arXiv subjects

Yueh-Lin Chiang

Publications and source records attributed to Yueh-Lin Chiang.

3 recordsLinked to original sources

arXiv2026

Bergman Kernel Asymptotics for Semipositive Line Bundles near Curvature-Degenerate Points

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.

math.CV↗

arXiv2024

Spectral Kernels and Holomorphic Morse Inequalities for Sequence of Line Bundles

Given a sequence of Hermitian holomorphic line bundles $(L_k,h_k)$ over a complex manifold $M$ which may not be compact, we generalize the scaling method in arXiv:2310.08048 to study the asymptotic behavior of the Bergman kernels and spectral kernels with respect to the space of global holomorphic sections of $L_k$ with $(0,q)$-forms. We derive the leading term of the Bergman and spectral kernels under the local convergence assumption in the sequence of Chern curvatures $c_1(L_k,h_k)$, inspired by arXiv:2012.12019. The manifold $M$ may be non-Kähler and $c_1(L_k,h_k)$ may be negative or degenerate. Moreover, we establish the $L_k$-asymptotic version of Demailly's holomorphic Morse inequalities as an application to compact complex manifolds.

math.CV↗

arXiv2023

Semi-Classical Asymptotics of Bergman and Spectral Kernels for (0,q)-forms

In this paper, we develop a new scaling method to study spectral and Bergman kernels for the k-th tensor power of a line bundle over a complex manifold under local spectral gap condition. In particular, we establish a simple proof of the pointwise asymptotics of spectral and Bergman kernels. As a new result, in the function case, we obtain the leading term of Bergman kernel under spectral gap with exponential decay. Moreover, in the general cases of (0,q)-forms, the asymptotics remain valid while the curvature of the line bundle is degenerate.

math.CV↗