AI 中文总结
本文研究半正线丛Bergman核在曲率退化点附近的渐近展开，通过谱假设和拟齐次结构获得完整展开与快速衰减，并应用于分支覆盖的拉回。
AI 中文摘要
我们研究了Hermitian流形上半正线丛高次张量幂的Bergman核的渐近行为。在曲率退化的点处，经典的渐近展开可能失效。在本文中，我们在度量允许局部解耦模型的退化点附近建立了完整的局部渐近展开和快速非对角衰减。更一般地，我们提出以下两个谱假设：Kodaira Laplacian的局部化温和谱隙，以及重标度局部模型的谱隙。在这些假设下，我们证明了Bergman核的局部化性质和快速非对角衰减。此外，如果度量具有局部拟齐次结构，我们在$C^\infty$-拓扑下获得完整的局部渐近展开。作为应用，我们研究了分支覆盖下正线丛的拉回。在光滑分支超曲面附近，所得的渐近展开反映了分支阶数。最后，对于某些非拟齐次模型，我们仍然获得局部化和主导阶渐近。
英文摘要
We study the asymptotic behavior of Bergman kernels for high tensor powers of semipositive line bundles over Hermitian manifolds. At points where the curvature degenerates, the classical asymptotic expansion may fail. In this paper, we establish a full local asymptotic expansion and rapid off-diagonal decay near degenerate points at which the metric admits a local decoupled model. More generally, we make the following two spectral hypotheses: a localized mild spectral gap for the Kodaira Laplacian and a spectral gap for the rescaled local model. Under these assumptions, we prove a localization property and the rapid off-diagonal decay for the Bergman kernel. Furthermore, if the metric has a local quasi-homogeneous structure, we obtain a full local asymptotic expansion in the $C^\infty$-topology. As an application, we study pull-backs of positive line bundles under branched coverings. Near a smooth ramification hypersurface, the resulting asymptotic expansion reflects the branching order. Finally, for certain non-quasi-homogeneous models, we still obtain localization and leading-order asymptotics.
