Infinite volumes of Bergman spaces on product manifolds.
problem Volume calculation of Bergman spaces on product manifolds.
method Calabi and Mabuchi volumes, inspired by Shiffman-Zelditch conjecture.
result Infinite volumes of Bergman spaces on product manifolds.
Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.
problem Proving non-vanishing of Poincaré series on finite-volume quotients of Hermitian symmetric spaces.
method Using Bergman kernels and averaging over discrete groups, proving non-vanishing of Poincaré series.
result Large class of relative Poincaré series does not vanish on general locally symmetric spaces of finite volume.
In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensor- products of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.
The paper proves bounded domains are biholomorphic to the unit ball.
problem Covering manifolds with finite volume.
method Analyzes C2 and C1,ε boundary conditions for bounded domains. result Bounded domains are biholomorphic to the unit ball under certain conditions.
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.
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group G⊂PU(n,1) acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…
The Bergman kernel's minimal point determines domain properties.
problem Characterizing domains based on the Bergman kernel's minimal point.
method Using Suita conjecture and properties of Bergman and Szegö kernels.
result The Bergman kernel's minimal point identifies specific domain types.
In this note we shall prove that the complete Kähler-Einstein volume form on a bounded strongly pseudoconvex domain with C∞-boundary is the normalized limit of a sequence of Bergman kernels.
Estimates Kähler metrics on compactified hyperbolic surfaces and their symmetric products.
problem Estimating Kähler metrics on compactified hyperbolic surfaces and their symmetric products.
method Deriving estimates for Bergman metrics and using them to estimate volume forms.
result Estimates for Kähler metrics on compactified hyperbolic surfaces and their symmetric products.
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.
Let X be a strictly pseudoconcave domain in a closed polarized complex manifold (Y,L) where L is a (semi-)positive line bundle over Y. Any given Hermitian metric on L, together with a volume form, induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k…
We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function F defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the…
Formula for Bergman kernel of complex hyperbolic manifolds proved.
problem Calculating the Bergman kernel of complex hyperbolic manifolds.
method Expressed as a sum over geodesic loops.
result Maximum and minimum of the Bergman kernel function proved.
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 continuity of Bergman kernels on degenerating varieties.
problem Continuity and uniform convergence of Bergman kernels on degenerating varieties.
method Introduced fiberwise Bergman kernel for flat families of polarized varieties, established continuity and uniform convergence results.
result Uniform convergence of Fubini-Study currents and continuity of fiberwise Bergman kernel on test configurations.
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 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.
Explicit formula for Bergman kernel of abelian varieties proved.
problem Explicit formula for Bergman kernel of polarized abelian varieties.
method Explicit formula for Bergman kernel of polarized abelian varieties.
result Explicit formula for Bergman kernel of polarized abelian varieties.
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.
Study on symplectic manifolds, focusing on Bergman kernels and their asymptotic behavior.
problem Asymptotic behavior of Bergman kernels on symplectic manifolds of bounded geometry.
method Establish off-diagonal exponential estimates, full asymptotic expansions, and Berezin-Toeplitz quantization.
result Improved remainder estimates for the asymptotic expansion of Bergman kernels.
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.
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.
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.
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.
Paper compares Bergman kernel and Masur-Veech measure on Teichmüller space.
problem Comparing Bergman kernel and Masur-Veech measure on Teichmüller space.
method Comparison between Bergman kernel form and pushforward measure of Masur-Veech measure.
result Obtained a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech measure.
The Bergman measure converges to the Zhang measure on a hybrid space.
problem Proving convergence of Bergman measures to Zhang measure.
method Analyzing convergence on a hybrid space and metrized curve complex.
result Bergman measure converges to Zhang measure on a hybrid space.
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 C1,α convergence. result Uniform C1,α convergence of Bergman metrics is demonstrated. Study shows how to calculate the volume of pseudoeffective line bundles on Kähler manifolds.
problem Calculating the volume of pseudoeffective line bundles on Kähler manifolds.
method Using a limit of section dimensions and a model potential associated to the line bundle.
result The limit of k−nh0(X,Lk⊗I(ku)) equals the non-pluripolar volume of P[u]I. New method for spectral and Bergman kernels under local spectral gap condition.
problem Analyzing spectral and Bergman kernels for complex manifolds.
method Developed a new scaling method to study spectral and Bergman kernels.
result Established pointwise asymptotics of spectral and Bergman kernels.
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.
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
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.
Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.
problem Weak holomorphic Morse inequalities on various types of manifolds.
method Asymptotic Bergman kernel functions and Bochner-Kodaira-Nakano formulas.
result Unified proofs of weak holomorphic Morse inequalities.
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.
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct …
Estimates Bergman kernel for positive line bundles.
problem Estimating the Bergman kernel for positive line bundles.
method Explicit estimate using positive line bundles.
result Explicit lower bound of the Bergman kernel.
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.
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.
Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.
problem Analyzing Bergman projections with Gevrey weights.
method Extending direct approach to semiclassical asymptotics to Gevrey weights using Fourier integral operators.
result Gevrey symbol amplitude of asymptotic Bergman projection with Gevrey weights and Gevrey-type growth rate.
Analytic Kähler potentials yield analytic Bergman kernels.
problem Characterizing Bergman kernels for analytic Kähler potentials.
method Linear recursive formula for Bergman kernel coefficients, simplified from Charles's work.
result Bergman kernels are analytic symbols with phase determined by Kähler potential polarization.
Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.
We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…
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.
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.
Upper bounds for Bergman kernels from smooth Kähler potentials.
problem Bounding Bergman kernels from smooth Kähler potentials.
method Using Taylor coefficients of the Kähler potential, we give upper bounds for Bergman kernels of tensor powers of a smooth positive line bundle.
result Improved off-diagonal rate of decay for analytic, quasi-analytic, and Gevrey potentials.
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 Bn+m are equivalent. Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.
problem Analyzing the asymptotics of weighted Bergman kernels for submultiplicative filtrations.
method Demonstrated that weight operator is a Toeplitz operator; analyzed asymptotics of weighted Bergman kernels.
result Local refinement of convergence of jumping measures towards geodesic ray pushforward measure.