scholarly journals Bergman kernel asymptotics for singular metrics on punctured Riemann surfaces

2019 ◽  
Vol 68 (2) ◽  
pp. 593-628
Author(s):  
Dan Coman ◽  
Semyon Klevtsov ◽  
George Marinescu
Keyword(s):  
Bergman Kernel ◽  
Riemann Surfaces ◽  
Singular Metrics ◽  
Bergman Kernel Asymptotics

scholarly journals A direct approach to Bergman kernel asymptotics for positive line bundles

2008 ◽  
Vol 46 (2) ◽  
pp. 197-217 ◽  
Author(s):  
Robert Berman ◽  
Bo Berndtsson ◽  
Johannes Sjöstrand
Keyword(s):  
Bergman Kernel ◽  
Direct Approach ◽  
Line Bundles ◽  
Bergman Kernel Asymptotics ◽  
Positive Line

scholarly journals Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

2017 ◽  
Vol 4 (1) ◽  
pp. 7-15 ◽  
Author(s):  
Robert Xin Dong
Keyword(s):  
Elliptic Curves ◽  
Bergman Kernel ◽  
Complex Structure ◽  
Elliptic Functions ◽  
Hyperelliptic Curves ◽  
Asymptotic Formulas ◽  
Holomorphic Family ◽  
Structure Information ◽  
Taylor Expansions ◽  
Bergman Kernel Asymptotics

Abstract We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

2019 ◽  
Vol 30 (14) ◽  
pp. 1950071
Author(s):  
Anilatmaja Aryasomayajula ◽  
Indranil Biswas
Keyword(s):  
Bergman Kernel ◽  
Riemann Surfaces ◽  
Symmetric Product ◽  
Volume Form ◽  
Bergman Metric ◽  
Poincaré Metric ◽  
Kähler Metric ◽  
Fixed Dimension ◽  
Holomorphic Sections ◽  
Kahler Metric

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].

Bergman kernel asymptotics for generalized Fock spaces

2001 ◽  
Vol 83 (1) ◽  
pp. 207-242 ◽  
Author(s):  
Finbarr Holland ◽  
Richard Rochberg
Keyword(s):  
Bergman Kernel ◽  
Fock Spaces ◽  
Bergman Kernel Asymptotics

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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