Off-diagonal estimates of the Bergman kernel associated to Siegel varieties
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| Format: | Preprint |
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2025
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| author | Aryasomayajula, Anilatmaja G, Harinarayanan |
| author_facet | Aryasomayajula, Anilatmaja G, Harinarayanan |
| contents | For $g\geq 2$, let $Γ\subset\mathrm{Sp}(2g,\mathbb{R})$ 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 $X_Γ:=Γ\backslash\mathbb{H}_{g}$ denote the quotient space, which is a complex manifold of dimension $g(g+1)/2$. Let $Ω_{X_Γ}$ denote the cotangent bundle, and let $\ell:=\mathrm{det}(Ω_{X_Γ})$ denote the determinant line bundle of $Ω_{X_Γ}$. For any $Z,W\in X_Γ$, let $d_{\mathrm{S}}(Z,W)$ denote the geodesic distance between the points $Z$ and $W$ on $X_Γ$.
\vspace{0.15cm}\noindent For any $k\geq 1$, let $H^{0}(X_Γ,\ell^{\otimes k})$ 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 $\mathcal{B}_{X_Γ}^{\ell^{ k}}$ denote the Bergman kernel associated to $H^{0}_{L^{2}}(X_Γ,\ell^{\otimes k})\subset H^{0}(X_Γ,\ell^{\otimes k})$, vector subspace of $L^2$ global sections. For any $k\gg 1$, and $Z,W\in X_Γ$ , we derive estimates of the Bergman kernel $\|\mathcal{B}_{X_Γ}^{\ell^{ k}}(Z,W)\|_{\ell^{k}}$, when $Γ$ is a cocompact subgroup and when $Γ$ is an arithmetic subgroup. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Off-diagonal estimates of the Bergman kernel associated to Siegel varieties Aryasomayajula, Anilatmaja G, Harinarayanan Complex Variables Differential Geometry 32A36, 32N10 For $g\geq 2$, let $Γ\subset\mathrm{Sp}(2g,\mathbb{R})$ 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 $X_Γ:=Γ\backslash\mathbb{H}_{g}$ denote the quotient space, which is a complex manifold of dimension $g(g+1)/2$. Let $Ω_{X_Γ}$ denote the cotangent bundle, and let $\ell:=\mathrm{det}(Ω_{X_Γ})$ denote the determinant line bundle of $Ω_{X_Γ}$. For any $Z,W\in X_Γ$, let $d_{\mathrm{S}}(Z,W)$ denote the geodesic distance between the points $Z$ and $W$ on $X_Γ$. \vspace{0.15cm}\noindent For any $k\geq 1$, let $H^{0}(X_Γ,\ell^{\otimes k})$ 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 $\mathcal{B}_{X_Γ}^{\ell^{ k}}$ denote the Bergman kernel associated to $H^{0}_{L^{2}}(X_Γ,\ell^{\otimes k})\subset H^{0}(X_Γ,\ell^{\otimes k})$, vector subspace of $L^2$ global sections. For any $k\gg 1$, and $Z,W\in X_Γ$ , we derive estimates of the Bergman kernel $\|\mathcal{B}_{X_Γ}^{\ell^{ k}}(Z,W)\|_{\ell^{k}}$, when $Γ$ is a cocompact subgroup and when $Γ$ is an arithmetic subgroup. |
| title | Off-diagonal estimates of the Bergman kernel associated to Siegel varieties |
| topic | Complex Variables Differential Geometry 32A36, 32N10 |
| url | https://arxiv.org/abs/2506.17583 |