A compact K¨ahler manifold admitting a com- plex symplectic form carries in his K¨ahler class a hyperk¨ahler metric, this is now a well- know consequence of the solution of the Calabi conjecture by S-T

GILLES CARRON

ABSTRACT. We show that on Hilbert scheme ofnpoints onC², the hyperk¨ahler metric construsted by H. Nakajima via hyperk¨ahler reduction is the Quasi-Asymptotically Locally Euclidean (QALE in short) metric constructed by D. Joyce.

1. INTRODUCTION

The Hilbert scheme (or Douady scheme) ofnpoints onC², denoted byHilbⁿ₀(C²), is a crepant resolution of the quotient of C²n

0 =n

q∈ C²n

,P

jqj= 0o

by the action of the symmetric groupSnwhich acts by permutation of the indices :

σ∈Sn, q∈ C²n

0, σ.q= (q_σ⁻¹₍₁₎, q_σ⁻¹₍₂₎, ..., q_σ⁻¹_(n)).

Hence we have a map

π : Hilbⁿ₀(C²)→ C²ⁿ

0/Sn.

The complex manifoldHilbⁿ₀(C²)carries a natural complex symplectic structure which comes from theSninvariant one of C²n

0. A compact K¨ahler manifold admitting a com- plex symplectic form carries in his K¨ahler class a hyperk¨ahler metric, this is now a well- know consequence of the solution of the Calabi conjecture by S-T. Yau (see [3]). However, Hilbⁿ₀(C²)is non compact, for instanceHilb²₀(C²) = T^∗P¹(C). There are many exten- sions of Yau’s result to non compact manifold (see for instance [2],[26],[27]) and in 1999, D. Joyce has introduced a new class of asymptotic geometry called Quasi-Asymptotically Locally Euclidean (QALE in short) ; this class is the extension of the class of ALE (for Asymptotically Locally Euclidean) ; roughly a complete manifold(M^d, g)is called ALE asymptotic toR^d/ΓwhereΓ⊂O(d)is a finite subgroup acting freely onS^d−1, if outside a compact setM is diffeomorphic to R^d\B

/Γand if on there the metric is asymptotic to the Euclidean metric (the precise definition requires estimates betweengand the Euclidean metric). WhenX^mis a crepant resolution ofC^m/ΓforΓ ⊂SU(m)a finite group, then roughly a K¨ahler metric onX^mis called QALE if firstly away from the pulled back of the singular set the metric is asymptotic to the Euclidean one and secondly on pieces of X^mwhich (up to a finite ambiguity) are diffeomorphic to a subset ofXA×Fix(A)where Ais a subgroup ofΓ, andXAis a crepant resolution of Fix(A)^⊥/A, then the metric is asymptotic to the sum of a QALE K¨ahler metric onXAand a Euclidean metric onFix(A).

And D. Joyce has proved the following ([15],[16][theorem 9.3.3 and 9.3.4]):

Theorem A. WhenΓ⊂SU(m)is a finite group andX^m→C^m/Γis a crepant resolution , then in any K¨ahler class of QALE metric there is a unique QALE K¨ahler Ricci flat metric.

Moreover ifΓ⊂Sp(m/2), then this metric is hyperk¨ahkler.

In particular, up to scaling,Hilbⁿ₀(C²)carries an unique hyperk¨ahler metric asymptotic to C²n

0/S_n.

1

Another fruitful construction of hyperk¨ahler metric is the hyperk¨ahler quotient con- struction of N. Hitchin, A. Karlhede, U. Lindstr¨om and M. Rocek [13]. In fact in 1999, H. Nakajima has constructed a hyperk¨ahler metric onHilbⁿ₀(C²)as a hyperk¨ahler quotient [24]. Moreover H. Nakajima asked wether this metric could be recover via a resolution of the Calabi conjecture ; also D. Joyce said hat it is likely that QALE hyperk¨ahler met- ric can be explicitly constructed using the hyperk¨ahker quotient , but outside the case of Γ⊂SU(2) =Sp(1)treated by Kronheimer [17], he has no examples. The main result of this paper is the following :

Theorem B. OnHilbⁿ₀(C²), up a scaling, D.Joyce’s and H.Nakajima’s metrics coincide.

It should be noted that a given complex manifold can carry two very different hy- perk¨ahler metrics ; for instance as it has been clearly explain by C. LebrunC² carries two quite different K¨ahler Ricci flat metrics, the Euclidean one and the Taub-Nut metric which has cubic volume growth [19].

The main evident idea of the proof of this result is to study the asymptotics of Naka- jima’s metric ; however in order to used D. Joyce unicity result, we would need also asymp- totics on the derivatives of Nakajima’s metric, this is probably possible but requires more estimates. Our analyze of the asymptotics of Nakajima’s metric gives that Joyce and Naka- jima’s metrics differ by O ρ⁻²σ⁻²

; whereρis the distance to a fixed point andσa regularized version of the distance to the singular set. And in order to used the classical argument of S-T. Yau giving the unicity of the solution to the Calabi conjecture, we need to find a functionϕvanishing at infinity such that, the difference between the two K¨ahler forms of Nakajima and Joyce’s metric isi∂∂ϕ. D. Joyce has developed elaborate tools to¯ solve the equation of the type∆u= f on QALE manifold ; but the decayO ρ⁻²σ⁻² is critical for this analysis. In fact, we have circumvent this difficulty using the Li-Yau’s estimates for the Green kernel of a manifold with non negative curvature [20], and we have obtained the following result, which has a independent interest and which can be generalize to other QALE manifolds :

Theorem C. Letf is a locally bounded function onHilbⁿ₀(C²), such that for someε >0 satisfies

f =O 1

ρ^εσ²

then the equation∆u=f has a unique solution such that

u=O

log(ρ+ 2) ρ^ε

.

For further more profound results on the analysis on QALE space, there is a very inter- esting work of A. Degeratu and R. Mazzeo [7].

In the physic litterature,Hilbⁿ₀(C²)is associated to the moduli space of instantons on noncommutativeR⁴[25]. Our motivation for the study of the asymptotic geometry of the Nakajima’s metric comes from a question of C.Vafa and E. Witten about the space ofL² harmonic forms onHilbⁿ₀(C²)endowed with the Nakajima’s metric. LetH^k be the space ofL²harmonick−forms onHilbⁿ₀(C²):

H^k =

α∈L² Λ^kT^∗Hilbⁿ₀(C²)

, dα=d^∗α= 0 .

In [28], see also the nice survey of T. Hausel [10], the following question is asked :

Conjecture D.

H^k=

{0} ifk6= 2(n−1) = dim_RHilbⁿ₀(C²) Im H_c^k(Hilbⁿ₀(C²))→H^k(Hilbⁿ₀(C²))

ifk= 2(n−1)

However, C. Vafa and E. Witten said ”Unfortunately, we do not understand the predic- tion ofS-duality on non-compact manifolds precisely enough to fully exploit them.”

In fact, N. Hitchin has shown that the vanishing of the space ofL²harmonicsk−forms outside middle degree is a general fact for hyperk¨ahler reduction of the flat quaternionic spaceH^mby a compact subgroup ofSp(m)[12] ; he obtained this result with a general- ization of an idea of M. Gromov ([9] see also related works by J.Jost, K. Zuo and J. Mc Neal [14],[21]). For the degreek= 2(n−1), the cohomology ofHilbⁿ₀(C²)is well known :

H_c²⁽ⁿ⁻¹⁾(Hilbⁿ₀(C²))'Im

H_c²⁽ⁿ⁻¹⁾(Hilbⁿ₀(C²))→H²⁽ⁿ⁻¹⁾(Hilbⁿ₀(C²)) 'H²⁽ⁿ⁻¹⁾(Hilbⁿ₀(C²))'R

and a dual class to the generator isπ⁻¹{0}. Moreover a general result of M. Anderson says that the image of the cohomology with compact support in the cohomology always injects inside the space ofL²harmonics forms [1]. Hence for the Hilbert scheme ofnpoints in C²endowed with Nakajima’s metric we always have

dimH²⁽ⁿ⁻¹⁾≥1

and the conjecture D predicts the equalitydimH²⁽ⁿ⁻¹⁾= 1.

There are many results on the topological interpretation of the space ofL² harmonic forms on non compact manifolds but all of them requires a little on the knowledge of the asymptotic geometry (see [11] for results related to some prediction from string theory and [6] for a list of such results) ; the rough idea is that this asymptotic geometry would provide a certain behavior ofL²harmonic forms (decay, polyhomogeneity in a good compactifica- tion) and that would imply a topological interpretation of this space with a cohomology of a compactification. With our paper [5], our main result implies :

Theorem E. The Vafa-Witten conjecture D conjecture is true whenn= 3.

The casen = 2can be treated by explicit computation (see [12] for clever computa- tions).

As the Vafa-Witten conjecture is in fact more general and concerns the quivers varieties constructed by H.Nakajima [23], a natural perspective is to understand the asymptotic geometry of the quivers varieties and the class of Quasi-asymptotically Conical manifolds introduced by R. Mazzeo should be usefull [22]. In a different direction it would be good to develop appropriate QALE tolls to settle the status of the Vafa-Witten conjecture.

Acknowledgements. It is a pleasure to thank A. Degeratu, P. Romon, R. Mazzeo, M. Singer, C. Sorger and Y. Rollin for interessing discussion related to this work ; a special thank is due to O. Biquard who suggested that I could used the classical proof of the unicity of the solution of the Calabi conjecture in place of difficult derivative estimate. This paper was finished during a stay at the MSRI, and I was partially supported by a joint NSF-CNRS project and the project ANR project GeomEinstein 06-BLAN-0154.

2. NAKAJIMA’S METRIC

In [24], H. Nakajima has shown that the Hilbert scheme ofnpoints inC² carries a natural hyperk¨ahler metric ; this metric is obtained from the Hyperk¨ahlerian quotient con- struction of N. Hitchin, A. Karlhede, U. Lindstr¨om and M. Rocek [13] : the complex vector space

Mn=M:={(A, B, x, y)∈ Mn(C)⊕ Mn(C)⊕Cⁿ⊕(Cⁿ)^∗,trA= trB = 0}

has a complex structure

J(A, B, x, y) = (B^∗,−A^∗, y^∗,−x^∗)

if we letK = iJ then(M, I = i, J, K := iJ)becomes a quarternionic vector space ; moreover the unitary groupU(n)acts linearly onM: ifg∈U(n)andz= (A, B, x, y)∈ Mthen

g.z= gAg⁻¹, gBg⁻¹, gx, yg⁻¹ . The real moment map associated to this action is

µ(A, B, x, y) = 1

2i([A, A^∗] + [B, B^∗] +xx^∗−yy^∗)∈u(n).

Ifh∈u(n)andz= (A, B, x, y)∈Mwe let l_z(h) = d

dt _t=0

e^th.z= ([h, A],[h, B], hx,−yh).

By definition, we have forz∈M, δz∈T_zM'M: hdµ(z)(δz), hi=hilz(h), δzi

The action ofGLn(C)onMpreserves the complex symplectic form ω_C(z, z⁰) = tr (A.B⁰−B.A⁰) +y⁰(x)−y(x⁰), and the associated complex moment map is :

µ_C(A, B, x, y) = [A, B] +xy∈ Mn(C).

Lett >0and defined

Lt(n) =Lt:=µ⁻¹ t

2i

∩µ⁻¹_C {0}, then the map

µµ:= (µ, µ_C) : M→u(n)⊕ Mn(C)

is a submersion nearLtandU(n)acts freely on it, hence the quotientHt:=Lt/U(n)is a smooth manifold, this manifold is endowed with the Riemannian metricgN which makes the submersionLt →Lt/U(n)Riemannian. By definition the tangent space ofU(n)zis naturally isometric to the orthogonal of the space

Iml_z⊕IIml_z⊕JIml_z⊕KIml_z.

In particular,Htis endowed with a quaternionic structure which is in fact integrable ; hence the metricg_nis hyperk¨ahler hence K¨ahler and Ricci flat.

2.1. Some remarks. Because forλ >0, we haveλLt=Lλ²t, all the spaces{Ht}t>0are isomorphic and their Riemannian metrics are proportional.

Fort= 0, the quotientL0/U(n)is not a smooth manifold. It is easy to show that (A, B, x, y)∈L0⇔(x= 0, y= 0,[A, B] = [A, A^∗] = [B, B^∗] = 0) ; hence to(A, B,0,0)∈L0we can associated their joint spectrum

Sn.((λ1, µ1),(λ2, µ2), ...,(λn, µn))∈ C²ⁿ

0/Sn

where C²n

0 :={(q1, ..., q_n ∈ C²n

,P

jq_j = 0}and the symmetric groupS_n acts on C²ⁿ

0 by permutation of the indices. We get an isomorphism (in fact an isometry) L0/U(n)' C²n

0/Sn. In fact fort >0, we still have

(2.1) (A, B, x, y)∈Lt⇒y= 0.

Hence forz:= (A, B, x,0)∈L^t, the joint spectrum of(A, B)is still defined and we can defined

π(U(n)z) =S_n.((λ₁, µ₁),(λ₂, µ₂), ...,(λ_n, µ_n))∈ C²ⁿ

0/S_n

where(λ₁, µ₁),(λ₂, µ₂), ...,(λ_n, µ_n)are such that for a g ∈ U(n), the matrixgAg⁻¹ (resp. gBg⁻¹) is upper triangular with diagonal(λ₁, λ₂, .., λ_n)(resp. (µ₁, µ₂, .., µ_n)).

Ht is isomorphic to the Hilbert scheme ofn points¹ inC² Hilbⁿ₀(C²). The mapπ : Hilbⁿ₀(C²)→ C²ⁿ

0/S_nis in fact a crepant resolution of C²ⁿ

0/S_n.

Remark 2.1. We also remark that ifv= (δA, δB, δx,0)∈TζLtis orthogonal to the range oflζ, thenJ vis also inTζLt, and henceδx= 0.

2.2. The geometry ofHilb²₀(C²). As an example, we look at the geometry ofHilb²₀(C²).

Letz= (A, B, x,0)∈ M2(C)⊕ M2(C)⊕C²⊕ C^2∗

such thattrA= trB = 0and [A, A^∗] + [B, B^∗] +xx^∗=tId

[A, B] = 0 .

WhendetA6= 0ordetB 6= 0then we can find ag∈U(2)such that gAg⁻¹=

λ a 0 −λ

, gBg⁻¹=

µ b 0 −µ

then letg(x) = (x1, x2). The equation[A, B] = 0implies that there is a numberρsuch thata=λρandb=µρ. Then the remaining equations are forR²:=|λ|²+|µ|²







|ρ|²R²+|x1|²=t

−|ρ|²R²+|x₂|²=t

−2R²ρ+x₁x₂= 0

We can always choosegsuch thatx1, x2∈R+then we obtainρ²= q

4 +_R^t24 −2Hence ρ= t

2R² +O 1

R⁶

1with the center of mass removed.

ifx1 = √

2tsin(φ), x2 = √

2tcos(φ), thentcos(2φ) = ρ²R² andtsin(2φ) = 2ρR² Henceφ= ^π₄+O _R¹

andx₁=q

t

2+O _R¹

,x₂=q

t

2+O _R¹

. Hence for(λ, µ)∈ C²\ {0} ' C²2

0we have found z(λ, µ) =

λ λρ(R)

0 −λ

,

µ µρ(R)

0 −µ

, x(R),0

∈Lt.

Moreover,z(λ, µ)andz(λ⁰, µ⁰)are in the sameU(2)orbit if and only if(λ, µ) =±(λ⁰, µ⁰)

; hence we have a map

z : C²\ {0}

/{±Id} →Lt/U(2).

From the exact value ofz, we can show that z^∗gN = 2

|dλ|²+|dµ|² +O

1 R⁴

.

This shows that(Hilb²₀(C²), gN)is a hyperk¨ahler metric which is Asymptotically Locally Euclidean asymptotic toC²/{±Id}. These manifolds has been classified by Kronheimer [18], so that in this case Nakajima’s metric is the Eguchi-Hansen metric onT^∗P¹(C).

2.3. A last useful remark. A priori, it is not clear wether the above mapzis holomorphic, this is in fact true as a consequence of the following useful lemma :

Lemma 2.2. Suppose that a compact Lie group Gacts onH^m by quaternionic linear maps and letµµ: H^m→g^∗⊗ImHbe the associated moment map. Assume that for some ζ= (ζ_R, ζ_C)∈g^∗⊗ImHthe hyperk¨ahler quotientQ:=µµ⁻¹{ζ}/Gis well defined. When Xis a complex manifold andΨ : X→µµ⁻¹{ζ}is a smooth map such that locally

Ψ(x) =g(x) ˜Ψ(x) whereg : X →G^Cis smooth andΨ :˜ X →µ⁻¹

C {ζ_C}is holomorphic, then the induced mapΨ :¯ X→Qis also holomorphic.

Proof. If q ∈ H^m letPq be the orthogonal projection onto the orthogonal of Imlq ⊕ IIml_q⊕JIml_q⊕KIml_q = Iml^C_q ⊕JIml^C_q wherel_q : k→H^qis defined as before by

lq(h) = d dt _t=0

e^th.q=h.q .

We must show that ifx∈X, then forq:= Ψ(x): Pq dΨ(x)(Iv)

=IPq dΨ(x)(v) . Butg(x) =˙ dg(x)(Iv)∈G^Cwe have

dΨ(x)(Iv) = ˙g(x).q+g(x).dΨ(x)(Iv) =˜ l^C_q( ˙g(x)) +g(x).dΨ(x)(Iv)˜ By definitionPq(l^C_q( ˙g(x))) = 0and becauseg(x)andPq are complex linear :

Pq dΨ(x)(Iv)

=Pq g(x).IdΨ(x)(v)˜

=IPq dΨ(x)(v) .

3. JOYCE’S METRIC

In [15, 16], D. Joyce has build many new K¨ahler, Ricci flat metrics on some crepant resolution of quotient ofC^mby a finite subgroup ofSU(m); his construction relies upon the resolution of a Calabi-Yau problem for a certain class of asymptotic geometry which is called QALE for Quasi Asymptotically Locally Euclidean. We will follow the presen- tation of D. Joyce for the Hilbert Scheme ofnpoints onC²and we will then describe the asymptotic geometry of these QALE metrics onHilbⁿ₀(C²).

3.1. The local product resolution ofHilbⁿ₀(C²). Ifp = (I1, I2, ..., Ik)is a partition of {1,2, .., n}², theI_l’s are called the cluster ofp. We will denote

V_p={q∈ C²ⁿ

0,∀l∈ {1, ..., k},∀i, j∈I_l:q_i=q_j}

andAp={γ∈Sn, γq=q∀q∈Vp} 'Sn₁×Sn₂×...×Sn_kwherenl= #Il.Then Wp=V_p^⊥'

k

M

l=1

C²ⁿl

0 .

Letm_p= codim_CV_p= dim_CW_p= 2(n−l(p))wherel(p) =k. The setPnof partitions of{1,2, ..., n}has the following partial order :

p≤q⇔Vq⊂Vp⇔Wp⊂Wq

Hencep≤qif and only ifpis a refinement ofq: i.e. ifq = (J1, J2, ..., Jr), then there are partitions(Il,1, Il,2, ..., Il,n_l)ofJl=Il,1∪...∪Il,n_lsuch that the cluster ofpare the Il,j’s. The smallest partition isp0={1} ∪ {2} ∪...∪ {n}withVp₀ = C²ⁿ

0, the largest partition isp∞ = {1,2, ..., n}withVp∞ = {0}.The fundamental partitions are thepi,j

such that

pi,j= {i, j},{k1},{k2}, ...,{k_n−2}

with{1,2, ..., n}\{i, j}={k1, k2, ..., k_n−2},thenVi,j:=Vp_i,j ={q∈ C²n

0, qi=qj}.

We have for any partitionp6=p0

V_p=∩pi,j≤pV_i,j

We will also denote ∆p = {(i, j) ∈ {1,2, ..., n}², pi,j 6≤ p} and ∆^c_p = {(i, j) ∈ {1,2, ..., n}², pi,j ≤ p}. The singular locus of C²n

0/Sn is the quotient of the gen- eralized diagonal

S=



 [

p6=p0

V_p



/S_n=



 [

i,j

V_i,j



/S_n.

Finally let

Sp=



 [

(i,j)∈∆p

Vi,j



/Ap

and forR >0, letTpbe theR-neighborhood ofSpis C²n 0/Ap: T_p:={q∈ C²ⁿ

0, ∃(i, j)∈∆_p|qi−q_j|< R}/Ap.

2theIl’s are disjoint and their reunion is{1,2, ..., n}.

The resolutionπ : Hilbⁿ₀(C²)→ C²n

0/Snis a local product resolution ; indeed there is a resolution ofW_p/A_pnamely

π_p : Hilb^p₀(C²) :=

k

Y

l=1

Hilbⁿ₀^l(C²)→W_p/A_p

such that forU_p= (π_p×Id)⁻¹(T_p)⊂Hilb^p₀(C²)×Vpandφ_p C²n

0/A_p→ C²n 0/S_n the natural map there is a local biholomorphism onto his imageψp : Up→Hilbⁿ₀(C²)for which the following diagram is commutative :

Hilb^p₀(C²)×V_p\U_p

πp×Id

ψp //Hilbⁿ₀(C²)

π

C²n

0/Ap\Tp

φp // C²n 0/Sn

In the hyperk¨ahlerian quotient description, the local biholomorphismψpis given as follows identifyingVpwith C²k

0, if we let

ζ= ((A₁, B₁, x₁,0),(A₂, B₂, x₂,0), ...,(A_k, B_k, x_k,0))∈

k

Y

j=1

Lt(n_j)

and η = ((λ1, µ1),(λ2, µ2), ...,(λk, µk)) ∈ C^2k

0 \ Up, we associated to (ζ, η), the vector(A, B, x,0)∈M(n)such thatAandBare block diagonal with respective diagonal (A1+λ1, A2+λ2, ..., Ak+λk)and(B1+µ1, B2+µ2, ..., Bk+µk)andx= (x1, x2, ..., xk) thenψp((U(n1)×U(n2)×...×U(nk)).ζ, η)is the set of points, in theGLn(C)-orbit of(A, B, x,0), satisfying the real moment map equation (see the part 4 for more details).

3.2. QALE metric onHilbⁿ₀(C²). We introduce several functions of distance’s type on Hilb^p₀(C²)×V_p

\U_p, ifz∈Hilb^p₀(C²)×V_p\U_pandv= (π_p×Id)(z), we note µp,q(z) = inf

γ∈Ap

d(γ.v, Vq) =d(v,(ApVq)/Ap)

and

νp(z) = 1 + inf

p6=p0

µp,q(z)

Then a Riemannian metricgonHilbⁿ₀(C²)is called QALE (asymptotic to C²n 0/Sn) if for each partitionpthere is a metricgponHilb^p₀(C²)such that for alll∈N:

(3.1) ∇^l ψ^∗_pg−(g_p+ eucl_Vp)

=X

q6≤p

O 1

ν_p^2+lµ^2m_p,q^q−2

!

However, ifq6≤pthere is always a(i, j)∈∆_psuch thatp_i,j6≤pandp_i,j ≤q, therefore µ^2m_p,p^p_i,j^i,j−2=µ²_p,pi,j ≤µ^2m_p,q^q−2

If we introduceρ_p(z) = inf_(i,j)∈∆pµ_p,pi,jthen in fact forv= (π_p×Id)(z)∈ C²ⁿ

0/A_p we have

ρp(z) = inf

(i,j)∈∆p

|vi−vj|

The asymptotic 3.1 are equivalent to

(3.2) ∇^l ψ^∗_pg−(g_p+ eucl_Vp)

=O 1

ν_p^2+lρ²_p

!

We can introduce two functions of distance’s type : whenz ∈Hilbⁿ₀(C²)andπ(z) = (v1, v2, ...vn)∈ C²ⁿ

0/Sn, then we let ρ(z) =

sX

i<j

|vi−vj|², and

σ(z) = inf

i6=j{|v_i−v_j|}+ 1

Ifpis a partition of{1,2, .., n}, and, τ, Rare positive real numbers, then we introduce Cˇp0 ={(v1, ..., v_n)∈ C²ⁿ

0/S_n,such that|v|> R

and∀i6=j, |vi−vj|> ε|v|}

Cˇp ={(v1, ..., vn)∈ C²ⁿ

0/Ap,such that|v|> R,

∀(i, j)∈∆p|vi−vj|>

s 2 n(n−1)|v|

and∀(i, j)∈∆^c_p, |vi−vj|<2|v|}

It is clear that ifis small enough then the C²\RBⁿ

0/S_n=∪_pφ_p( ˇC_p).

Moreover onCp:= (πp×Id)⁻¹ Cˇp

, the asymptotic3.2are (3.3) ∇^l ψ_p^∗g−(gp+ euclV_p)

=O 1

σ^2+lρ²

Remark 3.1. It can be shown that if all metricgp are QALE and if the estimate 3.3 is satisfied thengis also QALE.

3.3. Joyce’s result. The result of D. Joyce concerning the Hilbert scheme ofnpoints on C²is the following :

Theorem 3.2. Up to scaling,Hilbⁿ₀(C²)has a unique hyperk¨ahler metric which is QALE asymptotic to C²n

0/Sn.

4. ASYMPTOTIC OFNAKAJIMA’S METRIC

4.1. Induction’s hypothesis. In this part, we will prove the following result by induction onn:

i) OnHilbⁿ₀(C²), Nakajima’s metricg_N satisfies the estimate (3.3) forl= 0, more precisely ifg_pis the sum of Nakajima’s metric onHilb^p₀(C²), then for all partition pthen for >0small enough andRlarge enough, we have on(π_p×Id)⁻¹ Cˇ_p

ψ^∗_p(g_N)−g_p+ eucl_Vp=O 1

σ²ρ²

ii) There is a constantCsuch that ifz= (A, B, x,0)∈Ltthen

∀h∈un, klz(h)k²=k[h, A]k²+k[h, B]k²+khxk²≥Ckhk²

iii) There is a constantM such that for allz ∈ Lt and(δA, δB,0,0) ∈ TzLt

orthogonal toImlzthen

k[δA, δA^∗]k+k[δB, δB^∗]k ≤ M

σ² kδAk²+kδBk² .

It is easy to check these three conditions forHilb²₀(C²)thanks to the explicit description ofLtin this case. So we now assume that these induction hypothesis are true for allm < n.

4.2. The case of well separated points. We first examine the easiest case corresponding top0. More precisely, we consider q = (q1, q2, ..., qn)∈ C²n

0 such that for alli 6=j, then|qi−q_j|> R(Rwill be chosen large enough), the set of suchq’s will be denote by O₀. Ifq_j = (λ_j, µ_j)we search a solutionz= (A, B, x,0)∈Mof the equation

(4.1)

[A, A^∗] + [B, B^∗] +xx^∗=tId [A, B] = 0

WhereA,B are upper triangular matrices with respective diagonals(λ1, λ2, ..., λn)and (µ1, µ2, ..., µn)and upper diagonal coefficients a = (ai,j),b = (bi,j). We obtain the following equation³for the(i, j)coefficients of the equation (4.1) :

(4.2)

(¯λ_i−λ¯_j)a_i,j+ (¯µ_i−µ¯_j)b_i,j+P

k

¯a_k,ia_k,j+ ¯b_k,ib_k,j−a_i,k¯a_j,k−b_i,k¯b_j,k

=x_ix¯_j

−(¯µ_i−µ¯_j)a_i,j+ (λ_i−λ_j)b_i,j+P

k[a_i,kb_k,j−b_i,ka_k,j] = 0 And the equation for the diagonal coefficient(i, i)of (4.1) gives :

(4.3) X

k

|ai,k|²− |ak,j|²+|bi,k|²− |bk,j|²

+|xi|²=t

We letR_i,j =p

|λi−λ_j|²+|µi−µ_j|²and

(4.4)







x⁰_i =√ t

a⁰_i,j= (λi−λj)_R^t2 i,j

b⁰_i,j= (µi−µj)_R^t2 i,j

Then if we write the equations (4.2,4.3) in the synthetic form F(q,a,b,x) = 0 whereF : C²ⁿ

0 ×C^n(n−1)/2×C^n(n−1)/2×Cⁿ →C^n(n−1)/2×C^n(n−1)/2×Cⁿ. We have

F q,a⁰,b⁰,x⁰

=O σ⁻²

Moreover it is easy to check that whenσ is large enough, the partial derivative in the last three argumentD(a,b,x)F q,a⁰,b⁰,x⁰

is invertible and the norm of the inverse is uniformly bounded. The mapF being polynomial of degre2in its arguments, the implicit function theorem implies that the equations (4.2,4.3) have a unique solution such that (4.5) (a,b,x)(q) = (a⁰,b⁰,x⁰) +O σ⁻²

. MoreoverDq(a,b,x)(q) =O σ⁻²

. We have then build a map Ψb₀ : O0→Lt

q7→(A(q), B(q), x(q),0)

3with the convention thatai,j=bi,j= 0ifj≤i.

MoreoverΨb0(q)andΨb0(q’)live in the sameU(n)-orbit if and only if q and q’ live in the sameS_n-orbit henceΨb₀induces a map

Ψ₀ : O0/S_n →Hilbⁿ₀(C²)

which is holomorphic according to the lemma (2.2). With (4.5), we have

(4.6) |dΨ0(q).v|²=|v|²+O

|v|² σ⁴

The first term comes from the diagonals ofAandBthe second one from the off-diagonal terms and the derivative of x. In order to check the pointi)of the induction hypothesis we must show that

|dΨ0(q).v|²− |Πz(dΨ0(q).v)|²=|v|²+O |v|²

σ⁴

.

Where if Ψb0(q) = z ∈ Lt,Πz is the orthogonal projection onto the spaceImlz. But by construction, ifX ∈ Imlz thenIX is normal toTzLthencedΨ0(q).(Iv)⊥ IX in particularΠz(I.dΨ0(q).(Iv)) = 0;Hence

(4.7) Πz(dΨ0(q).v) = Πz(dΨ0(q).v+IdΨ0(q).Iv) = 2Πz ∂Ψ¯ 0(q).v . But by construction

(4.8)

∂Ψ¯ 0(q).v

2= ∂a¯

2+ ∂b¯

2+ ∂x¯

2=O |v|²

σ⁴

The assertion i) of the induction hypothesisi)follows from the estimates (4.6,4.7,4.8).

For the induction hypothesisii), we have forz=Ψb0(q)andh= (hi,j)∈un: kl_z(h)k²≥ 1

2



 X

i,j

R²_i,j|h_i,j|²+tX

i

|h_i,i|²



−Cσ⁻²khk²

Hence ifRis chosen large enough the induction hypothesisii)hold onO₀. Now we check the induction hypothesisiii)let

(δA, δB,0,0) =dΨ₀(q).v−Π_z(dΨ₀(q).v) we have just said that

kΠz(dΨ0(q).v)k=O σ⁻²

|v|.

Hence the off-diagonal part ofδAandδBare bounded byO σ⁻²

|v|, this implies that k[δA, δA^∗]k²+k[δB, δB^∗]k²≤O σ⁻⁴

|v|⁴.

4.3. The general case. We examine now the region Cp associated to another partition p6=p0., we can always assume that

p= ({m0= 1, ..., m1},{m1+ 1, m2}, ...,{m_k−1+ 1, .., n=mk}), letnl=ml−m_l−1. We consider the setOpof

(q,A,B,x)∈ C^2k

0×

k

M

j=1

M_nj(C)×

k

M

j=1

M_nj(C)×

k

M

j=1

C^nj

such that if q = (q1, q2, ..., qk)then for all i 6= j then |qi −qj| > q ₁

n(n−1)|q|and

|q| ≥ R and if A = (A1, A2, ..., Ak), B = (B1, B2, ..., Bk),x = (x1, x2, ..., xk)then each(Aj, Bj, xj)satisfiestrAj = trBj= 0and the moment map equation :

A_j, A^∗_j

+ B_j, B_j^∗

+x_jx^∗_j =tId_nj [Aj, Bj] = 0

and moreover

sup

j

kAjk²+kBjk²≤τ²|q|²

We will search a solutionz= (A, B, x,0)of the moment map equation which is approxi- matively

A'













A₁+λ₁ 0 . . . 0 0 A2+λ2 . .. ... ... . .. . .. ... 0 . . . Ak+λk











 ,

B'













B1+µ1 0 . . . 0 0 B2+µ2 . .. ... ... . .. . .. ... 0 . . . B_k+µ_k











 ,

x'(x1, x2, ..., xk)

We first fix someζ = (q,A,B,x) ∈ Opand we search az0 = (A⁰, B⁰, x⁰,0)where ifqj = (λj, µj)thenx⁰ = (x1, x2, ..., xk),A⁰(resp. B⁰) is upper block triangular with diagonal(A1+λ1, A2+λ2Id, ..., Ak+λk)(resp.(B1+µ1, B2+µ2, ..., Bk+µk)) and µ(z0), µ_C(z0)are block diagonal . Hence we search matricesAi,j, Bi,j ∈ Mni,nj(C), i <

jsuch that for alli < j: (4.9)







A^∗_i + ¯λ_i

A_i,j−A_i,j A^∗_j + ¯λ_j

+ (B_i^∗+ ¯µ_i)B_i,j−B_i,j B^∗_j + ¯µ_j +Q1(i, j) +Q2(i, j)2=xix^∗_j

−(B_i+µ_i)A_i,j+A_i,j(B_j+µ_j) + (A_i+λ_i)B_i,j−B_i,j(A_j+λ_j) +Q₃(i, j) = 0 whereQ₁(i, j)(resp.Q₂(i, j)) is a quadratic expression depending on theA_α,β’s (resp. in theB_α,β) andQ₃(i, j)is bilinear inA_α,β’s andB_α,β. Forτ >0small enough, with the same arguments given in the preceding paragraph, the implicit function theorem implies Lemma 4.1. The equations (4.9) has a solutionAi,j, Bi,j ∈ Mni,nj(C), i < j which depends smoothly onζ∈ Op, moreover we have that

X

i<j

kAi,jk²+kBi,jk²=O 1

|q|²

.

And the derivative of the mapζ7→(Ai,j, Bi,j)is bounded byO

1

|q|²

.

Then we obtainz0(ζ) = (A⁰, B⁰, x⁰,0)∈ Mn(C)× Mn(C)×Cⁿ×(Cⁿ)^∗an almost solution of the moment map equation :

( A⁰, B⁰

= 0 2iµ(z0)−t=O

1

|q|²

More precisely, the off block diagonal terms of the moment map equations are zero. We will now used an argument that we learned in a paper of S. Donaldson [8][Proposition 17]

: we will findh=ika Hermitian matrix such that if

zh=e^ik.z0= (e^hA⁰e^−h, e^hB⁰e^−h, e^h.x⁰,0)

then2iµ(zh)−tId = 0andµ_C(zh) = 0(this latter condition being obvious).

By the induction hypothesisii)and ifτis small enough andRlarge enough then we have

∀η∈un, klz0(η)k ≥C|η|

the constantCbeing uniform onOp. Hence ifh=iηwith kkk ≤δ:= min

1,Ce⁻² 4|z0|

then

∀η∈un, klz_h(η)k ≥ C 2|η|

So as soon as we have

µ(z₀)−_2i^t Id

< _C⁴₂δ, the proposition 17 in [8] furnishes ah=ik withµ(e^h.z0) =_2i^t Idwith

khk ≤ 4 C²

µ(z₀)− t 2iId

. But whenRis large enough, the condition

µ(z0)−_2i^t Id

< _C⁴2δis satisfied, hence there ish=ika Hermitian matrix such that2iµ(zh)−tId = 0andµ_C(zh) = 0.

We need to recall howh is found. For z ∈ M we have a linear map lz : un → TzM ' Mandl^∗_z its adjoint; by definition of the moment map we havel^∗_z = dµ(z)◦ I. The endomorphismQz ofun is given byQz = l^∗_zlz. Then for everyh = ik, with

|k| < δ,Qz_h is invertible andQ⁻¹_zh has a operator norm bounded by4C⁻². Leta(z) = Q⁻¹_z µ(z)−_2i^t Id

, we follow the maximal solution of the equation

(4.10) dz

ds =−ilz(a(z)), starting fromz0ats= 0. By definition we have

dµ(zs) ds =−

µ(z_s)− t 2iId

hence

µ(zs)− t

2iId =e^−s

µ(z0)− t 2iId

; in factzs=gs.z0where

dgs

ds =ia(zs).gs, gs∈GLn(C)

The arguments of [8] insures that the maximum solution of (4.10) is defined on[0,+∞[.

and ifgs=e^ηse^hswhereηs∈unandhsis Hermitian, then|hs| ≤δ. So that we also get : kg˙sk ≤ 4

C²

µ(z0)− t 2iId

e^−se^δ

henceg_∞= lim_s→+∞g_sexists and (4.11) kg_∞−Idk ≤ 4e^δ

C²

µ(z₀)− t 2iId

=O 1

|q|²

.

We clearly have2iµ(g∞.z0) = tIdandh= ikis given bye^2h =g^∗_∞g∞ i.e. the polar decomposition ofg∞ is g∞ = e^η∞e^h. Moreover if s ≥ 0, then the operator norm of lzsQ⁻¹_zz remains less than2/Chence

(4.12) kg∞.z0−z0k ≤ 2 C

µ(z0)− t 2iId

=O 1

|q|²

.

The implicit function theorem told us thathdepends smoothly onz₀hence onζ ∈ Op, indeed

d dt _t=0

µ(e^tik.z) =Qz(k).

This map will be called : ζ∈ O_p →h(ζ)∈iu_n. The following lemma gives an estimate of the size of the derivative ofh

Lemma 4.2. Let v∈TζOpbe a vector of unit length then

kdh(ζ).vk=O 1

|q|²

.

Proof. Let z˙0 = dz0(ζ0).v andh˙ = dh(ζ).v, we also letv ∈ Mbe the vector v :=

(δA, δB, δx,0)where if

v= (((δλ1, δµ1), ...,(δλk, δµk)),(δA1, ..., δAk),(δB1, ..., δBk),(δx1, ..δxk)) thenδA(resp.δB) is a block diagonal matrix with diagonal(δA₁+δλ₁Id_n1, ..., δA_k+ δλkIdn_k)(resp.(δB1+δµ1Idn₁, ..., δBk+δµkIdn_k)) andδx= ((δx1, ..., δxk).

We have

dµ(z_h).

Dexp(h) ˙h.z₀+e^h.z˙₀

= 0 Recall that :

Dexp(h) ˙h= e^adh−Id adh .h.e˙ ^h.

Letiη˙be the Hermitian part ofDexp(h) ˙handξ˙be its skew Hermitian part. Then dµ(z_h).(Dexp(h) ˙h.z₀) =dµ(z_h)(il_zhη) +˙ dµ(z_h)(l_zhξ) =˙ Q_zh( ˙η).

Moreover from the construction ofz0and the lemma (4.1), we obtain easily that dµ(z0)( ˙z0) =O

1

|q|²

;

and

˙

z₀=v+O 1

|q|²

|v|.

So ifk∈U(n)is such thatg_∞=ke^hthen Ad (k)dµ(zh). e^h.z˙0

=dµ(g_∞.z0).(g_∞.z˙0)

=dµ(g_∞.z0).((g_∞−Id ).z˙0) +dµ(g_∞.z0−z0).z˙0+dµ(z0)( ˙z0) Hence

Q_zh( ˙η) +dµ(z_h). k⁻¹. g_∞−Id ).z˙₀) =O 1

|q|²

.

We now make the scalar product of this quantity withη˙and we obtain : kl_zh( ˙η)k²≤O

1

|q|²

kηk − hl˙ _zh( ˙η), k⁻¹.I (g_∞−Id).z˙₀i

≤O 1

|q|²

(kηk˙ +klz_h( ˙η)k)

But our construction gives that

kηk ≤˙ 2

Cklz_h( ˙η)k, hence we obtain :

kηk ≤˙ 2

Cklz_h( ˙η)k=O 1

|q|²

. Nowh˙ is a Hermitian matrix andkhk=O |q|⁻²

hence by definition ofη˙andξ, we have˙ h˙ −iη˙ =O |q|⁻²

.

Hence the lemma.

We note that it is straightforward to verify the pointii)atzhbecause by construction

∀η∈un, lz_h(η)| ≥ C 2kηk.

We have build a mapfpfromOptoLtwhose value at a pointζ= (q,A,B,x)∈ Opis thezhconstructed before. This map isU(n1)×U(n2)×...×U(nk)-equivariant hence it induces a map

ψp : Op/(U(n1)×U(n2)×...×U(nk))→Hilbⁿ₀(C²) We remark that adjusting, R, τ, we have

C_p⊂ O_p/(U(n₁)×U(n₂)×...×U(n_k))⊂Hilb^p₀(C²)× C^2k

0. Where the last inclusion is an isometry ifHilb^p₀(C²)× C^2k

0is endowed with the product metric. We now want to compare the metricψ_p^∗g_N and the product metric onHilb^p₀(C²)×

C²k

0. Let v be a vector ofTζOpwhich is orthogonal to theU(n1)×U(n2)×...×U(nk) orbit ofζ. As before, we definedf(ζ) =zh=e^h.z0,h,˙ v..

Recall that we have denote byΠqthe orthogonal projection ontoImlq. Hence we need to compare

k(Id−Πz_h).dfp(ζ).vk²=kdfp(ζ).vk²− kΠz_h.dfp(ζ).vk² andkvk². But

dfp(ζ).v=lz_h( ˙ξ) +ilz_h( ˙η) +e^h.z˙0

Hence

(Id−Πz_h).dfp(ζ).v = (Id−Πz_h). ilz_h( ˙η) +e^h.z˙0

. But we have already seen that

kl_zh( ˙η)k²=O 1

|q|⁴

.

butilz_h( ˙η)is orthogonal toTLthence to the range ofΠz_h, we also have

h(Id−Πz_h) (ilz_h( ˙η)),(Id−Πz_h).(e^h.z˙0)i=h(Id−Πz_h).ilz_h( ˙η),(e^h.z˙0)i

=hilz_h( ˙η),(e^h.z˙0)i

=−

˙

η, dµ(z_h) e^h.z˙₀ But

dµ(zh) ((dfp(ζ).v) = 0 =dµ(zh) ilz_h( ˙η) +e^h.z˙0

so that

h(Id−Πz_h) (ilz_h( ˙η)),(Id−Πz_h).(e^h.z˙0)i=−

˙

η, dµ(zh) e^h.z˙0

=hη, dµ(z˙ _h) (il_zh( ˙η))i=kl_zh( ˙η)k

=O 1

|q|⁴

.

It remains now to estimate

k(Id−Π_zh) (e^h.z˙₀)k²=ke^h.z˙₀k²− kΠ_zh(e^h.z˙₀)k². ButΠz_h =lz_hQ⁻¹_zhl^∗_zh,and

l^∗_zh e^h.z˙0

=dµ(zh)(ie^h.z˙0).

We have already noticed that

I.z˙0=dz0(ζ).(I.v) +w wherew=O(|q|⁻². So

l^∗_zh e^h.z˙0

=dµ(zh) e^hdz0(ζ).(Iv)

+l_z^∗_h(w) And the proof of the lemma (4.2), furnishes aw⁰ =O(|q|⁻²)such that

dµ(zh) e^hdz0(ζ).(Iv)

=dµ(zh) (w⁰) +O(|q|⁻² as the operator norm ofl_zhQ⁻¹_z

h is bounded by2/Cwe have obtained : kΠzh(e^h.z˙₀)k²=O

1

|q|⁴

.

Hence we have obtain :

ψ_p^∗gN(v,v) =ke^h.z˙0k²+O 1

|q|⁴

kvk²

=kz˙0k²+ 2hz˙0, hz˙0i+O 1

|q|⁴

kvk².

By construction

|z˙₀|²=kvk²+O 1

|q|⁴

kvk²