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

Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and put $\omega=\ddbar\phi$. We obtain effective pointwise estimates for the Bergman form of $H^0(X,K_X\otimes L^m)$. If $\Ric\omega\leq\omega$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then $$K_{m\phi}\geq \frac{2m-1}{4\pi}\,\omega,$$ 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 $-\omega\leq\Ric\omega\leq\omega$ and the same closed-geodesic hypothesis, we also prove $$K_{m\phi}\leq \frac{m\omega}{2\pi} \left(1+\frac{54.8\log(2m)}{m-\frac{1}2}\right).$$ 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.