We establish an asymptotic expansion for families of Bergman kernels. The key
idea is to use the superconnection as in the local family index theorem.Comment: C. R. Math. Acad. Sci. Pari

Ma, Xiaonan

Zhang, Weiping

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arXiv

C. R. Math. Acad. Sci. ParisWe establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection as in the local family index theorem

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Superconnection and family Bergman kernelsXiaonan Ma, Weiping ZhangTo cite this version:Xiaonan Ma, Weiping Zhang. Superconnection and family Bergman kernels. C. R. Math.Acad. Sci. Paris. 2006. <hal-00085858>HAL Id: hal-00085858https://hal.archives-ouvertes.fr/hal-00085858Submitted on 16 Jul 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00085858, version 1 - 16 Jul 2006SUPERCONNECTION AND FAMILY BERGMAN KERNELSXIAONAN MA AND WEIPING ZHANGAbstract. We establish an asymptotic expansion for families of Bergman kernels.The key idea is to use the superconnection as in the local family index theorem.Superconnexion et noyaux de Bergman en familleRe´sume´. Nous annonc¸ons des re´sultats sur le de´veloppement asymptotique du noyaude Bergman en famille.Let W,S be smooth compact complex manifolds. Let pi : W → S be a holomorphicsubmersion with compact fiber X and dimC X = n.We will add a subscript R for the corresponding real objects. Thus TX is the holo-morphic relative tangent bundle of pi, and TRX is the corresponding real vector bundle.Let JTRX be the complex structure on TRX.Let E be a holomorphic vector bundle on W . Let L be a holomorphic line bundle onW . Let hL, hE be Hermitian metrics on L,E. Let ∇L,∇E be the holomorphic Hermitianconnections on (L, hL), (E, hE) with their curvatures RL, RE respectively. Setω =√−12piRL.(0.1)Then ω is a smooth real 2-form of complex type (1, 1) on W .We suppose that ω defines a fiberwise Ka¨hler form along the fiber X, i.e.gTRX(u, v) = ω(u, JTRXv)(0.2)defines a (fiberwise) Riemannian metric on TRX. We denote by hTX the correspondingHermitian metric on TX.Let dvX be the Riemannian volume form on (X, gTRX).By the Kodaira vanishing theorem, there exists p0 ∈ N such that H0(X, (Lp ⊗ E)|X)forms a vector bundle, denoted by H0(X,Lp ⊗ E), on S for p > p0. From now on, wealways assume p > p0.By the Grothendieck-Riemann-Roch Theorem, for p > p0, we havech(H0(X,Lp ⊗E)) =∫XTd(TX) ch(E) ch(Lp) in H•(S,R).(0.3)Date: July 18, 2006.12 XIAONAN MA AND WEIPING ZHANGThe component in H0(S,R) of (0.3) is the Hirzebruch-Riemann-Roch Theorem, andas p→ +∞,(0.4) dimH0(X,Lp ⊗ E) = rk(E)∫Xc1(L)nn!pn+∫X(c1(E) +rk(E)2c1(TX))c1(L)n−1(n− 1)! pn−1 + O(pn−2).For s ∈ S, let Pp,s be the orthogonal projection from C∞(Xs, (Lp ⊗ E)Xs) ontoH0(Xs, (Lp ⊗ E)|Xs). Let Pp,s(x, x′) (x, x′ ∈ Xs, s ∈ S) be the smooth kernel of Pp,swith respect to dvXs(x′). Then Pp,s(x, x′) is smooth on s ∈ S, and we denote it simplyby Pp(x, x′), especially, Pp(x, x) ∈ End(E)x.The results of [14], [16], [4] tell us that there exist br ∈ C∞(Xs,End(E|Xs)) such thatfor any k, l ∈ N, there exists C > 0 such that∣∣∣ 1pnPp,s(x, x)−k∑r=0br(x)p−r∣∣∣C l(Xs)≤ C p−k−1.(0.5)In [7], [15], Lu and Wang show that the first two coefficients b0, b1 coincide with thecorresponding terms in the local Hirzebruch-Riemann-Roch Theorem, i.e. the leadingterms in the Chern-Weil representative of Td(TX) ch(E) ch(Lp) with respect to the met-rics hTX , hL, hE. We refer to [5], [9], [10] for alternate approaches as well as extensionsto the symplectic case.By (0.3), in H2(S,R), as p→∞, we have(0.6) c1(H0(X,Lp ⊗E)) = rk(E)∫Xc1(L)n+1(n+ 1)!pn+1+∫X(c1(E) +rk(E)2c1(TX))c1(L)nn!pn + O(pn−1).Now, in view of the Bismut local family index theorem [2], it is nature to ask whetherthe analogue of (0.5) still holds on higher degree levels , which will involve the curvatureof the vector bundle H0(X,Lp ⊗ E) as in (0.6).In this note, we announce some results on the existence of such an expansion, andcompute the first two coefficients in the expansion.To define a canonical connection on H0(X,Lp⊗E), we need to introduce a horizontalsub-bundle THW of TW .Let THW be a sub-bundle of TW such thatTW = THW ⊕ TX.(0.7)Let P TX be the projection from TW onto TX. For U ∈ TS, let UH ∈ THW be the liftof U .Clearly, (0.7) induces canonically a decomposition Λ(T ∗RW ) = pi∗(Λ(T ∗RS))⊗̂Λ(T ∗RX).For a differential formA onW , we will denote by A(i) its component in Λi(T ∗RS)⊗̂Λ(T ∗RX).Then dvX = (ωn)(0)/n!.SUPERCONNECTION AND FAMILY BERGMAN KERNELS 3Let T ∈ Λ2(T ∗RW )⊗ TRX be the tensor defined in the following way: for U, V ∈ TS,X, Y ∈ TX,T (UH , V H) := −P TX [UH , V H ], T (X, Y ) := 0,T (UH , X) :=12(gTX)−1(LUHgTX)X.(0.8)Let ∇Lp⊗E be the connection on Lp ⊗ E induced by ∇L,∇E . For U ∈ TRS, σ ∈C∞(S,H0(X,Lp ⊗E)), we define∇H0(X,Lp⊗E)U σ = Pp∇Lp⊗EUHPpσ.(0.9)Then∇H0(X,Lp⊗E) is a holomorphic connection onH0(X,Lp⊗E) with curvature RH0(X,Lp⊗E),but it need not be a Hermitian connection with respect to the (usual) induced L2 metrichH0(X,Lp⊗E) on H0(X,Lp ⊗ E).Let k ∈ T ∗RW be such that for U ∈ TRS, X ∈ TRX,k(UH) =12(LUHdvX)/dvX , k(X) = 0.(0.10)Then∇Ker(Dp)U = Pp(∇Lp⊗EUH+ k(UH))Pp,(0.11)is a canonical Hermitian connection on (H0(X,Lp ⊗ E), hH0(X,Lp⊗E)) with curvatureRKer(Dp), but it need not be holomorphic.Let RH0(X,Lp⊗E)(x, x′), RKer(Dp)(x, x′) (x, x′ ∈ Xs, s ∈ S) be the smooth kernel of theoperator RH0(X,Lp⊗E), RKer(Dp) with respect to dvX(x′). ThenRH0(X,Lp⊗E)(x, x), RKer(Dp)(x, x) ∈ Λ2(T ∗S)⊗ End(Ex).(0.12)IfTHW = {u ∈ TW ;ω(u,X) = 0 for any X ∈ TX},(0.13)then the triple (pi, gTRX , THW ) defines a Ka¨hler fibration in the sense of [3, Definition1.4]. In this case, the connection ∇Ker(Dp) is the canonical holomorphic connection on(H0(X,Lp ⊗E), hH0(X,Lp⊗E)), andk = 0, ∇Ker(Dp) = ∇H0(X,Lp⊗E).(0.14)Let {wi} be an orthonormal frame of (TX, hTX). Let {ei} be an orthonormal frameof (TRX, gTX). Let {gα} be an frame of TS and {gα} its dual frame.Theorem 0.1. There exist smooth sections b2,r(x) ∈ C∞(W,Λ2(T ∗S)⊗End(Ex)) whichare polynomials in RTX , T , RE (and RL), their derivatives of order 6 2r− 1 (resp. 2r)along the fiber X, with(0.15) b2,0 = −2pi√−1(ωn+1)(2)(ωn)(0)IdE ,4 XIAONAN MA AND WEIPING ZHANGsuch that for any k, l ∈ N, there exists Ck, l > 0 such that for any x ∈W , p ∈ N, p > p0,∣∣∣ 1pn+1RH0(X,Lp⊗E)(x, x)−k∑r=0b2,r(x)p−r∣∣∣C l(W )6 Ck, l p−k−1.(0.16)For RKer(Dp)(x, x), we have the similar expansion as (0.16), with the same leading termb2,0 in (0.15), and the corresponding b2,r(x) depends also the derivative of dk of order6 2r − 1 along the fiber X.If (0.13) is verified, thenb2,1 =((12〈RTXwi, wi〉+RE +√−14gα ∧ gβ∆X(ω(gHα , gHβ )))ωn)(2)/(ωn)(0),(0.17)here ∆X = −∑i[(∇ei)2 −∇∇TXei ei] is the Bochner Laplacian along the fiber X.If we take the trace on E and integrate along X in (0.16), from (0.15) and (0.17),we get a refinement of (0.6) on the level of differential forms, in the spirit of the LocalFamily Index Theorem.From (0.15), we get(0.18) b2,0 = 2pigα ∧ gβ [−√−1ω(gHα , gHβ )− ω(gHα , wj)ω(gHβ , wj)] IdE .By (0.16), (0.18), the curvatures RH0(X,Lp⊗E)(x, x), RKer(Dp)(x, x) provide a naturalapproximation of the Monge-Ampe`re operator on the space of Ka¨hler metrics. It musthave relations with the existence problem of geodesics on the space of Ka¨hler metrics(cf. [6], [8], [13], [12]).The equation (0.18) gives also an exact local asymptotic behavior of the curvatureestimates in [1, §6].To prove Theorem 0.1, we will use the superconnection formation as in the local indextheory. This is the main idea of our work. An important feature of superconnectionis that its curvature is a second order differential operator along the fiber X, while thesuperconnection itself involves derivatives along the horizontal direction. This is alsoone of the points in the local index theory. Now, by combining the formal power seriestrick in [9], we get in fact a general and algorithmic way to compute the coefficients inthe expansion. More details will appear in [11].Remark. In this note, we have only formulated our results in the fiberwise positiveholomorphic line bundle case. Actually, the results hold also in the fiberwise symplecticcase, and we have the off-diagonal expansion results too.Acknowledgements. The work of the second author was partially supported byMOEC and NNSFC. Part of work was done while the first author was visiting Centrede Recerca Matema`tica (CRM) in Barcelona during June and July, 2006. He would liketo thank CRM for hospitality.SUPERCONNECTION AND FAMILY BERGMAN KERNELS 5References[1] Berndtsson B., Curvature of vector bundles associated to holomorphic fibrations, math.CV/0511225(2005).[2] J.-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equationproofs, Invent. Math. 83 (1986), no. 1, 91–151.[3] J.-M. Bismut, H. Gillet, and C. Soule´, Analytic torsion and holomorphic determinant bundles. II.Direct images and Bott-Chern forms, Comm. Math. Phys. 115 (1988), no. 1, 79–126.[4] D. Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complexvariables (Katata, 1997), Trends Math., Birkha¨user Boston, Boston, MA, 1999, pp. 1–23.[5] X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom.72 (2006), no. 1, 1–41; announced in C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193–198.[6] S. K. Donaldson, Symmetric spaces, Ka¨hler geometry and Hamiltonian dynamics, Northern Cali-fornia Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc.,Providence, RI, 1999, pp. 13–33.[7] Z. Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math.122 (2000), no. 2, 235–273.[8] T. Mabuchi, Some symplectic geometry on compact Ka¨hler manifolds. I, Osaka J. Math. 24 (1987),no. 2, 227–252.[9] X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad.Sci. Paris 339 (2004), no. 7, 493–498. The full version: math.DG/0411559.[10] , Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics,Birkha¨user Boston Inc., Boston, MA, 2007.[11] X. Ma and W. Zhang, Superconnection and family Bergman kernels, to appear.[12] D. Phong and J. Sturm, The Monge-Ampe`re operator and geodesics in the space of Ka¨hler potentials,math.DG/0504157 (2005).[13] S. Semmes, Complex Monge-Ampe`re and symplectic manifolds, Amer. J. Math. 114 (1992), no. 3,495–550.[14] G. Tian, On a set of polarized Ka¨hler metrics on algebraic manifolds, J. Differential Geom. 32(1990), no. 1, 99–130.[15] X. Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13 (2005), no. 2, 253–285.[16] S. Zelditch, Szego˝ kernels and a theorem of Tian, Internat. Math. Res. Notices (1998), no. 6, 317–331.Centre de Mathe´matiques Laurent Schwartz, UMR 7640 du CNRS, Ecole Polytech-nique, 91128 Palaiseau Cedex, France (ma@math.polytechnique.fr)Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R.China. (weiping@nankai.edu.cn)

Superconnection and family Bergman kernels

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 344 (2007) 41–44http://france.elsevier.com/direct/CRASS1/Differential GeometrySuperconnection and family Bergman kernelsXiaonan Ma a, Weiping Zhang ba Centre de mathématiques Laurent-Schwartz, UMR 7640 du CNRS, École polytechnique, 91128 Palaiseau cedex, Franceb Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. ChinaReceived 6 November 2006; accepted 8 November 2006Available online 11 December 2006Presented by Jean-Michel BismutAbstractWe establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection formalism asin the local family index theorem. To cite this article: X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 344 (2007).© 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméSuperconnexion et noyaux de Bergman en famille. Nous annonçons des résultats sur le développement asymptotique du noyaude Bergman en famille. L’idée principale est d’utiliser le formalisme des superconnexions comme dans la preuve du théorème del’indice local en famille. Pour citer cet article : X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 344 (2007).© 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeSoit π :W → S une submersion holomorphe de variétés compactes de fibre X, et soit L un fibré en droites ho-lomorphes sur W qui est positif le long de la fibre X. Soit E un fibré vectoriel holomorphe sur W . Pour p assezgrand, la première classe de Chern du fibré vectoriel holomorphe H 0(X,Lp ⊗ E) est calculé par le théorème deGrothendieck–Riemann–Roch (G.R.R.).En considérant la courbure de H 0(X,Lp ⊗E) comme un opérateur agissant le long de la fibre, nous étudions danscette Note le développement asymptotique de son noyau quand la puissance p tend vers +∞. Nos résultats raffinent ledéveloppement asymptotique de la première classe de Chern donnée par le théorème de G.R.R. au niveau des formesdifférentielles comme dans la version locale du théorème de l’indice en famille. L’idée principale est d’utiliser « unmorceau » de la superconnexion introduite par Bismut dans la preuve du théorème de l’indice local en famille.Les résultats annoncés dans cette Note sont démontrés dans [12].1. IntroductionLet W,S be smooth compact complex manifolds. Let π :W → S be a holomorphic submersion with compact fiberX and dimC X = n. Let E be a holomorphic vector bundle on W . Let L be a holomorphic line bundle on W .E-mail addresses: ma@math.polytechnique.fr (X. Ma), weiping@nankai.edu.cn (W. Zhang).1631-073X/$ – see front matter © 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2006.11.01342 X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 41–44We suppose that L is positive along the fiber X.We will add a subscript R for the corresponding real objects. Thus T X is the holomorphic relative tangent bundleof π , and TRX is the corresponding real vector bundle. Let J TRX be the complex structure on TRX.By the Kodaira vanishing theorem, there exists p0 ∈ N such that the higher fiberwise cohomologies vanish and thatH 0(X, (Lp ⊗ E)|X) forms a vector bundle, denoted by H 0(X,Lp ⊗ E), on S for p > p0. From now on, we alwaysassume p > p0.By the Grothendieck–Riemann–Roch Theorem, in H 2(S,R), as p → +∞, we havec1(H 0(X,Lp ⊗ E)) = rk(E)∫Xc1(L)n+1(n + 1)! pn+1 +∫X(c1(E) + rk(E)2c1(T X))c1(L)nn! pn + O(pn−1). (1)Now, in view of the Bismut local family index theorem [2], it is natural to ask whether a local version of (1) stillholds which involves the curvature of the vector bundle H 0(X,Lp ⊗ E).Let us introduce our geometric data now. Let hL be a Hermitian metric on L such that the restriction of√−1RLalong the fiber X is a positive (1,1)-form, here RL is the curvature of the holomorphic Hermitian connection ∇L on(L,hL). Let ω := c1(L,hL) be the Chern–Weil representative of the first Chern class c1(L) of (L,hL), thenω = c1(L,hL) = √−12πRL. (2)Thus ω is a smooth real 2-form of complex type (1,1) on W . Moreover, ω defines a Kähler form along the fiber X,i.e.gTRX(u, v) = ω(u,J TRXv) (3)defines a Riemannian metric on TRX. We denote by hT X the corresponding Hermitian metric on T X.Let dvX be the Riemannian volume form on (X,gTRX).Let hE be a Hermitian metric on E. Let ∇E be the holomorphic Hermitian connection on (E,hE) with its curva-ture RE .Let hH0(X,Lp⊗E) be the L2-metric on H 0(X,Lp ⊗ E) induced by hL,hE and gTRX . Let ∇H 0(X,Lp⊗E) be theholomorphic Hermitian connection on (H 0(X,Lp ⊗ E),hH 0(X,Lp⊗E)). Let RH 0(X,Lp⊗E) = (∇H 0(X,Lp⊗E))2 be thecurvature of ∇H 0(X,Lp⊗E). ThenRH0(X,Lp⊗E) ∈ Λ2(T ∗RS) ⊗ End(H 0(X,Lp ⊗ E)).For s ∈ S, let Pp,s be the orthogonal projection from C ∞(Xs, (Lp ⊗ E)|Xs ) onto H 0(Xs, (Lp ⊗ E)|Xs ). In thesequel, we write instead Pp .We will identify RH0(X,Lp⊗E) toPpRH 0(X,Lp⊗E)Pp ∈ Λ2(T ∗RS) ⊗ End(C ∞(X, (Lp ⊗ E)∣∣X)).Let RH0(X,Lp⊗E)(x, x′) (x, x′ ∈ Xs, s ∈ S) be the smooth kernel of the operator RH 0(X,Lp⊗E) with respect todvXs (x′). ThenRH0(X,Lp⊗E)(x, x) ∈ π∗(Λ2(T ∗RS)) ⊗ End(Ex). (4)The purpose of this Note is to evaluate the asymptotics as p → ∞ of the kernel of RH 0(X,Lp⊗E).2. Main resultLet T H W be the orthogonal bundle to T X with respect to ω. Then T H W is a sub-bundle of T W such thatT W = T H W ⊕ T X. (5)Let P T X be the projection from T W = T H W ⊕ T X onto T X. For U ∈ TRS, let UH ∈ T HR W be the horizontal liftof U .X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 41–44 43Let T ∈ Λ2(T ∗RW) ⊗ TRX be the tensor defined in the following way: for U,V ∈ TRS, X,Y ∈ TRX,T(UH ,V H) := −P T X[UH ,V H ], T (X,Y ) := 0,T(UH ,X) := 12(gTRX)−1(LUH gTRX)X. (6)Let RT X be the curvature of the holomorphic Hermitian connection ∇T X on (T X,hT X). Then the Chern–Weilrepresentative of the first Chern class of (T X,hT X) is c1(T X,hT X) =√−12π Tr[RT X].Let {gα} be a frame of T S and {gα} its dual frame.Clearly, (5) induces canonically a decomposition Λ(T ∗RW) = π∗(Λ(T ∗RS))⊗̂Λ(T ∗RX). We will denote by A(i) thecomponent in π∗(Λi(T ∗RS))⊗̂Λ(T ∗RX), of a differential form A on W . Then dvX = (ωn)(0)/n!.Theorem 2.1. There exist smooth sections b2,r (x) ∈ C ∞(W,Λ2(T ∗RS) ⊗ End(Ex)) which are polynomials in RT X ,T , RE (and RL), and their derivatives of order  2r − 1 (resp. 2r) along the fiber X such that for any k, l ∈ N, thereexists Ck,l > 0 such that for any p ∈ N, p > p0,∣∣∣∣∣RH 0(X,Lp⊗E)(x, x) −k∑r=0b2,r (x)pn−r+1∣∣∣∣∣C l (W) Ck,lpn−k, (7)with √−12πb2,0 = (ωn+1)(2)(n + 1)(ωn)(0) IdE = gα ∧ ḡβω(gHα , ḡHβ ) IdE,√−12πb2,1 =((12c1(T X,hT X) + √−12πRE − 18πgα ∧ ḡβX(ω(gHα , ḡHβ)))ωn)(2)/(ωn)(0),(8)where X is the (positive) Laplace operator of the fiber Xs .If we take the trace of this asymptotic (7) on E and integrate along X, we get a refinement of (1) on the level ofdifferential forms, in the spirit of the local family index theorem.3. Idea of the proofProof. By using the full off-diagonal asymptotic expansion of the Bergman kernel [6] with the parameter s ∈ S, it isnot hard to prove the existence of an expansion with leading term pn+2, but further work is needed to get the vanishingof the first coefficient, and it is difficult to compute the other coefficients this way.Our main idea here is to use the superconnection formalism to prove Theorem 2.1. This gives us a conceptually clearway to get our result: an important feature of our superconnection is that its curvature is a second order differentialoperator along the fiber X, while the superconnection itself involves derivatives along the horizontal direction. Justas in the Bismut local family index theorem [2], this property of our superconnection plays an important role in ourproof.We now explain briefly the superconnection formalism.Let ∂̄Lp⊗E,∗ be the formal adjoint of the fiberwise Dolbeault operator ∂̄Lp⊗E on the Dolbeault complexΩ0,•(X,Lp ⊗ E). SetDp =√2(∂̄Lp⊗E + ∂̄Lp⊗E,∗). (9)Let ∇Ep be the connection on Ep := Λ(T ∗(0,1)X) ⊗ Lp ⊗ E induced by the holomorphic Hermitian connections∇T X , ∇L, ∇E on T X, L, E respectively.For U ∈ TRS, if σ is a smooth section of Ω0,•(X,Lp ⊗ E) over S, i.e. σ ∈ C ∞(W,Ep), set∇ΩU σ = ∇EpUH σ. (10)Then ∇Ω is a Hermitian connection on Ω0,•(X,Lp ⊗ E) over S. Let Bp be the superconnection onΛ(T ∗S) ⊗̂ Ω0,•(X,Lp ⊗ E) defined byR44 X. Ma, W. Zhang / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 41–44Bp = Dp + ∇Ω. (11)We now describe the explicit geometric construction of ∇H 0(X,Lp⊗E) given in [4, Theorem 3.4] (cf. [3, Theo-rem 3.11]). Let ∇Lp⊗E be the connection on Lp ⊗E induced by ∇L,∇E . For U ∈ TRS, σ ∈ C ∞(S,H 0(X,Lp ⊗E)),then∇H 0(X,Lp⊗E)U σ = Pp∇Lp⊗EUHσ, (12)where σ is considered as a section of Lp ⊗ E on W .From (11), (12) and the spectral gap property of D2p (cf. [5,9]), for p large enough, we haveRH0(X,Lp⊗E) = 12π√−1[ ∫|λ|=2πp(λ − B2p)−1λdλ](2). (13)Now, by using the formal power series trick developed in [10], we get a general and algorithmic way to computethe coefficients in the expansion. More details will appear in [12]. Remark 3.1. In this Note, we have only formulated our results in the case of holomorphic line bundles which arefiberwise positive. Actually, the results hold also for symplectic line bundles. In [12], we also prove the existence ofan off-diagonal asymptotic expansion which implies, for example, that RH0(X,Lp⊗E) is a Toeplitz operator with valuesin Λ2(T ∗RS) in the sense of [11, Chapter 7].By (7), (8), the curvatures RH0(X,Lp⊗E)(x, x) provide a natural approximation of the Monge–Ampère operator onthe space of Kähler metrics. It should have relations with the existence problem of geodesics on the space of Kählermetrics (cf. [7,8,14,13]).From Eq. (8), for large p, we can obtain more precise positivity estimates for H 0(X,Lp ⊗ E) than in [1, §6].AcknowledgementsThe work of the second author was partially supported by NNSFC. Part of the work was done while the first authorwas visiting Centre de Recerca Matemàtica (CRM) in Barcelona during June and July, 2006. He would like to thankCRM for hospitality.References[1] B. Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, math.CV/0511225, 2005.[2] J.-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1) (1986) 91–151.[3] J.-M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants,Comm. Math. Phys. 115 (2) (1988) 301–351.[4] J.-M. Bismut, K. Köhler, Higher analytic torsion forms for direct images and anomaly formulas, J. Algebraic Geom. 1 (4) (1992) 647–684.[5] J.-M. Bismut, É. Vasserot, The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle, Comm.Math. Phys. 125 (2) (1989) 355–367.[6] X. Dai, K. Liu, X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (1) (2006) 1–41; announced in C. R. Math.Acad. Sci. Paris 339 (3) (2004) 193–198.[7] S.K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, in: Northern California Symplectic Geometry Seminar, in:Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 13–33.[8] T. Mabuchi, Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (2) (1987) 227–252.[9] X. Ma, G. Marinescu, The Spinc Dirac operator on high tensor powers of a line bundle, Math. Z. 240 (3) (2002) 651–664.[10] X. Ma, G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris 339 (7) (2004) 493–498. The fullversion: math.DG/0411559.[11] X. Ma, G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, vol. 254, Birkhäuser Boston, Boston,MA, 2007.[12] X. Ma, W. Zhang, Superconnection and family Bergman kernels, in press.[13] D. Phong, J. Sturm, The Monge–Ampère operator and geodesics in the space of Kähler potentials, math.DG/0504157, 2005.[14] S. Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (3) (1992) 495–550.

Xiaonan Ma

Weiping Zhang

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Superconnection and family Bergman kernels

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