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 $k\geq 1$, 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 $k\geq 1$, let $\mathcal{B}_{\Lambda}^{ k}$ denote the Bergman kernel of the complex vector space $H^{0}(X_{\Gamma},\Lambda^{\otimes k })$. For any $k\geq 3$, 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 $k\geq 1$, 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_{\Gamma}}\bigg|\frac{\mu_{\mathrm{ber}}^{k,\mathrm{vol}}(z)}{\mu_{\mathrm{hyp}}^{\mathrm{vol}}}\bigg|=O_{X_{\Gamma},\epsilon}\big(k^{4+\epsilon}\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}