Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

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 Ω X 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.

Estimates of automorphic cusp forms over quaternion algebras

“…Estimates of Bergman metric in the setting of noncompact manifolds are difficult to obtain, as the Bergman metric and the ambient metric on the manifold, both admit singularities. Improving and improvising arguments from [ARS24], we derive estimates of the Bergman metric associated to high tensor-powers of the cotangent bundle derived in [AB19] and [ARS24], to the setting of noncompact Picard varieties, of finite hyperbolic volume, which is the second main result of this article.…”

“…Another problem of interest in complex geometry, is deriving estimates of Bergman metric, associated to tensor powers of a given holomorphic line bundle. In [AB19], Biswas and the first named author have derived estimates of the Bergman metric associated to high tensor-powers of the cotangent bundle, defined over a compact hyperbolic Riemann surface. In [AM23], the first named author and Mukherjee have extended estimates from [AB19] to the setting of noncompact hyperbolic Riemann surfaces.…”

“…In [AB19], Biswas and the first named author have derived estimates of the Bergman metric associated to high tensor-powers of the cotangent bundle, defined over a compact hyperbolic Riemann surface. In [AM23], the first named author and Mukherjee have extended estimates from [AB19] to the setting of noncompact hyperbolic Riemann surfaces. In [ARS24], the first named author, Roy, and Sadhukhan have extended the estimates from [AB19] to the setting of Picard surfaces.…”

“…In [AM23], the first named author and Mukherjee have extended estimates from [AB19] to the setting of noncompact hyperbolic Riemann surfaces. In [ARS24], the first named author, Roy, and Sadhukhan have extended the estimates from [AB19] to the setting of Picard surfaces.…”