Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

Author:

Aryasomayajula Anilatmaja1,Biswas Indranil2ORCID

Affiliation:

1. Indian Institute of Science Education and Research (IISER) Tirupati, Karakambadi Road, Mangalam (P.O.), Tirupati – 517507, Andhra Pradesh, India

2. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

Abstract

Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

Funder

INSPIRE research

Publisher

World Scientific Pub Co Pte Lt

Subject

General Mathematics

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Estimates of Bergman kernels and Bergman metric on compact Picard surfaces;Journal of Mathematical Analysis and Applications;2024-06

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