Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

2023/05/19 by Anilatmaja Aryasomayajula, Arijit Mukherjee, Aryasomayajula, Anilatmaja +1 · 2 citations
Mathematics · #11F11 #32A25 #32N05 #53C07 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2305.11609

openalex publication_date 2023/05/19 · openalex created_date 2023/05/23 · openalex updated_date 2026/07/28

Abstract

Let X denote a noncompact finite volume hyperbolic Riemann surface of genus g≥ 2, with only one puncture at i∞ (identifying X with its universal cover ℍ). Let X:=X∪\lbrace i∞\rbrace denote the Satake compactification of X. Let Ω_X denote the cotangent bundle on X. For k≫1, we derive an estimate for μ_X^Ber,k, the Bergman metric associated to the line bundle L^k:=Ω_X⊗ O_X((k-1)∞). For a given d≥ 1, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by μ_Sym^d(X)^FS,k, defines a Kähler metric on Sym^d(X), the d-fold symmetric product of X. Using our estimates of μ_X^Ber,k, as an application, we derive an estimate for μ_Sym^d(X),vol^FS,k, the volume form associated to the (1,1)-form μ_Sym^d(X)^FS,k.

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