Explicit Estimates for the Bergman Kernel Form
Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $\omega=i\partial\overline\partial\phi$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\omega\leq\omega$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then \[ K_{m\phi}\geq \frac{2m-1}{4\pi}\,\omega, \] with sharpness holding for $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. 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. Under the two-sided bound $-\omega\leq\mathrm{Ric}\omega\leq\omega$ 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 version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound is obtained by observing the submean inequality with quantitative isothermal coordinates that were obtained in recent work by Eilat.