Estimates of Picard modular cusp forms

In this article, for $n\ge 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{S}\mathrm{U}((n,1),\mathbb{C})$. The main result of the article is the following result. Let $\mathrm{\Gamma }\subset \mathrm{S}\mathrm{U}((2,1),{\mathcal{O}}_K)$ be a torsion-free subgroup of finite index, where $K$ is a totally imaginary field. Let ${\mathcal{B}}_{\mathrm{\Gamma }}^k$ denote the Bergman kernel associated to the ${\mathcal{S}}_k(\mathrm{\Gamma })$, complex vector space of weight-$k$ cusp forms with respect to $\mathrm{\Gamma }$. Let ${\mathbb{B}}^2$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric, and let $X_{\mathrm{\Gamma }}:=\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{B}}^2$ denote the quotient space, which is a noncompact complex manifold of dimension $2$. Let $\mid \cdot {\mid }_{\mathrm{p}\mathrm{e}\mathrm{t}}$ denote the point-wise Petersson norm on ${\mathcal{S}}_k(\mathrm{\Gamma })$. Then, for $k\ge 6$, we have the following estimate \begin{equation*} \sup_{z\in X_Γ}\big|$\mathcal{B}$Γ^{k}(z)\big|{$\mathrm{p}\mathrm{e}\mathrm{t}$}=O_Γ\big(k^{\frac{5}{2}}\big), \end{equation*} where the implied constant depends only on $\mathrm{\Gamma }$.