In this paper we will discuss local coordinates canonically corresponding to a Kahler metric. We will also discuss and prove the convergence of Bergman metrics following Tian's result on convergence of Bergman metrics. At the end, we present an interesting characterization of ample line bundle that cou…
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48 results for “canonical Bergman metrics”
Study Bergman metric on Cartan-Hartogs domains and their duals.
problem Characterize Bergman metrics and duals on Cartan-Hartogs domains.
method Define Bergman dual, prove equivalence of conditions, compare with other metrics.
result Equivalence of Bergman metric properties and duals on Cartan-Hartogs domains.
In this article, we construct the canonical semipositive current or the canonical measure ( the potential of the canonical semipositive current) on a smooth projective variety of nonnegative Kodaira dimension in terms of a dynamical system of Bergman kernels. This current is considered to be a generalization of a Kä…
Study constructs complete Kahler-Einstein metrics near isolated log-canonical singularities.
problem Constructing complete Kahler-Einstein metrics near isolated log-canonical singularities.
method Two approaches: uniformization by a complex ball and Calabi Ansatz.
result Two metrics are shown to be the same, providing a local model.
Study quantization of Kähler-Einstein metrics using balanced metrics.
problem Approximating Kähler-Einstein metrics by balanced metrics.
method Use canonical Bergman metrics and introduce algebro-geometric obstructions.
result Existence and weak convergence of balanced metrics for CKE manifolds.
The canonical metric on a Riemann surface is the pullback from the Euclidean metric on the Jacobian variety via the period map. We study its induced L^2 metric on Teichmuller space via a variational approach.
In this paper, I give a new construction of a Kähler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular Kähler-Einstein metric on a smooth projective variety of general type which gives an AZD of the canonical bundle. Also the va…
Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.
problem Understanding Bergman kernels on Kähler manifolds and their properties.
method Localization and expansion analysis of Bergman kernels.
result Answered Lu-Tian's question about Bergman kernels having no logarithmic singularity.
Study on holomorphic isometries between complex domains, revealing geometric properties.
problem Characterizing holomorphic isometries between bounded symmetric domains.
method Analyzing holomorphic isometries between complex unit ball and other bounded symmetric domains, using classical results for complex-analytic subvarieties of Stein manifolds.
result Images of holomorphic isometries have specific geometric properties, including intersections with affine-linear subspaces.
In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the c…
Paper introduces hybrid curves moduli space and canonical measures.
problem Asymptotic geometry of Riemann surfaces and their moduli spaces.
method Definition of hybrid curves and canonical measures.
result Continuous variation of canonical measures over hybrid curves moduli space.
Survey on the Canonical Metrics on the Teichmüller Spaces and the Moduli Spaces of Riemann Surfacesmath.DG
This thesis surveys various metrics on Riemann surface spaces.
problem Various metrics on Riemann surface spaces.
method Survey of metrics and their properties.
result Equivalence of Kähler-Einstein metric to Teichmüller metric.
On a polarized manifold , the Bergman iteration is defined as a sequence of Bergman metrics on with two integer parameters . We study the relation between the Kähler-Ricci flow at any time and the limiting behavior of metrics when and the ratio ap…
We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existe…
The Bergman metric on symmetrized bidisc has negative curvature properties.
problem Holomorphic curvature properties of the Bergman metric on symmetrized bidisc.
method Analysis of holomorphic sectional and bisectional curvatures.
result Negative pinched holomorphic sectional curvature and non-positive holomorphic bisectional curvature.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Study on volumes and scalar curvature in complex geometry, proving new bounds and conditions.
problem Understanding volumes and scalar curvature in complex geometry.
method Asymptotic behavior of Bergman kernel, Kähler-Ricci flow, singular Kähler-Einstein metrics, and gluing techniques.
result New bounds and conditions for volumes and scalar curvature in complex manifolds.
The paper studies invariant weighted Bergman metrics on domains.
problem Investigating invariant weighted Bergman metrics under biholomorphisms.
method Introducing invariant weight assignments, using Bergman's minimum integral method and domain version of Tian-Yau-Zelditch expansion.
result Uniform convergence of weighted Bergman kernels and metrics on uniform squeezing domains.
In this paper we continue our study on the canonical metrics on the Teichmüller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carathéodory metric to the Kähler-Einstein metric, solving another old conjecture of Yau. We then prove that the Ricci curvature of the p…
Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.
problem Equivalence of invariant metrics on noncompact Kähler manifolds with bounded curvature.
method Use boundedness of ratio between Bergman kernel and wedge product of metric in fundamental domain.
result Equivalence of Bergman metric and Kähler-Einstein metric under specific conditions.
The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.
problem Characterizing domains with Bergman metrics of constant holomorphic sectional curvature.
method Using the Bergman-Calabi diastasis and its connection to the Bergman representative coordinate, the paper derives explicit formulas and proves properties of these metrics.
result Domains with Bergman metrics of constant holomorphic sectional curvature are hyperconvex or exhaustively biholomorphic to a ball.
New approach finds analytic interpretation of algebraic invariants for balanced metrics.
problem Finding analytic interpretation of algebraic invariants for balanced metrics.
method Using log canonical thresholds and basis divisors, the approach involves quantized Ding functionals on Bergman spaces.
result Each δ_m is the coercivity threshold of a quantized Ding functional on the m-th Bergman space, characterizing the existence of balanced metrics.
In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and…
Study on convergence rate of Bergman metrics on Kähler manifolds.
problem Analyzing convergence rate of Bergman metrics on Kähler manifolds.
method Using Tian's peak section method to show uniform convergence.
result Uniform convergence of Bergman metrics is demonstrated.
Study shows Bergman metric is non-Einstein for certain domains.
problem Characterizing the Bergman metric of specific domains.
method Analyzing pseudoconvex domains with strongly pseudoconvex polyhedral boundaries.
result Bergman metric is not Einstein for the studied domains.
Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.
problem Characterizing domains with specific metric properties.
method Analyzing Carathéodory and Bergman metrics on strictly pseudoconvex domains.
result Domains with specific metric properties are biholomorphically equivalent to balls.
Study shows convergence of Bergman kernels on covering spaces of Kähler manifolds.
problem Analyzing convergence of Bergman kernels on covering spaces of Kähler manifolds.
method Proving convergence of Bergman kernels and -Hodge numbers on a tower of coverings.
result Sections of canonical line bundles give rise to immersions into projective spaces.
Proper holomorphic isometries between Bergman domains are biholomorphisms.
problem Characterizing isometries between Bergman domains.
method New method from Information Geometry.
result Proper holomorphic local isometries are biholomorphisms.
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.
Study Bergman kernel and Kähler metrics on Riemann surfaces and symmetric products.
problem Estimating metrics on Riemann surfaces and symmetric products.
method Investigates Bergman kernels and Kähler metrics on Riemann surfaces and symmetric products.
result Estimates the Bergman metric and Kähler metric on symmetric products in terms of the Bergman kernel and Poincaré metric.
Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
problem Characterizing domains with Kähler-Einstein Bergman metrics.
method Asymptotics of derivatives of the Bergman kernel along critically tangent paths.
result Two-dimensional pseudoconvex domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
Study on when Bergman metrics of domains are induced by balls.
problem When does a domain's Bergman metric match a ball's up to a constant factor?
method Holomorphic isometric immersions, Calabi's diastasis criterion, explicit Bergman kernel formulas, algebraic arguments.
result For strictly pseudoconvex domains in \(\mathbb{C}^2\), if the immersion extends smoothly and transversally past the boundary and the scaling factor meets certain conditions, the domain is biholomorphic to the ball.
On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metricsmath.CV
The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.
In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the p…
Study Bergman kernel metrics on degenerating hyperelliptic surfaces.
problem Asymptotic behavior of Bergman kernels near singularities.
method Taylor expansion for Abelian differentials and period matrices.
result Explicit coefficients in asymptotic formulas for Bergman kernels.
This paper is a sequel to \cite{Xu}. In this paper, an estimation of the Bergman Kernel of Kähler hyperbolic manifold is given by the estimate and the Bochner formula. As an application, an effective criterion of the very ampleness of the canonical line bundle of Kähler hyperbolic manifold is given, which is a ge…
Study on symmetric domains with Bergman metric properties.
problem Properties of Bergman metrics in symmetric domains.
method Combining Hermitian symmetric spaces theory, Kähler immersions, and analytic/pluripotential tools.
result Rigidity results for Bergman metrics in bounded domains.
We prove that the Bergman and the Teichmuller metrics are equivalent on Teichmuller spaces.
Study shows quantum behavior near infinity in metric asymptotics.
problem Quantum behavior of metrics near infinity on quasi-projective manifolds.
method Analysis of Bergman kernel function near smooth divisor at infinity of Cheng-Yau metric.
result Quantum phenomenon observed for points very close to the divisor at infinity.
Rigidity theorem for Bergman metric on Hartogs domains over bounded homogeneous domains.
problem Proving rigidity of Bergman metric on Hartogs domains.
method Combining explicit formula for Bergman kernel and structural invariants of bounded homogeneous base.
result Conditions for Bergman metric to be Kähler-Einstein, homogeneous, and biholomorphic to are equivalent.
Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.
problem Localization of Bergman kernels on Kähler manifolds with complex hyperbolic cusps.
method Revisiting Tian's peak section method, applying to Kähler-Einstein metrics and quotients of complex balls.
result Partial localization result for Poincaré type cusps.
Statistical Bergman geometrymath.CV
The paper connects Bergman geometry with information geometry.
problem Exploring the Bergman geometry of complex domains.
method Introducing a mapping Φ and using Fisher information metrics.
result Established a new statistical curvature formula for the Bergman metric.
New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
Consider the sum of the first eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree to be thos…
The paper introduces new metrics on complex domains with specific geometric properties.
problem Developing metrics on complex domains with strong pseudoconvexity and holomorphic invariance.
method Explicit construction of holomorphic invariant strongly pseudoconvex complex Finsler metrics via deformation of Bergman metrics.
result These metrics have bounded curvature properties similar to Bergman metrics.
Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincaré metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels of punctured Riemann surfaces near singularities.
Localization of Bergman Kernels and the Cheng-Yau Conjecture on Real Analytic Pseudoconvex Domainsmath.CV
The paper proves a conjecture about the Bergman metric of real analytic domains.
problem Proving the Cheng-Yau conjecture for real analytic pseudoconvex domains.
method Localization of Bergman kernels, extension theorem, and Einstein metrics.
result The Bergman metric of a bounded pseudoconvex domain with real-analytic boundary is Einstein if and only if the domain is biholomorphic to the unit ball.
We show that the exponential map of the Bochner connection on the restricted holomorphic tangent bundle of a complex manifold admitting the positive-definite Bergman metric coincides with the inverse of Bergman's representative map. We also present a generalization of the Lu theorem, as an application.