Bergman kernels and symplectic reduction Xiaonan Ma

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Differential Geometry

Bergman kernels and symplectic reduction

Xiaonan Ma

, Weiping Zhang

aCentre de mathématiques, UMR 7640 du CNRS, École polytechnique, 91128 Palaiseau cedex, France bNankai Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China

Received 9 July 2005; accepted 18 July 2005 Available online 24 August 2005 Presented by Jean-Michel Bismut

Abstract

We present several results concerning the asymptotic expansion of the invariant Bergman kernel of the spin^cDirac operator associated with high tensor powers of a positive line bundle on a compact symplectic manifold. To cite this article: X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Résumé

Noyaux de Bergman et réduction symplectique. Nous annonçons des résultats sur le développement asymptotique du noyau de BergmanG-invariant de l’opérateur de Dirac spin^cassocié à une puissance tendant vers l’infini d’un fibré en droites positif sur une variété symplectique compacte. Pour citer cet article : X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Version française abrégée

Soit (X, ω) une variété symplectique compacte, et soit (L, h^L) un fibré en droites hermitien muni d’une connexion hermitienne∇^Ltelle que

√−1

2π (∇^L)²=ω. Soit(E, h^E)un fibré vectoriel hermitien surXmuni d’une connexion hermitienne ∇^E. Soit g^{T X} une métrique riemannienne surX, et soit J une structure presque com- plexe compatible séparément àg^{T X}etω. Alors les données géométriques ci-dessus définissent canoniquement un opérateur de Dirac spin^cD_pagissant surΩ^0,•(X, L^p⊗E), l’espace de(0,•)-formes à valeurs dansL^p⊗E.

SoitGun groupe de Lie compact connexe et soitgson algèbre de Lie. On suppose queGagit surX, et que son action se relève àLetE en préservantJ, les métriques et les connexions ci-dessus. Alors KerD_p est une G-représentation de dimension finie. Soit (KerD_p)^G la partie G-invariante de KerD_p. Le noyau de Bergman

E-mail addresses: [email protected] (X. Ma), [email protected] (W. Zhang).

1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.

doi:10.1016/j.crma.2005.07.009

G-invariantP_p^G(x, x) (x, x∈X) est le noyauC^∞ de la projection orthogonaleP_p^G de Ω^0,•(X, L^p⊗E)sur (KerD_p)^Gassocié à la forme de volume riemannienne dv_X(x).

Dans cette Note, nous annonçons des résultats sur le développement asymptotique deP_p^G(x, x)quandptend vers l’infini. Le détail des démonstrations et des applications de nos résultats est donné dans [6].

1. Introduction

Let(X, ω)be a compact symplectic manifold of real dimension 2n. Assume that there exists a Hermitian line bundleLoverXendowed with a Hermitian connection∇^Lwith the property that

√−1

2π R^L=ω, whereR^L=(∇^L)² is the curvature of(L,∇^L). Let(E, h^E)be a Hermitian vector bundle onXwith Hermitian connection∇^Eand its curvatureR^E.

Letg^{T X} be a Riemannian metric onX. Let J :T X→T Xbe the skew-adjoint linear map which satisfies the relationω(u, v)=g^{T X}(Ju, v)for u, v∈T X. LetJ be an almost complex structure such that g^{T X}(J u, J v)= g^{T X}(u, v), ω(J u, J v)=ω(u, v), and that ω(·, J·) defines a metric on T X. Then J commutes with J, thus J =J(−J²)^−1/2. Let ∇^{T X} be the Levi-Civita connection on (T X, g^{T X}) with curvature R^{T X} and scalar cur- vature r^X. Then∇^{T X} induces a natural connection∇^det on det(T^(1,0)X) with curvatureR^det, and the Clifford connection∇^Cliffon the Clifford moduleΛ(T^∗(0,1)X)with curvatureR^Cliff. The spin^cDirac operatorD_pacts on Ω^0,•(X, L^p⊗E)=_n

q=0Ω^0,q(X, L^p⊗E), the direct sum of spaces of (0, q)-forms with values in L^p⊗E.

We denote byD_p⁺the restriction ofD_p onΩ^0,even(X, L^p⊗E). By [4, Theorem 2.5], whenp is large enough, CokerD⁺_p vanishes.

LetGbe a compact connected Lie group with Lie algebragand dimG=n₀. Suppose thatGacts onXand its action onXlifts onL, E. Moreover, we assume theG-action preserves the above connections and metrics on T X, L, EandJ. Then KerDpis a finite dimensional representation space ofG.

The action ofGonLinduces naturally a moment mapµ:X→g^∗. Now we assume that 0∈g^∗is a regular value ofµ. Then the Marsden–Weinstein symplectic reduction(X_G=µ⁻¹(0)/G, ω_XG)is a symplectic manifold whenGacts freely onµ⁻¹(0). Moreover,(L,∇^L),(E,∇^E)descend to(L_G,∇^LG),(E_G,∇^EG)overX_Gso that the corresponding curvature condition

√−1

2π R^LG=ωGholds (cf. [3]). TheG-invariant almost complex structureJ also descends to an almost complex structureJGonT XG, andh^L, h^E, g^{T X}descend toh^LG,h^EG, g^{T XG} respectively.

Thus we can construct the corresponding spin^cDirac operatorDG,ponXG.

Let (KerD_p)^G denote the G-invariant part of KerD_p. Let P_p^G be the orthogonal projection fromΩ^0,•(X, L^p⊗E) on(KerD_p)^G. TheG-invariant Bergman kernel is P_p^G(x, x) (x, x∈X), the smooth kernel ofP_p^G with respect to the Riemannian volume form dv_X(x). Let pr₁ and pr₂be the projections fromX×X onto the first and second factorX respectively. Then P_p^G(x, x)is a smooth section of pr^∗₁E_p⊗pr^∗₂E_p^∗ onX×X with E_p=Λ(T^∗(0,1)X)⊗L^p⊗E. In particular,P_p^G(x, x)∈End(E_p)_x=End(Λ(T^∗(0,1)X)⊗E)_x.

TheG-invariant Bergman kernelP_p^G(x, x)is a local analytic version of(KerD_p)^G.

In this Note, we present several results concerning the asymptotic expansions ofP_p^G(x, x)asp→ +∞. More details will appear in [6].

2. Main results

The first result shows that one can reduce our problem to a problem nearµ⁻¹(0).

Theorem 2.1. For any openG-neighborhoodU ofµ⁻¹(0)inX,ε₀>0,l, m∈N, there existsC_l,m>0 (depend onU,ε₀) such that forp1,x, x∈X,d(Gx, x)ε₀orx, x∈X\U,

P_p^G(x, x)

C^mC_l,mp^−l, (1)

whereC^mis theC^m-norm induced by∇^L,∇^E,∇^{T X},h^L, h^E, g^{T X}.

Assume for simplicity thatGacts freely onµ⁻¹(0). LetU be an open neighborhood ofµ⁻¹(0)such thatG acts freely onU. For anyG-equivariant bundle(F,∇^F)onU, we denote byF_Bthe bundle onU/G=Binduced naturally byG-invariant sections ofF onU. The connection∇^F induces canonically a connection∇^FB onF_B. LetR^FB be its curvature. We denote also byµ^F(K)= ∇_K^F−L_K∈End(F )forK∈g. Note thatP_p^G∈(C^∞(U× U,pr^∗₁E_p⊗pr^∗₂E_p^∗))^G×G, thus we can viewP_p^G(x, x)as a smooth section of(pr^∗₁E_p)_B⊗(pr^∗₂E_p^∗)_BonB×B.

Letg^{T B} be the Riemannian metric onU/G=B induced byg^{T X}. Let∇^{T B} be the Levi-Civita connection on (T B, g^{T B})with curvatureR^{T B}. LetN_G be the normal bundle toX_G inB. We identifyN_G with the orthogonal complement ofT XGin(T B|X_G, g^{T B}). LetP^{T XG},P^NGbe the orthogonal projections fromT B|X_GonT XG,NG

respectively. Set

∇^NG=P^NG

∇^{T B}|X_G

P^NG, ⁰∇^{T B}=P^{T XG}

∇^{T B}|X_G

P^{T XG}⊕ ∇^NG, A= ∇^{T B}−⁰∇^{T B}. (2)

Then∇^NG,⁰∇^{T B} are Euclidean connections onN_G,T B|XG onX_G, andAis the associated second fundamental form. We denote by vol(Gx) (x ∈U ) the volume of the orbit Gx equipped with the metric induced by g^{T X}. Following [9, (3.10)], leth(x)be the function onUdefined by

h(x)=

vol(Gx)1/2

. (3)

Thenhreduces to a function onB. We denote byI_C⊗E the projection fromΛ(T^∗(0,1)X)⊗EontoC⊗Eunder the decompositionΛ(T^∗(0,1)X)=C⊕Λ^>0(T^∗(0,1)X), andI_C⊗EB the corresponding projection onB.

For anyx₀∈X_G,Z∈T_x0B, we writeZ=Z⁰+Z^⊥, withZ⁰∈T_x0X_G,Z^⊥∈N_G,x0. Letτ_Z0Z^⊥∈N

G,exp^XG_x

0 (Z⁰)

be the parallel transport of Z^⊥ with respect to the connection ∇^NG along the geodesic in X_G, [0,1] t → exp^X_x0^G(t Z⁰). Forε₀>0 small enough, we identifyZ∈T_x0B,|Z|< ε₀with exp^B

exp^XGx0 (Z⁰)(τ_Z0Z^⊥)∈B, then for x₀∈X_G,Z, Z∈T_x0B,|Z|,|Z|< ε₀, the map

Ψ:T B|XG×T B|XG→B×B, Ψ (Z, Z)= exp^B

exp^XG_x

0 (Z⁰)(τ_Z0Z^⊥),exp^B

exp^XG_x

0 (Z⁰)(τ_Z0Z ^⊥)

is well defined. We identify(E_p)_B,Z to(E_p)_B,x0 by using parallel transport with respect to∇^(Ep)B along[0,1] u→uZ. LetπB:T B|X_G×T B|X_G→XGbe the natural projection from the fiberwise product ofT B|X_G onXG

ontoXG. From Theorem 2.1, we only need to understandP_p^G◦Ψ, and under our identification,P_p^G◦Ψ (Z, Z)is a smooth section ofπ_B^∗(End(Ep)B)=π_B^∗(End(Λ(T^∗(0,1)X)⊗E)B)onT B|X_G×T B|X_G. Let| |_C^m_(XG)be the C^m-norm onC^∞(XG,End(Λ(T^∗(0,1)X)⊗E)B)induced by∇^CliffB,∇^EB,h^E andg^{T X}. The norm| |_C^m_(XG) induces naturally aC^m-norm alongXGonC^∞(T B|X_G×T B|X_G, π_B^∗(End(Λ(T^∗(0,1)X)⊗E)B)), we still denote it by| |_C^m_(XG).

Letg^{T XG},g^NG be the metric onT X_G, N_Ginduced byg^{T X}. Let dv_XG, dv_NG be the Riemannian volume forms on(X_G, g^{T XG}),(N_G, g^NG). Letκ∈C^∞(T B|XG,R), withκ=1 onX_G, be defined by that forZ∈T_x0B,x₀∈X_G,

dv_B(x₀, Z)=κ(x₀, Z)dv_Tx

0B(Z)=κ(x₀, Z)dv_XG(x₀)dv_NG,x

0. (4)

Theorem 2.2. Assume that G acts freely on µ⁻¹(0) and J=J on µ⁻¹(0). Then there exist Qr(Z, Z)∈ End(Λ(T^∗(0,1)X)⊗E)_B,x0 (x₀∈X_G, r∈N), polynomials inZ, Zwith the same parity asr, whose coefficients are polynomials inA,R^{T B},R^CliffB,R^EB,µ^E,µ^Cliff (resp.r^X,R^det,R^E; resp.h,R^L,R^LB; resp.µ) and their derivatives atx₀to orderr−1 (resp.r−2; resp.r, resp.r+1), such that if we denote by

P_x^(r)

0 (Z, Z)=Qr(Z, Z)P (Z, Z), Q0(Z, Z)=I_C⊗EB, (5)

with

P (Z, Z)=exp

−π

2Z⁰−Z⁰²−π√

−1

J_x0Z⁰, Z⁰

2^n0/2exp

−π

|Z^⊥|²+ |Z ^⊥|²

, (6)

then there existsC>0 such that for anyk, m, m, m∈N, there existsC >0 such that forx₀∈XG,Z, Z∈Tx₀B,

|Z|,|Z|ε₀, 1+√

p|Z^⊥| +√

p|Z ^⊥|m

sup

|α|+|α|m

∂^|α|+|α|

∂Z^α∂Z^α p^−n+n0/2

hκ^1/2 (Z)

hκ^1/2

(Z)P_p^G◦Ψ (Z, Z)− k

r=0

P_x^(r)

0 (√ p Z,√

p Z)p^−r/2

C^m(X_G)

Cp^{−(k+1−m)/2} 1+√

pZ⁰+√

pZ^02(n+k+2)+mexp

−√ C√

p|Z−Z| +O

p^−∞

. (7)

Furthermore, the expansion is uniform in the following sense: for any fixed k, m, m, m∈N, assume that the derivatives ofg^{T X},h^L,∇^L,h^E,∇^E, andJwith order2n+2k+m+m+4 run over a set bounded in theC^m- norm taken with respect to the parameters and, moreover,g^{T X}runs over a set bounded below. Then the constant Cis independent ofg^{T X}; and theC^m-norm in (7) includes also the derivatives on the parameters.

In (7), the termO(p^−∞)means that for anyl, l₁∈N, there existsC_l,l1>0 such that itsC^l1-norm is dominated byC_l,l1p^−l. The kernelP (Z, Z)is the product of two kernels: alongT_x0X_G, it is the classical Bergman kernel on T_x0X_Gwith complex structureJ_x0, while alongN_G, it is the kernel of a harmonic oscillator onN_G,x0.

Remark 1. (i) WhenG= {1}, Theorem 2.2 has been proved in [2, Theorem 3.18].

(ii) If we takeZ=Z=0 in (7), then we get forx₀∈X_G, P_x⁽⁰⁾

0 (0,0)=2^n0/2I_C⊗EB, (8)

and

p^−n+n0/2h²(x₀)P_p^G(x₀, x₀)− k

r=0

P_x^(2r)

0 (0,0)p^−r C^m(XG)

Cp^−k−1. (9)

In fact, (8) and (9) can be obtained as direct consequences of the full off-diagonal asymptotic expansion of the Bergman kernel proved in [2, Theorem 3.18].

Remark 2. Assume that(X, ω)is a Kähler manifold and J=J onX. Assume also that (L,∇^L),(E,∇^E)are holomorphic vector bundles with holomorphic Hermitian connections. ThenD_p² preserves the Z-graduation of Ω^0,•(X, L^p⊗E)and KerD_p=H⁰(X, L^p⊗E)whenpis large enough, and soP_p^G(x₀, x₀)∈End(E). In particu- larP_x⁽⁰⁾₀ (0,0)=2^n0/2Id_Ein (8). In the special case ofE=C,P_p^G(x₀, x₀)is a function onX_G, (9) has been proved in [7, Theorem 1] without knowing the informations onP_x^(2r)₀ (0,0), while in [8, Theorem 1], it was claimed that P_x⁽⁰⁾₀ (0,0)=1.

LetIpbe a section of End(Λ(T^∗(0,1)X)⊗E)_BonX_Gdefined by Ip(x₀)=

Z∈NG,|Z|ε0

h²(x₀, Z)P_p^G◦Ψ

(x₀, Z), (x₀, Z)

κ(x₀, Z)dv_NG(Z). (10)

By Theorem 2.1, moduloO(p^−∞),Ip(x₀)does not depend onε₀, and dim(KerD_p)^G=

X

Tr

P_p^G(y, y)

dv_X(y)=

XG

Tr

Ip(x₀)

dv_XG(x₀)+O p^−∞

. (11)

A direct consequence of Theorem 2.2 is the following corollary.

Corollary 2.3. TakenZ=Z∈N_G,x0,m=0 in (7), we get

p^−n+n0/2h²(Z)P_p^G(Z, Z)− k

r=0

P_x^(r)

0 (√ p Z,√

p Z)p^−r/2 C^m(XG)

Cp^−(k+1)/2 1+√

p|Z|₋m

+O p^−∞

. (12)

In particular, there existΦ_r∈End(Λ(T^∗(0,1)X)⊗E)_B,x0 (r∈N)which are polynomials inA,R^{T B},R^CliffB,R^EB, µ^E,µ^Cliff, (resp. r^X, R^det,R^E; resp. h,R^LB,R^L; resp. µ), and their derivatives at x₀ to order 2r−1 (resp.

2r−2; resp. 2r; resp. 2r+1), andΦ₀=I_C⊗EB, such that for anyk, m∈N, there existsC_k,m >0 such that for anyx₀∈XG,p∈N,

p⁻ⁿ⁺ⁿ⁰Ip(x₀)− k

r=0

Φ_r(x₀)p^−r C^m

C_k,mp^−k−1. (13)

Theorem 2.4. If(X, ω)is a Kähler manifold andL, E are holomorphic vector bundles with holomorphic Her- mitian connections∇^L,∇^E, J=J on U, and Gacts freely onµ⁻¹(0), then in (13),Φ_r(x₀)∈End(E_G)_x0 are polynomials inA,R^{T B},R^EB,µ^E,R^E (resp.h,R^LB; resp.µ) and their derivatives atx₀to order 2r−1 (resp. 2r;

resp. 2r+1), andΦ₀=Id_EG. Moreover Φ₁(x₀)= 1

8πr_x^XG

0 + 3

4π_XGlog(h|XG)+

√−1 4π R_x^EG

0

e_j⁰, J_Ge⁰_j

. (14)

Herer^XG is the Riemannian scalar curvature of(T X_G, g^{T XG}),_XG is the Bochner–Laplacian onX_G, and{e⁰_j} is an orthonormal basis ofT X_G.

Still making the same assumptions as in Theorem 2.4, leth˜denote the restriction toX_Gof the functionhdefined in (3). In view of [9, (3.11), (3.54)], seth˜^EG= ˜h²h^EG.

LetP_p^XG denote the orthogonal projection fromC^∞(X_G, L^p_G⊗E_G)ontoH⁰(X, L^p_G⊗E_G)associated to the metricsh^LG,h˜^EG,g^{T XG}. LetPp^XG(x0, x₀) (x0, x₀ ∈XG)denote the corresponding Bergman kernel with respect to dv_XG(x₀).

Then by [2, Theorem 1.3], we have the following theorem.

Theorem 2.5. Under the assumption of Theorem 2.4, there exist smooth coefficientsΦ_r(x₀)∈End(E_G)_x0 which are polynomials inR^{T XG},R^EG (resp.h), and their derivatives atx₀to order 2r−1 (resp. 2r), andΦ₀=IdE_G, such that for anyk, l∈N, there existsCk,l>0 such that for anyx0∈XG,p∈N,

p⁻ⁿ⁺ⁿ⁰P_p^XG(x₀, x₀)− k

r=0

Φr(x₀)p^−r

_ClCk,lp^−k−1. (15) Moreover, the following identity holds,

Φ₁(x₀)= 1 8πr_x^XG

0 + 1

2π_XGlogh˜+

√−1 4π R^E_x^G

0

e⁰_j, J_Ge⁰_j

. (16)

Remark 3. From (14) and (16), one sees that in generalΦ₁=Φ₁, ifhis not constant onX_G. This reflects a subtle difference between the Bergman kernel and the geometric quantization.

The proof of the above theorems uses techniques adapted from [1, §11], [2,5], along with a deformation ofD²_p by the Casimir operator (i.e., to considerD²_p−pCas, which plays a key role in proving Theorems 2.1, 2.2). We refer to [6] for more details.

Acknowledgements

The work of the second author was partially supported by MOEC and MOSTC. Part of this Note was written while the second author was visiting IHES during February and March, 2005. He would like to thank Professor Jean-Pierre Bourguignon for the hospitality.

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