2025/06/21 by Aryasomayajula, Anilatmaja, G, Harinarayanan
#32A36 #32N10 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2506.17583
For g≥ 2, let Γ\subsetSp(2g,ℝ) be a discrete subgroup, which is either a cocompact subgroup or an arithmetic subgroup without torsion elements, and let ℍg denote the Siegel upper half space of genus g. Let XΓ:=Γ\backslashℍg denote the quotient space, which is a complex manifold of dimension g(g+1)/2. Let ΩXΓ denote the cotangent bundle, and let ℓ:=det(ΩXΓ) denote the determinant line bundle of ΩXΓ. For any Z,W∈ XΓ, let dS(Z,W) denote the geodesic distance between the points Z and W on XΓ. \vspace0.15cm\noindent For any k≥ 1, let H0(XΓ,ℓ⊗ k) denote the complex vector space of global sections of the line bundle ℓ⊗ k, and let ‖⋅‖k denote the point-wise norm on ℓ⊗ k. Let BXΓ^ℓ k denote the Bergman kernel associated to H0L2(XΓ,ℓ⊗ k)⊂ H0(XΓ,ℓ⊗ k), vector subspace of L2 global sections. For any k≫ 1, and Z,W∈ XΓ , we derive estimates of the Bergman kernel ‖BXΓ^ℓ k(Z,W)‖ℓk, when Γ is a cocompact subgroup and when Γ is an arithmetic subgroup.