Explicit Estimates for the Bergman Kernel Form on Polarized Riemann Surfaces for General Tensor Powers

Abstract

Let $(L,e^{-\varphi })$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and put $ω=\ddbarϕ$. We obtain effective pointwise estimates for the Bergman form of $H^0(X,K_X\otimes L^m)$. If $\Ricω\leqω$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then \[ K_{mϕ}\geq \frac{2m-1}{4π}\,ω, \] and the constant is sharp on $({\mathbb{P}}^1,{\mathcal{O}}_{{\mathbb{P}}^1}(2))$. A local version, depending on an upper curvature bound and the injectivity radius, recovers the first two terms of the Bergman expansion when the curvature is constant. Under the two-sided bound $-ω\leq\Ricω\leqω$ and the same closed-geodesic hypothesis, we also prove \[ K_{mϕ}\leq \frac{mω}{2π} \left(1+\frac{54.8\log(2m)}{m-\frac{1}2}\right). \] The lower estimates use the deformation-to-the-tangent-space form of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound combines a weighted submean inequality with quantitative isothermal coordinates which was obtained in recent work by Eilat.

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