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.
Study Kähler-Ricci solitons on bounded domains, proving they are Kähler-Einstein.
problem Characterize Kähler-Ricci solitons on bounded pseudoconvex domains.
method Prove solitons are Kähler-Einstein under suitable assumptions, using Huang and Xiao's resolution of Cheng's conjecture.
result Kähler-Ricci solitons on bounded pseudoconvex domains are Kähler-Einstein.
The paper extends results on Bach-flat solitons to new types.
problem Analyzing Bach-like tensors on complete gradient Ricci solitons.
method Extending previous results to new types of solitons.
result Results on Bach-flat solitons extended to new types.
In this paper, we give a lower bound of Bergman kernels for a sequence of almost Kähler-Einstein Fano manifolds, or more general, a sequence of Fano manifolds with almost Kähler-Ricci solitons. This generalizes a result by Donaldson-Sun, Tian for Kähler-Einstein manifolds sequence with positive scalar curvature. As an …
We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…
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.
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.
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.
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…
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.
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.
The paper studies Ricci curvature on Kähler-Ricci flow.
problem Analyzing Ricci curvature on Kähler-Ricci flow.
method Examining n-dimensional compact Kähler manifolds with semi-ample canonical line bundles under Kähler Ricci Flow.
result Ricci curvature converges to negative of generalized Kähler Einstein metric ωB locally away from singular set. 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. We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtai…
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.
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.
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.
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.
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.
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 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.
The paper proves properties of Bergman spaces and Schiffer operators on Riemann surfaces with quasicircles.
problem Properties of Bergman spaces and Schiffer operators on Riemann surfaces with quasicircles.
method Proof of isomorphism of Schiffer operators and application to Plemelj-Sokhotski isomorphism and jump decomposition.
result Schiffer integral operator is an isomorphism between Bergman spaces on different subsets of a Riemann surface.
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.
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 an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line b…
The study shows algebraic Bergman kernels imply finite type boundaries in complex domains.
problem Understanding the relationship between algebraic Bergman kernels and the finite type of boundaries in complex domains.
method Analyzing algebraic Bergman kernels and their implications on the finite type of boundaries in smoothly bounded pseudoconvex domains in C2. result The boundary of a smoothly bounded pseudoconvex domain with an algebraic Bergman kernel of degree d is of finite type with type r≤2d.