Explicit Estimates for the Bergman Kernel Form
Abstract
Let (L,e−ϕ) be a positive Hermitian holomorphic line bundle over a compact Riemann surface X, and let ω=i∂∂̅ϕ. We obtain explicit pointwise estimates for the Bergman form of the tensor power mL. If Ric ω≤ω and the shortest nonconstant closed geodesic has length at least 2π, then Kmϕ≥ (2m−1)/4π ω, with sharpness holding for (ℙ1,𝒪ℙ1(2)). We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound −ω≤Ric ω≤ω and the same closed-geodesic hypothesis, we also prove Kmϕ≤ mω/2π (1+3/2m). The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki--Zwonek and isoperimetric inequalities.
Related
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BibTeX
@misc{testorf2026explicit,
title = {{Explicit Estimates for the Bergman Kernel Form}},
author = {Johannes Testorf},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03493},
eprint = {2608.03493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03493},
}
RIS
TY - UNPB
TI - Explicit Estimates for the Bergman Kernel Form
AU - Johannes Testorf
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.03493
UR - https://arxiv.org/abs/2608.03493
AB - Let (L,e^−ϕ) be a positive Hermitian holomorphic line bundle over a compact Riemann surface X, and let ω=i∂∂̅ϕ. We obtain explicit pointwise estimates for the Bergman form of the tensor power mL. If Ric ω≤ω and the shortest nonconstant closed geodesic has length at least 2π, then K_mϕ≥ (2m−1)/4π ω, with sharpness holding for (ℙ^1,𝒪_ℙ^1(2)). We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound −ω≤Ric ω≤ω and the same closed-geodesic hypothesis, we also prove K_mϕ≤ mω/2π (1+3/2m). The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki--Zwonek and isoperimetric inequalities.
ER -