For $g\geq 2$, 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 $\ell:=\mathrm{ det}(\Omega_{ X_{ \Gamma}})$ 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 $k\geq 1$, 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 $\|\cdot\|_{ k}$ denote the point-wise norm on $\ell^{ \otimes k}$. Let $\mathcal{ B}_{ X_{ \Gamma}}^{ \ell^{ k}}$ denote the Bergman kernel associated to $H^{ 0}_{ L^{ 2}}(X_{ \Gamma},\ell^{ \otimes k})\subset H^{ 0}(X_{ \Gamma},\ell^{ \otimes k})$, 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 $\|\mathcal{ B}_{ X_{ \Gamma}}^{ \ell^{ k}}(Z,W)\|_{ \ell^{ k}}$, when $\Gamma$ is a cocompact subgroup and when $\Gamma$ is an arithmetic subgroup.
Categories: math.CV math.DG