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

TL;DR

This paper derives off-diagonal estimates for the Bergman kernel associated with line bundles over Siegel varieties, providing new bounds in the context of cocompact and arithmetic subgroups of the symplectic group.

Contribution

It introduces novel off-diagonal estimates for the Bergman kernel on Siegel varieties, extending understanding of their asymptotic behavior for large tensor powers.

Findings

Derived bounds for the Bergman kernel off the diagonal

Established estimates for cocompact subgroups

Extended results to arithmetic subgroups

Abstract

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.