Estimates of Bergman Kernels and Bergman metric on compact Picard surfaces

Abstract

Let $\Gamma \subset \mathrm{SU}((2,1),\mathbb{C})$ be a torsion-free cocompact subgroup. Let ${\mathbb{B}}^2$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric ${\mu }_{\mathrm{hyp}}$, and let $X_{\Gamma }:=\Gamma \backslash {\mathbb{B}}^2$ denote the quotient space, which is a compact complex manifold of dimension $2$. Let $\Lambda :={\Omega }_{X_{\Gamma }}^2$ denote the line bundle on $X_{\Gamma }$, whose sections are holomorphic $(2,0)$-forms. For any $k\ge 1$, the hyperbolic metric induces a point-wise metric on $H^0(X_{\Gamma },{\Lambda }^{\otimes k})$, which we denote by $|\cdot {|}_{\mathrm{hyp}}$. For any $k\ge 1$, let ${\mathcal{B}}_{\Lambda }^k$ denote the Bergman kernel of the complex vector space $H^0(X_{\Gamma },{\Lambda }^{\otimes k})$. For any $k\ge 3$, and $z,w\in X_{\Gamma }$, the first main result of the article is an off-diagonal estimate of the Bergman kernel ${\mathcal{B}}_{\Lambda }^k$. For any $k\ge 1$, let ${\mu }_{\mathrm{ber}}^k(z):=-\frac{i}{2\pi }{\partial }_z{\partial }_{\overline{z}}\log |{\mathcal{B}}_{\Lambda }^k(z,z){|}_{\mathrm{hyp}}$ denote the Bergman metric associated the line bundle ${\Lambda }^{\otimes k}$, and let ${\mu }_{\mathrm{ber}}^{k,\mathrm{vol}}(z)$ denote the associated volume form. For $k\gg 1$ sufficiently large, and $\epsilon >0$, the second main result of the article is the following estimate \begin{align*} \sup_{z\in X_Γ}\bigg|\frac{μ_{\mathrm{ber}}^{k,\mathrm{vol}}(z)}{μ_{\mathrm{hyp}}^{\mathrm{vol}}}\bigg|=O_{X_Γ,ε}\big(k^{4+ε}\big), \end{align*} where ${\mu }_{\mathrm{hyp}}^{\mathrm{vol}}$ denotes the volume form associated to the hyperbolic metric ${\mu }_{\mathrm{hyp}}$, and the implied constant depends on the Picard surface $X_{\Gamma }$, and on the choice of $\epsilon >0$. \end{abstract}

To appear in Journal of Mathematical Analysis and Applications