Abstract We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact symplectic manifold and the generalized Bergman kernel on the base. Then we state the full off-diagonal asymptotic expansion of the generalized Bergman kernel, improving the remainder estimate known in the compact case to an exponential decay. Finally, we establish the theory of Berezin-Toeplitz quantization on symplectic orbifolds associated with the renormalized Bochner-Laplacian.

中文翻译：

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有界几何辛流形上的广义伯格曼核

摘要 我们研究了重整化 Bochner-Laplacian 的广义 Bergman 核在有界几何辛流形上正线丛的高张量幂上的渐近行为。首先，我们建立广义伯格曼核的非对角指数估计。作为应用，我们获得了紧致辛流形的Galois覆盖上的广义Bergman核与基上的广义Bergman核之间的关系。然后我们陈述广义伯格曼核的完全非对角渐近展开，将紧凑情况下已知的剩余估计改进为指数衰减。最后，我们建立了与重整化 Bochner-Laplacian 相关的辛 orbifolds 的 Berezin-Toeplitz 量化理论。