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On the asymptotic expansion of Bergman kernel
"... Abstract. We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle. 1. ..."
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Cited by 47 (18 self)
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Abstract. We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle. 1.
Generalized Bergman kernels on symplectic manifolds,
 Adv. Math.
, 2008
"... ABSTRACT. We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized BochnerLaplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a clo ..."
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Cited by 33 (16 self)
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ABSTRACT. We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized BochnerLaplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As consequence, we calculate the density of states function of the BochnerLaplacian and establish a symplectic version of the convergence of the induced FubiniStudy metric. We also discuss generalizations of the asymptotic expansion for noncompact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of BismutLebeau. INTRODUCTION s1 The Bergman kernel for complex projective manifolds is the smooth kernel of the orthogonal projection from the space of smooth sections of a positive line bundle L on the space of holomorphic sections of L, or, equivalently, on the kernel of the Tian,Ru,Zelditch,Catlin,BSZ,Lu, [46, 41, 50, 17, D [27] where the existence of Kähler metrics with constant scalar curvature is shown to be closely related to ChowMumford stability. In DLM [20], Dai, Liu and Ma studied the asymptotic expansion of the Bergman kernel of the spin c Dirac operator associated to a positive line bundle on a compact symplectic manifold, and related it to that of the corresponding heat kernel. As a by product, they gave a new proof of the above results. This approach is inspired by Local Index Theory, especially by the analytic localization techniques of BismutLebeau Another natural generalization of the operator L in symplectic geometry was initiated by Guillemin and Uribe
Szegö kernels and a theorem of Tian
 Int. Math. Res. Notices
, 1998
"... A variety of results in complex geometry and mathematical physics depend upon the analysis of holomorphic sections of high powers L ⊗N of holomorphic line bundles L → M over compact Kähler manifolds ([A][Bis][Bis.V] [Bou.1][Bou.2][B.G][D][Don] [G][G.S][K][Ji] [T] [W]). The principal tools have been ..."
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Cited by 29 (2 self)
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A variety of results in complex geometry and mathematical physics depend upon the analysis of holomorphic sections of high powers L ⊗N of holomorphic line bundles L → M over compact Kähler manifolds ([A][Bis][Bis.V] [Bou.1][Bou.2][B.G][D][Don] [G][G.S][K][Ji] [T] [W]). The principal tools have been Hörmander’s L 2estimate on
WEIGHTED BERGMAN KERNELS AND QUANTIZATION
, 2000
"... Let Ω be a bounded pseudoconvex domain in C N, φ, ψ two positive functions on Ω such that − log ψ, − log φ are plurisubharmonic, z ∈ Ω a point at which − log φ is smooth and strictly plurisubharmonic, and M a nonnegative integer. We show that as k → ∞, the Bergman kernels with respect to the weig ..."
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Cited by 25 (3 self)
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Let Ω be a bounded pseudoconvex domain in C N, φ, ψ two positive functions on Ω such that − log ψ, − log φ are plurisubharmonic, z ∈ Ω a point at which − log φ is smooth and strictly plurisubharmonic, and M a nonnegative integer. We show that as k → ∞, the Bergman kernels with respect to the weights φ k ψ M have an asymptotic expansion K φ k ψ M (x, y) = k N π N φ(x, y) k ψ(x, y) M bj(x, y) k −j, j=0 b0(x, x) = det − ∂2 log φ(x)
Toeplitz operators on symplectic manifolds
 J. Geom. Anal
"... Dedicated to Professor Gennadi Henkin with the occasion of his 65th anniversary ..."
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Cited by 21 (3 self)
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Dedicated to Professor Gennadi Henkin with the occasion of his 65th anniversary
Geometric quantization and nogo theorems
 of Banach Center Publ
, 1998
"... Abstract. A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a ..."
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Abstract. A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a natural (projectively) flat connection on this vector bundle. We prove that for a broad class of manifolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist. This is a consequence of a “no go ” theorem claiming that the entire Lie algebra of smooth functions on a compact symplectic manifold cannot be quantized, i.e., it has no essentially nontrivial finitedimensional representations. 1.
THE SPIN c DIRAC OPERATOR ON HIGH TENSOR POWERS OF A LINE BUNDLE
, 2001
"... Abstract. We study the asymptotic of the spectrum of the spin c Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Mon ..."
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Cited by 7 (4 self)
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Abstract. We study the asymptotic of the spectrum of the spin c Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10]. 1.