Bergman kernel on Riemann surfaces and Kaehler metric on symmetric products

2019/09/09 by Anilatmaja Aryasomayajula, Indranil Biswas, Aryasomayajula, Anilatmaja +1
Mathematics · #11F03 #32A25 #32N05 #53C07 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1909.03776

openalex publication_date 2019/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer k≥ 2, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle Ω^{⊗ k}_X, where Ø is the holomorphic cotangent bundle of X. Our first main result estimates the corresponding Bergman metric on X in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of X into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of Ω^{⊗ k}_X. The Fubini-Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of X. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of Ω^{⊗ k}_X and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on X.

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