We generalize several recent results concerning the asymptotic expansions of
Bergman kernels to the framework of geometric quantization and establish an
asymptotic symplectic identification property. More precisely, we study the
asymptotic expansion of the $G$-invariant Bergman kernel of the spin^c Dirac
operator associated with high tensor powers of a positive line bundle on a
symplectic manifold. We also develop a way to compute the coefficients of the
expansion, and compute the first few of them, especially, we obtain the scalar
curvature of the reduction space from the $G$-invariant Bergman kernel on the
total space. These results generalize the corresponding results in the
non-equivariant setting, which has played a crucial role in the recent work of
Donaldson on stability of projective manifolds, to the geometric quantization
setting. As another kind of application, we generalize some Toeplitz operator
type properties in semi-classical analysis to the framework of geometric
quantization. The method we use is inspired by Local Index Theory, especially
by the analytic localization techniques developed by Bismut and Lebeau.Comment: 132 page

Ma, Xiaonan

Zhang, Weiping

English

arXiv

132 pages.We generalize several recent results concerning the asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. More precisely, we study the asymptotic expansion of the $G$-invariant Bergman kernel of the spin^c Dirac operator associated with high tensor powers of a positive line bundle on a symplectic manifold. We also develop a way to compute the coefficients of the expansion, and compute the first few of them, especially, we obtain the scalar curvature of the reduction space from the $G$-invariant Bergman kernel on the total space. These results generalize the corresponding results in the non-equivariant setting, which has played a crucial role in the recent work of Donaldson on stability of projective manifolds, to the geometric quantization setting. As another kind of application, we generalize some Toeplitz operator type properties in semi-classical analysis to the framework of geometric quantization. The method we use is inspired by Local Index Theory, especially by the analytic localization techniques developed by Bismut and Lebeau

HAL-Polytechnique

Bergman kernels and symplectic reduction

Xiaonan Ma

Weiping Zhang

Comptes Rendus Mathématique

1/ue duoitese-ent unnC. R. Acad. Sci. Paris, Ser. I 341 (2005) 297–302http://france.elsevier.com/direct/CRASSDifferential GeometryBergman kernels and symplectic reductionXiaonan Maa, Weiping Zhangba Centre de mathématiques, UMR 7640 du CNRS, École polytechnique, 91128 Palaiseau cedex, Franceb Nankai Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR ChinaReceived 9 July 2005; accepted 18 July 2005Available online 24 August 2005Presented by Jean-Michel BismutAbstractWe present several results concerning the asymptotic expansion of the invariant Bergman kernel of the spinc Dirac operatorassociated 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.RésuméNoyaux de Bergman et réduction symplectique. Nous annonçons des résultats sur le développement asymptotiqnoyau de BergmanG-invariant de l’opérateur de Dirac spinc associé à une puissance tendant vers l’infini d’un fibré en drpositif 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 abrégéeSoit (X,ω) une variété symplectique compacte, et soit(L,hL) un fibré en droites hermitien muni d’unconnexion hermitienne∇L telle que√−12π (∇L)2 = ω. Soit (E,hE) un fibré vectoriel hermitien surX muni d’uneconnexion hermitienne∇E . Soit gT X une métrique riemannienne surX, et soitJ une structure presque complexe compatible séparément àgT X etω. Alors les données géométriques ci-dessus définissent canoniquemopérateur de Dirac spinc Dp agissant surΩ0,•(X,Lp ⊗ E), l’espace de(0,•)-formes à valeurs dansLp ⊗ E.Soit G un groupe de Lie compact connexe et soitg son algèbre de Lie. On suppose queG agit surX, et queson action se relève àL et E en préservantJ , les métriques et les connexions ci-dessus. Alors KerDp est uneG-représentation de dimension finie. Soit(KerDp)G la partieG-invariante de KerDp. Le noyau de BergmaE-mail addresses:ma@math.polytechnique.fr (X. Ma), weiping@nankai.edu.cn (W. Zhang).1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.07.009298 X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 297–302nethe-,s onG-invariantP Gp (x, x′) (x, x′ ∈ X) est le noyauC ∞ de la projection orthogonaleP Gp de Ω0,•(X,Lp ⊗ E) sur(KerDp)G associé à la forme de volume riemannienne dvX(x′).Dans cette Note, nous annonçons des résultats sur le développement asymptotique deP Gp (x, x′) quandp tendvers l’infini. Le détail des démonstrations et des applications de nos résultats est donné dans [6].1. IntroductionLet (X,ω) be a compact symplectic manifold of real dimension 2n. Assume that there exists a Hermitian libundleL overX endowed with a Hermitian connection∇L with the property that√−12π RL = ω, whereRL = (∇L)2is the curvature of(L,∇L). Let (E,hE) be a Hermitian vector bundle onX with Hermitian connection∇E and itscurvatureRE .Let gT X be a Riemannian metric onX. Let J :T X → T X be the skew-adjoint linear map which satisfiesrelationω(u, v) = gT X(Ju,v) for u,v ∈ T X. Let J be an almost complex structure such thatgT X(Ju,Jv) =gT X(u, v), ω(Ju,Jv) = ω(u, v), and thatω(·, J ·) defines a metric onT X. Then J commutes withJ, thusJ = J(−J2)−1/2. Let ∇T X be the Levi-Civita connection on(T X,gT X) with curvatureRT X and scalar curvaturerX . Then∇T X induces a natural connection∇det on det(T (1,0)X) with curvatureRdet, and the Cliffordconnection∇Cliff on the Clifford moduleΛ(T ∗(0,1)X) with curvatureRCliff . The spinc Dirac operatorDp acts onΩ0,•(X,Lp ⊗ E) = ⊕nq=0 Ω0,q (X,Lp ⊗ E), the direct sum of spaces of(0, q)-forms with values inLp ⊗ E.We denote byD+p the restriction ofDp on Ω0,even(X,Lp ⊗ E). By [4, Theorem 2.5], whenp is large enoughCokerD+p vanishes.Let G be a compact connected Lie group with Lie algebrag nd dimG = n0. Suppose thatG acts onX andits action onX lifts on L,E. Moreover, we assume theG-action preserves the above connections and metricT X,L,E andJ . Then KerDp is a finite dimensional representation space ofG.The action ofG on L induces naturally a moment mapµ :X → g∗. Now we assume that 0∈ g∗ is a regularvalue ofµ. Then the Marsden–Weinstein symplectic reduction(XG = µ−1(0)/G,ωXG) is a symplectic manifoldwhenG acts freely onµ−1(0). Moreover,(L,∇L), (E,∇E) descend to(LG,∇LG), (EG,∇EG) overXG so that thecorresponding curvature condition√−12π RLG = ωG holds (cf. [3]). TheG-invariant almost complex structureJ alsodescends to an almost complex structureJG on T XG, andhL,hE,gT X descend tohLG , hEG,gT XG respectively.Thus we can construct the corresponding spinc Dirac operatorDG,p onXG.Let (KerDp)G denote theG-invariant part of KerDp. Let P Gp be the orthogonal projection fromΩ0,•(X,Lp ⊗ E) on (KerDp)G. The G-invariant Bergman kernel isP Gp (x, x′) (x, x′ ∈ X), the smooth kernel ofP Gpwith respect to the Riemannian volume form dvX(x′). Let pr1 and pr2 be the projections fromX × X onto thefirst and second factorX respectively. ThenP Gp (x, x′) is a smooth section of pr∗1 Ep ⊗ pr∗2 E∗p on X × X withEp = Λ(T ∗(0,1)X) ⊗ Lp ⊗ E. In particular,P Gp (x, x) ∈ End(Ep)x = End(Λ(T ∗(0,1)X) ⊗ E)x .TheG-invariant Bergman kernelP Gp (x, x′) is a local analytic version of(KerDp)G.In this Note, we present several results concerning the asymptotic expansions ofP Gp (x, x′) asp → +∞. Moredetails will appear in [6].2. Main resultsThe first result shows that one can reduce our problem to a problem nearµ−1(0).Theorem 2.1. For any openG-neighborhoodU of µ−1(0) in X, ε0 > 0, l,m ∈ N, there existsCl,m > 0 (dependonU , ε0) such that forp  1, x, x′ ∈ X, d(Gx,x′)  ε0 or x, x′ ∈ X \ U ,X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 297–302 299nltalss∣∣P Gp (x, x′)∣∣C m  Cl,mp−l , (1)whereC m is theC m-norm induced by∇L,∇E , ∇T X , hL,hE,gT X .Assume for simplicity thatG acts freely onµ−1(0). Let U be an open neighborhood ofµ−1(0) such thatGacts freely onU . For anyG-equivariant bundle(F,∇F ) onU , we denote byFB the bundle onU/G = B inducednaturally byG-invariant sections ofF on U . The connection∇F induces canonically a connection∇FB on FB .Let RFB be its curvature. We denote also byµF (K) = ∇FK − LK ∈ End(F ) for K ∈ g. Note thatP Gp ∈ (C ∞(U ×U,pr∗1 Ep ⊗ pr∗2 E∗p))G×G, thus we can viewP Gp (x, x′) as a smooth section of(pr∗1 Ep)B ⊗ (pr∗2 E∗p)B onB × B.Let gT B be the Riemannian metric onU/G = B induced bygT X. Let ∇T B be the Levi-Civita connection o(T B,gT B) with curvatureRT B . Let NG be the normal bundle toXG in B. We identifyNG with the orthogonacomplement ofT XG in (T B|XG,gT B). Let P T XG , P NG be the orthogonal projections fromT B|XG onT XG, NGrespectively. Set∇NG = P NG(∇T B |XG)P NG, 0∇T B = P T XG(∇T B |XG)P T XG ⊕ ∇NG, A = ∇T B − 0∇T B. (2)Then∇NG, 0∇T B are Euclidean connections onNG, T B|XG on XG, andA is the associated second fundamenform. We denote by vol(Gx) (x ∈ U) the volume of the orbitGx equipped with the metric induced bygT X .Following [9, (3.10)], leth(x) be the function onU defined byh(x) = (vol(Gx))1/2. (3)Thenh reduces to a function onB. We denote byIC⊗E the projection fromΛ(T ∗(0,1)X) ⊗ E ontoC ⊗ E underthe decompositionΛ(T ∗(0,1)X) = C ⊕ Λ>0(T ∗(0,1)X), andIC⊗EB the corresponding projection onB.For anyx0 ∈ XG, Z ∈ Tx0B, we writeZ = Z0+Z⊥, with Z0 ∈ Tx0XG, Z⊥ ∈ NG,x0. Let τZ0Z⊥ ∈ NG,expXGx0 (Z0)be the parallel transport ofZ⊥ with respect to the connection∇NG along the geodesic inXG, [0,1]  t →expXGx0 (tZ0). For ε0 > 0 small enough, we identifyZ ∈ Tx0B, |Z| < ε0 with expBexpXGx0 (Z0)(τZ0Z⊥) ∈ B, then forx0 ∈ XG, Z,Z′ ∈ Tx0B, |Z|, |Z′| < ε0, the mapΨ :T B|XG × T B|XG → B × B, Ψ (Z,Z′) =(expBexpXGx0 (Z0)(τZ0Z⊥),expBexpXGx0 (Z′0)(τZ′0Z′⊥))is well defined. We identify(Ep)B,Z to (Ep)B,x0 by using parallel transport with respect to∇(Ep)B along[0,1] u → uZ. Let πB :T B|XG × T B|XG → XG be the natural projection from the fiberwise product ofT B|XG on XGontoXG. From Theorem 2.1, we only need to understandP Gp ◦ Ψ , and under our identification,P Gp ◦ Ψ (Z,Z′) isa smooth section ofπ∗B(End(Ep)B) = π∗B(End(Λ(T ∗(0,1)X) ⊗ E)B) on T B|XG × T B|XG . Let | |C m′ (XG) be theC m′-norm onC ∞(XG,End(Λ(T ∗(0,1)X) ⊗ E)B) induced by∇Cliff B , ∇EB , hE andgT X. The norm| |C m′ (XG)induces naturally aC m′-norm alongXG onC ∞(T B|XG × T B|XG,π∗B(End(Λ(T ∗(0,1)X)⊗ E)B)), we still denoteit by | |C m′ (XG).Let gT XG , gNG be the metric onT XG,NG induced bygT X. Let dvXG , dvNG be the Riemannian volume formon(XG,gT XG ), (NG,gNG). Letκ ∈ C ∞(T B|XG,R), with κ = 1 onXG, be defined by that forZ ∈ Tx0B, x0 ∈ XG,dvB(x0,Z) = κ(x0,Z)dvTx0B(Z) = κ(x0,Z)dvXG(x0)dvNG,x0 . (4)Theorem 2.2. Assume thatG acts freely onµ−1(0) and J = J on µ−1(0). Then there existQr (Z,Z′) ∈End(Λ(T ∗(0,1)X) ⊗ E)B,x0 (x0 ∈ XG, r ∈ N), polynomials inZ,Z′ with the same parity asr , whose coefficientare polynomials inA, RT B , RCliff B , REB , µE , µCliff (resp.rX , Rdet, RE ; resp.h, RL, RLB ; resp.µ) and theirderivatives atx0 to orderr − 1 (resp.r − 2; resp.r , resp.r + 1), such that if we denote byP (r)x (Z,Z′) = Qr (Z,Z′)P (Z,Z′), Q0(Z,Z′) = IC⊗EB , (5)0300 X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 297–302tantnn of thedtwithP(Z,Z′) = exp(−π2∣∣Z0 − Z′0∣∣2 − π√−1〈Jx0Z0,Z′0〉)2n0/2 exp(−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′|∂Zα∂Z′α′(p−n+n0/2(hκ1/2)(Z)(hκ1/2)(Z′)P Gp ◦ Ψ (Z,Z′) −k∑r=0P (r)x0 (√p Z,√p Z′)p−r/2)∣∣∣∣∣C m′(XG) Cp−(k+1−m)/2(1+ √p ∣∣Z0∣∣ + √p ∣∣Z′0∣∣)2(n+k+2)+m exp(−√C′′ √p |Z − Z′|) + O(p−∞). (7)Furthermore, the expansion is uniform in the following sense: for any fixedk,m,m′,m′′ ∈ N, assume that thederivatives ofgT X, hL, ∇L, hE , ∇E , andJ with order 2n+2k +m+m′ +4 run over a set bounded in theC m′ -norm taken with respect to the parameters and, moreover,gT X runs over a set bounded below. Then the consC is independent ofgT 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 existsCl,l1 > 0 such that itsC l1-norm is dominatedby Cl,l1p−l . The kernelP(Z,Z′) is the product of two kernels: alongTx0XG, it is the classical Bergman kernel oTx0XG with complex structureJx0, while alongNG, it is the kernel of a harmonic oscillator onNG,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 ∈ XG,P (0)x0 (0,0) = 2n0/2IC⊗EB , (8)and ∣∣∣∣∣p−n+n0/2h2(x0)P Gp (x0, x0) −k∑r=0P (2r)x0 (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 expansioBergman kernel proved in [2, Theorem 3.18′].Remark 2. Assume that(X,ω) is a Kähler manifold andJ = J on X. Assume also that(L,∇L), (E,∇E) areholomorphic vector bundles with holomorphic Hermitian connections. ThenD2p preserves theZ-graduation ofΩ0,•(X,Lp ⊗E) and KerDp = H 0(X,Lp ⊗E) whenp is large enough, and soP Gp (x0, x0) ∈ End(E). In particu-lar P (0)x0 (0,0) = 2n0/2IdE in (8). In the special case ofE = C, P Gp (x0, x0) is a function onXG, (9) has been provein [7, Theorem 1] without knowing the informations onP (2r)x0 (0,0), while in [8, Theorem 1], it was claimed thaP(0)x0 (0,0) = 1.Let Ip be a section of End(Λ(T ∗(0,1)X) ⊗ E)B onXG defined byIp(x0) =∫h2(x0,Z)PGp ◦ Ψ((x0,Z), (x0,Z))κ(x0,Z)dvNG(Z). (10)Z∈NG, |Z|ε0X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 297–302 301r-epectBy Theorem 2.1, moduloO(p−∞), Ip(x0) does not depend onε0, anddim(KerDp)G =∫XTr[P Gp (y, y)]dvX(y) =∫XGTr[Ip(x0)]dvXG(x0) + O(p−∞). (11)A direct consequence of Theorem 2.2 is the following corollary.Corollary 2.3. TakenZ = Z′ ∈ NG,x0, m = 0 in (7), we get∣∣∣∣∣p−n+n0/2h2(Z)P Gp (Z,Z) −k∑r=0P (r)x0 (√p Z,√p Z)p−r/2∣∣∣∣∣C m′(XG) Cp−(k+1)/2(1+ √p |Z|)−m′′ + O(p−∞). (12)In particular, there existΦr ∈ End(Λ(T ∗(0,1)X)⊗E)B,x0 (r ∈ N) which are polynomials inA, RT B , RCliff B , REB ,µE , µCliff , (resp.rX , Rdet, RE ; resp.h, RLB , RL; resp.µ), and their derivatives atx0 to order 2r − 1 (resp.2r − 2; resp.2r ; resp.2r + 1), andΦ0 = IC⊗EB , such that for anyk,m′ ∈ N, there existsCk,m′ > 0 such that foranyx0 ∈ XG, p ∈ N,∣∣∣∣∣p−n+n0Ip(x0) −k∑r=0Φr(x0)p−r∣∣∣∣∣C m′ Ck,m′p−k−1. (13)Theorem 2.4. If (X,ω) is a Kähler manifold andL,E are holomorphic vector bundles with holomorphic Hemitian connections∇L,∇E , J = J on U , and G acts freely onµ−1(0), then in(13), Φr(x0) ∈ End(EG)x0 arepolynomials inA, RT B , REB , µE , RE (resp.h, RLB ; resp.µ) and their derivatives atx0 to order2r − 1 (resp.2r ;resp.2r + 1), andΦ0 = IdEG . MoreoverΦ1(x0) = 18πrXGx0 +34πXG log(h|XG) +√−14πREGx0(e0j , JGe0j). (14)Here rXG is the Riemannian scalar curvature of(T XG,gT XG), XG is the Bochner–Laplacian onXG, and{e0j }is an orthonormal basis ofT XG.Still making the same assumptions as in Theorem 2.4, leth̃ d note the restriction toXG of the functionh definedin (3). In view of [9, (3.11), (3.54)], set̃hEG = h̃2hEG .Let P̃ XGp denote the orthogonal projection fromC ∞(XG,LpG ⊗ EG) ontoH 0(X,LpG ⊗ EG) associated to thmetricshLG , h̃EG , gT XG . Let P̃ XGp (x0, x′0) (x0, x′0 ∈ XG) denote the corresponding Bergman kernel with resto dvXG(x′0).Then by [2, Theorem 1.3], we have the following theorem.Theorem 2.5. Under the assumption of Theorem2.4, there exist smooth coefficients̃Φr(x0) ∈ End(EG)x0 whichare polynomials inRT XG , REG (resp.h), and their derivatives atx0 to order2r − 1 (resp.2r), andΦ̃0 = IdEG ,such that for anyk, l ∈ N, there existsCk,l > 0 such that for anyx0 ∈ XG, p ∈ N,∣∣∣∣∣p−n+n0P̃ XGp (x0, x0) −k∑r=0Φ̃r (x0)p−r∣∣∣∣∣C l Ck,lp−k−1. (15)Moreover, the following identity holds,Φ̃1(x0) = 18πrXGx0 +12πXG logh̃ +√−14πREGx0(e0j , JGe0j). (16)302 X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 297–302n ofWewrittenfessor. (1992).he full38.498. The8) 229–Remark 3. From (14) and (16), one sees that in generalΦ1 = Φ̃1, if h̃ is not constant onXG. This reflects a subtledifference 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 deformatioD2pby the Casimir operator (i.e., to considerD2p − pCas, which plays a key role in proving Theorems 2.1, 2.2).refer to [6] for more details.AcknowledgementsThe work of the second author was partially supported by MOEC and MOSTC. Part of this Note waswhile the second author was visiting IHES during February and March, 2005. He would like to thank ProJean-Pierre Bourguignon for the hospitality.References[1] J.-M. Bismut, G. Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math. (1991), no. 74, ii+298 pp[2] X. Dai, K. Liu, X. Ma, On the asymptotic expansion of Bergman kernel, C. R. Math. Acad. Sci. Paris 339 (3) (2004) 193–198. Tversion: J. Differential Geom., preprint, math.DG/0404494.[3] V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (3) (1982) 515–5[4] X. Ma, G. Marinescu, The Spinc Dirac operator on high tensor powers of a line bundle, Math. Z. 240 (3) (2002) 651–664.[5] X. Ma, G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C.R. Math. Acad. Sci. Paris 339 (7) (2004) 493–full version: math.DG/0411559.[6] X. Ma, W. Zhang, Bergman kernels and symplectic reduction, preprint.[7] R. Paoletti, Moment maps and equivariant Szegö kernels, J. Symplectic Geom. 2 (2003) 133–175.[8] R. Paoletti, The Szegö kernel of a symplectic quotient, Adv. Math. (2005), math.SG/0404549.[9] Y. Tian, W. Zhang, An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg, Invent. Math. 132 (2) (199259.

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Bergman kernels and symplectic reduction

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