω ) ω m − πm ( m − ε m − R M ω m − ε m = s + O ( ε m − ) , and(41) Z Bl p M l ( h v ) ω mε = − ε m h v ( p ) − ε m +2 m + 1 ∆ h v ( p ) , Z Bl p M l ( h v ) s ( ω ε ) ω mε = Z M h v s ( ω ) ω m − πm ( m − ε m − h v ( p ) − π ( m − ε m ∆ h v ( p ) . LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 31
Combining these, and using thatFut( M, [ ω ] , v ) = − Z h v s ( ω ) ω m since h v has integral zero, we getFut(Bl p M, [ ω ε ] , ˆ v ) = Z Bl p M l ( h v )( s ε − s ( ω ε )) ω mε = s ε Z Bl p M l ( h v ) ω mε − Z Bl p M l ( h v ) s ( ω ε ) ω mε = s ε (cid:18) − ε m h v ( p ) − ε m +2 m + 1 ∆ h v ( p ) (cid:19) − Z M h v s ( ω ) ω m + 2 πm ( m − ε m − h v ( p ) + 2 π ( m − ε m ∆ h v ( p )= Fut( M, [ ω ] , v ) + A ε h v ( p ) + B ε ∆ h v ( p ) . Here A ε = 2 πm ( m − ε m − − ε m s ε ,B ε = 2 π ( m − ε m − ε m +2 m + 1 s ε . Using this together with the formula (40), we can obtain all the results that we aretrying to prove. (cid:3)
We need one more result, relating the inner product of holomorphic Killing fields,introduced by Futaki-Mabuchi [13]. This inner product is simply the L productof the Hamiltonian functions, which are normalized to have zero mean. So if v, w have Hamiltonians h v , h w , normalized to have zero mean on M , then h v, w i = Z M h v h w ω m . If v, w vanish at p ∈ M , then the product of the lifts to Bl p M is given by(42) h ˆ v, ˆ w i = Z Bl p M l ( h v ) l ( h w ) ω mε − V ε Z Bl p M l ( h v ) ω mε Z Bl p M l ( h w ) ω mε , where V ε is the volume of Bl p M with respect to ω ε . The crucial property of thisinner product is that it is independent of the representative ω ε of its K¨ahler class. Proposition 37.
Assume that h v, w i = 0 . Then for any δ > we have on theblowup Bl p M , that h ˆ v, ˆ w i = O ( ε m − δ ) . If in addition h v is normalized to have zero mean on M , and h v ( p ) = 0 , then forany δ > we have h ˆ v, ˆ w i = O ( ε m +2 − δ ) . In fact we could even take δ = 0 in both formulas, but we will not need this.Proof. We can assume that h v and h w are normalized to have zero mean on M .Then from (35) we know that the averages of l ( h v ) and l ( h w ) on Bl p M are of order ε m , so in the formula (42) for h ˆ v, ˆ w i we can ignore the integrals of l ( h v ) and l ( h w ). Since l ( h v ) l ( h w ) = h v h w and ω ε = ω outside B r ε , we just need to estimate theintegrals on B r ε with respect to the different metrics ω and ω ε . The volume of B r ε is O ( r mε ) with respect to both ω and ω ε so we obtain h ˆ v, ˆ w i = O ( r mε ) . If in addition h v ( q ) = 0, then we have h v ∈ C , since also ∇ h v ( q ) = 0 by ourassumption. It follows that also l ( h v ) ∈ C . This implies that Z B rε h v h w ω m C Z r ε r r m − dr = O ( r m +2 ε ) , and also Z B rε h v h w ω mε C (cid:18)Z r ε ε r r m − dr + ε ε m (cid:19) = O ( r m +2 ε ) . We can do the construction with r ε = ε α for any α <
1, and choosing α sufficientlyclose to 1 we obtain the results we want. Note that one can do the analogous cal-culation algebro-geometrically and get a more precise result like in Proposition 35,but we will not need this. (cid:3) We can now improve Proposition 34 following Stoppa [27] and [29] to get
Proposition 38.
Suppose that M admits an extremal metric ω . Then ( M, [ ω ]) isrelatively K-stable (with respect to smooth test-configurations). The proof is essentially identical to the argument in [29], using Proposition 34together with the formulas that we have shown in this section. In fact our situationis simpler since we are only dealing with test-configurations with smooth centralfiber.As an application we have the following proposition, which shows that we canonly hope to construct extremal metrics on blowups Bl p M , for which ∇ s ( ω ) van-ishes at p . Proposition 39.
Suppose that ( M, ω ) is an extremal K¨ahler manifold, and p ∈ M is such that ∇ s ( ω ) does not vanish at p . Then (Bl p M, [ ω ε ]) is relatively K-unstablefor all sufficiently small ε > .Proof. Let T ⊂ G p be a maximal torus, with Lie algebra t . By our assumption, s t . We have an orthogonal decomposition s = s ⊥ + s t , where s t ∈ t and s ⊥ ⊥ t . Then ∇ s ⊥ ( p ) = 0, and we can assume that ∇ s ⊥ generatesa C ∗ -action. If it did not, we could approximate s ⊥ with elements of g orthogonalto t , which do generate C ∗ -actions.Suppose then that − s ⊥ generates the C ∗ -action λ ( t ), and let q = lim t → λ ( t ) · p .This way we obtain a test-configuration for (Bl p M, π ∗ [ ω ] − ε [ E ]) with central fiberBl q M . The Futaki invariant of the test-configuration is given byFut(Bl q M, π ∗ [ ω ] − ε [ E ] , − ˆ s ⊥ ) = Fut( M, [ ω ] , − s ⊥ ) + O ( ε m − ) , using a calculation similar to Corollary 36. SinceFut( M, [ ω ] , − s ⊥ ) = h− s ⊥ , s i = −k s ⊥ k , LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 33 we have a constant c > q M, π ∗ [ ω ] − ε [ E ] , − ˆ s ⊥ ) < − c , for sufficiently small ε . In order to show that (Bl p M, L ε ) is relatively K-unstable,we still need to adjust this test-configuration to be orthogonal to t . Since s ⊥ isorthogonal to t , it follows from Proposition 37 that for any v ∈ t we have h ˆ s ⊥ , v i = O ( ε m − δ ) , for any δ >
0. This means that after modifying the test-configuration with anelement in t to make it orthogonal to t on Bl q M , the Futaki invariant will still benegative for sufficiently small ε . It follows that (Bl p M, L ε ) is relatively K-unstablefor sufficiently small ε > (cid:3) Test-configurations for blowups.
In this section we will give the proof ofTheorem 2. Suppose that (
M, ω ) is cscK , and suppose that v is a Hamiltonianholomorphic vector field on M generating a C ∗ -action λ ( t ). Then for any p ∈ M , λ ( t ) induces a test-configuration for (Bl p M, [ ω ε ]). The total space of this test-configuration is simply the blowup of the product M × C along the closure of theorbit { ( λ ( t ) · p, t ) : t ∈ C ∗ } . If q = lim t → λ ( t ) · p , then the central fiber of the test-configuration is (Bl q M, ω ε ).The induced C ∗ -action on Bl q M is given by λ ( t ), which lifts to Bl q M since q is afixed point. The formula (39) can be used to compute the Futaki invariant of thistest-configuration.When combined with Theorem 1, the following proposition implies Theorem 2.This proposition generalizes [32, Theorem 5] to K¨ahler manifolds. Proposition 40.
Suppose that n > , and for some ε > there does not exist q ∈ G c · p with µ ( q ) + ε ∆ µ ( q ) = 0 for any ε ∈ (0 , ε ) . Then (Bl p M, [ ω ε ]) isK-unstable for all sufficiently small ε > .Proof. By moving p in its G c -orbit, we can assume that G p is a maximal compactsubgroup of G cp . Then if T ⊂ G p is a maximal torus, then T c ⊂ G cp is also a maximaltorus. As in Section 2 we will work with the group G T ⊥ . The corresponding momentmap µ T ⊥ is simply the orthogonal projection of µ onto g T ⊥ . Our assumption saysthat p is unstable for the action of G cT ⊥ with respect to the moment map µ T ⊥ + ε ∆ µ T ⊥ for all sufficiently small ε >
0. There are several cases to consider separately. • Suppose that p is strictly unstable for the moment map µ T ⊥ . This meansthat there is a v ∈ g T ⊥ generating a C ∗ -action λ ( t ), such thatlim t → h µ ( λ ( t ) · p ) , v i < . Write q = lim t → λ ( t ) · p . We then have h v ( q ) <
0. From Corollary 36 itfollows that the corresponding test-configuration for (Bl p M, [ ω ε ]) has Futakiinvariant(43) Fut(Bl q M, [ ω ε ] , ˆ v ) < − c ε m − , for some c >
0. This means that (Bl p M, [ ω ε ]) is K-unstable. • Suppose that p is semistable for the moment map µ T ⊥ , and we can find a v ∈ g T ⊥ generating a C ∗ -action λ ( t ), such thatlim t → h µ ( λ ( t ) · p ) , v i = 0lim t → h ∆ µ ( λ ( t ) · p ) , v i < . Writing again q = lim t → λ ( t ) · p we then have h v ( q ) = 0 and ∆ h v ( q ) < q M, [ ω ε ] , ˆ v ) < − c ε m , for some c >
0, and so it follows that (Bl p M, [ ω ε ]) is K-unstable. • In the remaining case p is semistable with respect to µ T ⊥ + ε ∆ µ T ⊥ for allsufficiently small ε >
0. This implies that we can find q ε in the boundary ∂G c · p of the G c -orbit such that µ T ⊥ ( q ε ) + ε ∆ µ T ⊥ ( q ε ) = 0 . Since ∂G c · p is a finite union of orbits, at least one orbit must containinfinitely many q ε . Choose q ε and q ε to be in the same orbit. Since themoment maps are equivariant and T fixes q ε i , we have µ ( q ε i ) + ε i ∆ µ ( q ε i ) ∈ t . In addition, the projection of the moment map to the stabilizer is an in-variant of the orbit, so if q is in the same G cT ⊥ orbit as the q ε i , thenpr g q ( µ ( q ) + ε i ∆ µ ( q )) ∈ t . Since this holds for at least two different ε i , we must have(44) pr g q µ ( q ) , pr g q ∆ µ ( q ) ∈ t . It follows that the stabilizer of q ε in G cT ⊥ is reductive and so there is alocal slice for the action of G cT ⊥ near q ε (see Sjamaar [25] or Snow [26]).Using the Hilbert-Mumford criterion applied to the action of the stabilizeron the tangent space at q ε , we can find a v ∈ g T ⊥ generating a C ∗ -action λ ( t ), and a point p ′ ∈ G c · p , such that q ε = lim t → λ ( t ) · p ′ . This meansthat there exists a test-configuration for (Bl p M, [ ω ε ]), whose central fiberis (Bl q M, [ ω ε ]), writing q = q ε .We claim that this test-configuration has zero Futaki invariant. In-deed, since v ∈ t ⊥ , it follows from (44) that the Hamiltonian h v satisfies h v ( q ) = ∆ h v ( q ) = 0. In addition since s ∈ t , we have Fut( M, [ ω ] , v ) = 0.Corollary 36 then implies that Fut(Bl q M, [ ω ε ] , ˆ v ) = 0. It follows that(Bl p M, [ ω ε ]) is K-unstable. (cid:3) Combining our results we can prove Theorem 2.
Proof of Theorem 2. (1) ⇒ (2): This follows from Proposition 38.(2) ⇒ (3): This is the statement of Proposition 40.(3) ⇒ (1): It follows from Theorem 1, that under the assumption the blowupBl p M admits an extremal metric in the class [ ω ε ]. We just need to check that thismetric has constant scalar curvature. To do this we need to compute the Futakiinvariant Fut(Bl p M, [ ω ε ] , ˆ v ) for all v ∈ g p . Since ( M, ω ) is cscK we know that
LOWING UP EXTREMAL K¨AHLER MANIFOLDS II 35
Fut( M, [ ω ] , v ) = 0. In addition if ε is sufficiently small, then an argument similarto the proof of Proposition 7 shows thatpr g p µ ( p ) , pr g p ∆ µ ( p ) = 0 . Therefore h v ( p ) = ∆ h v ( p ) = 0, so from Corollary 36 we get Fut(Bl p M, [ ω ε ] , ˆ v ) = 0.It follows that the extremal metric constructed using Theorem 1 has constant scalarcurvature. (cid:3) References [1] C. Arezzo and F. Pacard. Blowing up and desingularizing constant scalar curvature K¨ahlermanifolds.
Acta Math. , 196(2):179–228, 2006.[2] C. Arezzo and F. Pacard. Blowing up K¨ahler manifolds with constant scalar curvature II.
Ann. of Math. (2) , 170(2):685–738, 2009.[3] C. Arezzo, F. Pacard, and M. A. Singer. Extremal metrics on blow ups.
Duke Math. J. ,157(1):1–51, 2011.[4] O. Biquard and Y. Rollin. Smoothing singular extremal K¨ahler surfaces and minimal la-grangians. arXiv:1211.6957 .[5] S. Bochner and W. T. Martin.
Several complex variables , volume 10 of
Princeton Mathemat-ical Series . Princeton University Press, Princeton, N.J., 1948.[6] E. Calabi. Extremal K¨ahler metrics. In S. T. Yau, editor,
Seminar on Differential Geometry .Princeton, 1982.[7] E. Calabi. Extremal K¨ahler metrics II. In
Differential geometry and complex analysis , pages95–114. Springer, 1985.[8] X. X. Chen and G. Tian. Geometry of K¨ahler metrics and foliations by holomorphic discs.
Publ. Math. Inst. Hautes ´Etudes Sci. , (107):1–107, 2008.[9] A. Clarke and C. Tipler. Lower bounds on the modified K-energy and complex deformations. preprint .[10] A. Della Vedova. CM-stability of blow-ups and canonical metrics. preprint (2008).[11] S. K. Donaldson. Scalar curvature and stability of toric varieties.
J. Differential Geom. ,62:289–349, 2002.[12] A. Futaki. An obstruction to the existence of Einstein-K¨ahler metrics.
Invent. Math. , 73:437–443, 1983.[13] A. Futaki and T. Mabuchi. Bilinear forms and extremal K¨ahler vector fields associated withK¨ahler classes.
Math. Ann. , 301:199–210, 1995.[14] P. Gauduchon. Invariant scalar-flat K¨ahler metrics on O ( − l ). preprint , 2012.[15] F. C. Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry . PrincetonUniversity Press, 1984.[16] C. LeBrun. Counter-examples to the generalized positive action conjecture.
Comm. Math.Phys. , 118(4):591–596, 1988.[17] C. LeBrun and S. R. Simanca. Extremal K¨ahler metrics and complex deformation theory.
Geom. and Func. Anal. , 4(3):298–336, 1994.[18] C. LeBrun and M. A. Singer. Existence and deformation theory for scalar-flat K¨ahler metricson compact complex surfaces.
Invent. Math. , 112(2):273–313, 1993.[19] H. Li and Y. Shi. The Futaki invariant on the blowup of K¨ahler surfaces. arXiv:1211.2954 .[20] I. Mundet i Riera. A Hitchin-Kobayashi correspondence for K¨ahler fibrations.
J. Reine Angew.Math. , 528:41–80, 2000.[21] F. Pacard. Constant scalar curvature and extremal K¨ahler metrics on blow ups. In
Proceedingsof the International Congress of Mathematicians. Volume II , pages 882–898, New Delhi,2010. Hindustan Book Agency.[22] F. Pacard and X. Xu. Constant mean curvature spheres in Riemannian manifolds.
Manuscripta Math. , 128(3):275–295, 2009.[23] Y. Rollin and M. A. Singer. Non-minimal scalar-flat K¨ahler surfaces and parabolic stability.
Invent. Math. , 162(2):235–270, 2005.[24] S. R. Simanca. K¨ahler metrics of constant scalar curvature on bundles over C P n − . Math.Ann. , 291(2):239–246, 1991. [25] R. Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations.
Ann. of Math. (2) , 141(1):87–129, 1995.[26] D. M. Snow. Reductive group actions on Stein spaces.
Math. Ann. , 259(1):79–97, 1982.[27] J. Stoppa. K-stability of constant scalar curvature K¨ahler manifolds.
Adv. Math. , 221(4):1397–1408, 2009.[28] J. Stoppa. Unstable blowups.
J. Algebraic Geom. , 19(1):1–17, 2010.[29] J. Stoppa and G. Sz´ekelyhidi. Relative K-stability of extremal metrics.
J. Eur. Math. Soc. ,13(4):899–909, 2011.[30] G. Sz´ekelyhidi.
Extremal metrics and K -stability . PhD thesis, Imperial College, London,2006.[31] G. Sz´ekelyhidi. Extremal metrics and K -stability. Bull. Lond. Math. Soc. , 39(1):76–84, 2007.[32] G. Sz´ekelyhidi. On blowing up extremal K¨ahler manifolds.
Duke Math. J. , 161(8):1411–1453,2012.[33] A. Teleman. Symplectic stability, analytic stability in non-algebraic complex geometry.
In-ternat. J. Math. , 15(2):183–209, 2004.[34] G. Tian. K¨ahler-Einstein metrics with positive scalar curvature.
Invent. Math. , 137:1–37,1997.[35] C. Tipler. Extremal K¨ahler metrics on blow-ups of parabolic ruled surfaces. arXiv:1104.4315 .[36] V. Tosatti.
The K-energy on small deformations of constant scalar curvature K¨ahler mani-folds , volume 21 of
Advanced Lectures in Math. , pages 139–150. International Press, 2012.[37] S.-T. Yau. Open problems in geometry.
Proc. Symposia Pure Math. , 54:1–28, 1993.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46615
E-mail address ::