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Generalized Bergman kernels on symplectic manifolds. (English) Zbl 1141.58018

In this paper, the authors study the asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner Laplacian introduced by V. Guillemin and A. Uribe [Asymptotic Anal. 1, 105–113 (1988; Zbl 0649.53026)]. Consider a compact symplectic manifold \((X,\omega)\), with \((L,h_L)\) and \((E,h_E)\) two Hermitian bundles on \(X\) endowed with Hermitian connexions \(\nabla_L\) and \(\nabla_E\). Assume that \(L\) is a line bundle and a polarisation for \((M,\omega)\) in the sense that the curvature of \(\nabla_L\) is precisely \(\omega\). One can define a skew-adjoint linear map \(\mathbf{J}:TX \rightarrow TX\) by \(\omega(u,v)=g(\mathbf{J}u,v)\) where \(g\) is the Riemannian metric on \(X\). There is also an almost complex structure \(\mathcal{J}\) satisfying \(g(\mathcal{J}u,\mathcal{J}v)=g(u,v)\), \(\omega(\mathcal{J}u,\mathcal{J}v)=\omega(u,v)\).
One can define now \(\tau=-\pi \mathrm{tr}(\mathcal{J}\mathbf{J})=-\pi\mathrm{tr}(\mathbf{J}^2(-\mathbf{J}^2)^{-1/2})>0\) (which is just \(\pi\dim(X)\) in the Kähler case). Choose also \(\Phi\) a Hermitian section of \(\text{End}(E)\). The Bochner Laplacian \(\Delta\) acting on \(C^{\infty}(E\otimes L^k)\) induces the renormalized Bochner Laplacian \[ \Delta_{k,\Phi}=\Delta -k\tau +\Phi \] which is an elliptic operator. It has a spectral gap when \(k\) tends to infinity, i.e., \[ Sp(\Delta_{k,\Phi})\subset [-C,C]\cup [k\mu - C, \infty[. \] One can form a generalized Bergman kernel by considering the eigenvalues of \((\lambda_{i,k})_{i\geq 1}\) of \(\Delta_{k,\Phi}\). With an orthonormal basis \(S_{i,k}\) of the direct sum of eigenspaces corresponding to the eigenvalues \(\lambda\) with \(\lambda\in [-C,C]\), one defines the Bergman function as \[ B_{q,k}(x)= \sum_i \lambda^q_{i,k}S_{i,k}(x)\otimes \left(S_{i,k}(x)\right)^{*} \] with \(x \in X\) and \(q\in \mathbb{N}\). Here the sections \(S_{i,k}\) are orthonormal with respect to the \(L2\)-inner structure induced from the metric on \(E \otimes L^k\) and the volume form induced by \(g\).
The main result of the paper is the existence of an asymptotic for \(B_{q,k}\) when \(k\) tends to infinity and the computation of the first terms of this asymptotic. Roughly speaking,
\[ B_{q,k}(x)= k^nb_{q,0}(x)+b_{q,1}k^{n-1}(x)+\cdots \]
and \(b_{0,0}=\det(\mathbf{J})^{1/2}Id_{E}\). Furthermore, if \(\mathcal{J}=\mathbf{J}\), highly technical computations lead to \[ b_{0,1}=\frac{1}{8\pi}\left(\text{scal}(g) + \frac{1}{4}| \nabla_X J| ^2 + 2R_E\right) \] (where \(R_E\) is the contracted curvature of \(E\)), and \[ b_{q,0}=\left(\frac{1}{24}| \nabla_X J| ^2 + \frac{1}{2}R_E + \Phi\right)^q. \]
The proof is based on the spectral gap result and a localization technique inspired from the work of Bismut and Lebeau. We refer to the recent book of the authors [Holomorphic Morse inequalities and Bergman kernels. Basel: Birkhäuser (2007; Zbl 1135.32001)] where a similar strategy has been presented.
The last section of the paper is dedicated to applications. One can in particular find a symplectic version of the Kodaira embedding theorem by applying the previous results with \(E\) trivial. The Kodaira map \(\psi_k\) is given by considering the sections with eigenvalues in \([-C,C]\). Note that \(\psi_k\) is asymptotically symplectic and isometric. Moreover, \(\psi_k\) is nearly holomorphic since \(\frac{1}{k}| | \partial \psi_k | | \geq c>0\) and \(\frac{1}{k}| | \bar\partial\psi_k| | =O(1/k)\).
Finally, the authors give some partial results on non compact manifolds or for singular polarisations with strictly positive curvature. On complete Hermitian manifolds with certain bounds on the curvatures, the asymptotic still holds. On the other hand, using Demailly’s approximation procedure and by considering \(L^2\) sections (with respect to a natural induced Poincaré metric), one can obtain again the asymptotic for a singular polarisation after resolutions of singularities. This extends the previous work of the authors in that context.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53D50 Geometric quantization
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53D05 Symplectic manifolds (general theory)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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References:

[1] Berline, N.; Getzler, E.; Vergne, M., Heat Kernels and Dirac Operators (1992), Springer-Verlag · Zbl 0744.58001
[2] Bismut, J.-M., Demailly’s asymptotic inequalities: A heat equation proof, J. Funct. Anal., 72, 263-278 (1987) · Zbl 0649.58030
[3] Bismut, J.-M., A local index theorem for non-Kähler manifolds, Math. Ann., 284, 4, 681-699 (1989) · Zbl 0666.58042
[4] Bismut, J.-M.; Lebeau, G., Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math., 74, ii+298 (1991), (1992)
[5] Bismut, J.-M.; Vasserot, E., The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys., 125, 355-367 (1989) · Zbl 0687.32023
[6] Bleher, P.; Shiffman, B.; Zelditch, S., Universality and scaling of correlations between zeros on complex manifolds, Invent. Math., 142, 2, 351-395 (2000) · Zbl 0964.60096
[7] Bleher, P.; Shiffman, B.; Zelditch, S., Universality and scaling of zeros on symplectic manifolds, (Random Matrix Models and Their Applications. Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., vol. 40 (2001), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 31-69 · Zbl 1129.58303
[8] Bonavero, L., Inégalités de Morse holomorphes singulières, J. Geom. Anal., 8, 409-425 (1998) · Zbl 0966.32011
[9] Borthwick, D.; Uribe, A., Almost complex structures and geometric quantization, Math. Res. Lett.. Math. Res. Lett., Math. Res. Lett., 5, 211-212 (1998), Erratum:
[10] Borthwick, D.; Uribe, A., Nearly Kählerian embeddings of symplectic manifolds, Asian J. Math., 4, 3, 599-620 (2000) · Zbl 0990.53086
[11] Borthwick, D.; Uribe, A., The spectral density function for the Laplacian on high tensor powers of a line bundle, Ann. Global Anal. Geom., 21, 269-286 (2002) · Zbl 1004.53065
[12] Bouche, T., Convergence de la métrique de Fubini-Study d’un fibré linéare positif, Ann. Inst. Fourier (Grenoble), 40, 117-130 (1990) · Zbl 0685.32015
[13] Boutet de Monvel, L.; Guillemin, V., The spectral theory of Toeplitz operators, Ann. of Math. Stud., vol. 99 (1981), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0469.47021
[14] Boutet de Monvel, L.; Sjőstrand, J., Sur la singularité des noyaux de Bergman et de Szegő, Journées: Équations aux Dérivées Partielles de Rennes (1975). Journées: Équations aux Dérivées Partielles de Rennes (1975), Astérisque, 34-35, 123-164 (1976) · Zbl 0344.32010
[15] Braverman, M., Vanishing Theorems on Covering Manifolds, Contemp. Math., vol. 231 (1999), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, pp. 1-23 · Zbl 0940.32011
[16] Carlson, J.; Griffiths, P., A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. (2), 95, 557-584 (1972) · Zbl 0248.32018
[17] Catlin, D., The Bergman kernel and a theorem of Tian, (Analysis and Geometry in Several Complex Variables. Analysis and Geometry in Several Complex Variables, Katata, 1997. Analysis and Geometry in Several Complex Variables. Analysis and Geometry in Several Complex Variables, Katata, 1997, Trends Math. (1999), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 1-23 · Zbl 0941.32002
[18] Cornalba, M.; Griffiths, P., Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math., 28, 1-106 (1975) · Zbl 0293.32026
[19] Chazarain, J.; Piriou, A., Introduction à la théorie des équations aux dérivées partielles linéaires (1981), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0446.35001
[20] Dai, X.; Liu, K.; Ma, X., On the asymptotic expansion of Bergman kernel, J. Differential Geom.. J. Differential Geom., C. R. Math. Acad. Sci. Paris, 339, 3, 193-198 (2004), announced in · Zbl 1057.58018
[21] Demailly, J.-P., Estimations \(L^2\) pour l’opérateur \(\overline{\partial}\) d’un fibré holomorphe semipositif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup., 15, 457-511 (1982) · Zbl 0507.32021
[22] Demailly, J.-P., Champs magnétiques et inegalités de Morse pour la \(d''\)-cohomologie, Ann. Inst. Fourier (Grenoble), 35, 189-229 (1985) · Zbl 0565.58017
[23] Demailly, J.-P., Sur l’identité de Bochner-Kodaira-Nakano en géométrie hermitienne, Lecture Notes in Math., vol. 1198 (1985), Springer-Verlag, pp. 88-97
[24] Demailly, J.-P., Regularization of closed positive currents and Intersection Theory, J. Algebraic Geom., 1, 361-409 (1992) · Zbl 0777.32016
[25] Donaldson, S. K., Remarks on gauge theory, complex geometry and 4-manifold topology, (Fields Medallists’ Lectures. Fields Medallists’ Lectures, World Sci. Ser. 20th Century Math., vol. 5 (1997), World Sci. Publishing: World Sci. Publishing River Edge, NJ), 384-403
[26] Donaldson, S. K., Scalar curvature and projective embeddings. I, J. Differential Geom., 59, 479-522 (2001) · Zbl 1052.32017
[27] P. Gauduchon, Calabi’s extremal Kähler metrics: an elementary introduction, book in preparation, 2005; P. Gauduchon, Calabi’s extremal Kähler metrics: an elementary introduction, book in preparation, 2005
[28] Guillemin, V.; Uribe, A., The Laplace operator on the \(n\)-th tensor power of a line bundle: Eigenvalues which are bounded uniformly in \(n\), Asymptot. Anal., 1, 105-113 (1988) · Zbl 0649.53026
[29] Ji, S.; Shiffman, B., Properties of compact complex manifolds carrying closed positive currents, J. Geom. Anal., 3, 1, 37-61 (1993) · Zbl 0784.32009
[30] Karabegov, A. V.; Schlichenmaier, M., Identification of Berezin and Toeplitz deformation quantization, J. Reine Angew. Math., 540, 49-76 (2001) · Zbl 0997.53067
[31] Lu, Z., On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math., 122, 2, 235-273 (2000) · Zbl 0972.53042
[32] Lu, Z.; Tian, G., The log term of the Szegő kernel, Duke Math. J., 125, 2, 351-387 (2004) · Zbl 1072.32014
[33] Ma, X.; Marinescu, G., The \(Spin^c\) Dirac operator on high tensor powers of a line bundle, Math. Z., 240, 3, 651-664 (2002) · Zbl 1027.58025
[34] Ma, X.; Marinescu, G., Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris, 339, 7, 493-498 (2004), Full version available at arXiv: · Zbl 1066.32006
[35] X. Ma, G. Marinescu, Toeplitz operators on symplectic manifolds, preprint, 2005; X. Ma, G. Marinescu, Toeplitz operators on symplectic manifolds, preprint, 2005 · Zbl 1152.81030
[36] Ma, X.; Marinescu, G., Holomorphic Morse Inequalities and Bergman Kernels, Progr. Math., vol. 254 (2007), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1135.32001
[37] Marinescu, G., Asymptotic Morse inequalities for pseudoconcave manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci., 23, 1, 27-55 (1996) · Zbl 0867.32004
[38] Marinescu, G.; Yeganefar, N., Embeddability of some strongly pseudoconvex CR manifolds, Trans. Amer. Math. Soc., 359, 4757-4771 (2007) · Zbl 1122.32026
[39] Nadel, A.; Tsuji, H., Compactification of complete Kähler manifolds of negative Ricci curvature, J. Differential Geom., 28, 3, 503-512 (1988) · Zbl 0661.53047
[40] Napier, T.; Ramachandran, M., The \(L^2\)-method, weak Lefschetz theorems and the topology of Kähler manifolds, J. Amer. Math. Soc., 11, 2, 375-396 (1998) · Zbl 0897.32007
[41] Ohsawa, T., Isomorphism theorems for cohomology groups of weakly 1-complete manifolds, Publ. Res. Inst. Math. Sci., 18, 191-232 (1982) · Zbl 0526.32016
[42] Ruan, W.-D., Canonical coordinates and Bergman metrics, Comm. Anal. Geom., 6, 3, 589-631 (1998) · Zbl 0917.53026
[43] Shiffman, B.; Zelditch, S., Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math., 544, 181-222 (2002) · Zbl 1007.53058
[44] Shiffman, B.; Zelditch, S., Number variance of random zeros on complex manifolds, preprint, arXiv: · Zbl 1168.32009
[45] Takayama, S., A differential geometric property of big line bundles, Tôhoku Math. J., 46, 2, 281-291 (1994) · Zbl 0802.32027
[46] Taylor, M. E., Partial Differential Equations. 2: Qualitative Studies of Linear Equations, Appl. Math. Sci., vol. 115 (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0869.35003
[47] Taylor, M. E., Pseudodifferential Operators (1981), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0453.47026
[48] Tian, G., On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., 32, 99-130 (1990) · Zbl 0706.53036
[49] Todor, R.; Chiose, I.; Marinescu, G., Asymptotic Morse inequalities for covering manifolds, Nagoya Math. J., 163, 145-165 (2001) · Zbl 1018.32022
[50] Wang, X., Canonical metrics on stable vector bundles, Comm. Anal. Geom., 13, 253-285 (2005) · Zbl 1093.32008
[51] Zelditch, S., Szegő kernels and a theorem of Tian, Int. Math. Res. Not. (6), 317-331 (1998) · Zbl 0922.58082
[52] Zucker, S., Hodge theory with degenerating coefficients: \(L_2\)-cohomology in the Poincaré metric, Ann. of Math., 109, 415-476 (1979) · Zbl 0446.14002
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