Off-diagonal estimates of the Bergman kernel associated to Siegel varieties

Abstract

For $g\ge 2$, let $\Gamma \subset \mathrm{Sp}(2g,\mathbb{R})$ be a discrete subgroup, which is either a cocompact subgroup or an arithmetic subgroup without torsion elements, and let ${\mathbb{H}}_g$ denote the Siegel upper half space of genus $g$. Let $X_{\Gamma }:=\Gamma \backslash {\mathbb{H}}_g$ denote the quotient space, which is a complex manifold of dimension $g(g+1)/2$. Let ${\Omega }_{X_{\Gamma }}$ denote the cotangent bundle, and let $\ell :=\det ({\Omega }_{X_{\Gamma }})$ denote the determinant line bundle of ${\Omega }_{X_{\Gamma }}$. For any $Z,W\in X_{\Gamma }$, let $d_{\mathrm{S}}(Z,W)$ denote the geodesic distance between the points $Z$ and $W$ on $X_{\Gamma }$. \vspace{0.15cm}\noindent For any $k\ge 1$, let $H^0(X_{\Gamma },{\ell }^{\otimes k})$ denote the complex vector space of global sections of the line bundle ${\ell }^{\otimes k}$, and let $\Vert \cdot {\Vert }_k$ denote the point-wise norm on ${\ell }^{\otimes k}$. Let ${\mathcal{B}}_{X_{\Gamma }}^{{\ell }^k}$ denote the Bergman kernel associated to $H_{L^2}^0(X_{\Gamma },{\ell }^{\otimes k})\subset H^0(X_{\Gamma },{\ell }^{\otimes k})$, vector subspace of $L^2$ global sections. For any $k\gg 1$, and $Z,W\in X_{\Gamma }$ , we derive estimates of the Bergman kernel $\Vert {\mathcal{B}}_{X_{\Gamma }}^{{\ell }^k}(Z,W){\Vert }_{{\ell }^k}$, when $\Gamma$ is a cocompact subgroup and when $\Gamma$ is an arithmetic subgroup.