Explicit Estimates for the Bergman Kernel Form

Abstract

Let $(L,e^{-\varphi })$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $\omega =i\partial \overline{\partial }\varphi$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\omega \le \omega$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then \[ K_{mϕ}\geq \frac{2m-1}{4π}\,ω, \] 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. We also find a higher dimensional version. Under the two-sided bound $-\omega \le \mathrm{Ric}\omega \le \omega$ and the same closed-geodesic hypothesis, we also prove \[ K_{mϕ}\leq \frac{mω}{2π} \left(1+\frac{3}{2m}\right). \] 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 via Błocki--Zwonek and isoperimetric inequalities.

arXiv comment13 Pages. v3. Improved upper bound