Study of rational Picard group in moduli space of holomorphic n-differentials.
problem Understanding the rational Picard group of moduli space of holomorphic n-differentials.
method Two approaches: analytic using Bergman tau function and algebro-geometric cohomological computations.
result Express Prym-Tyurin classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero.
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…
The paper calculates determinants for Laplacians on spinor bundles over surfaces with flat metrics.
problem Calculating determinants for Laplacians on spinor bundles over surfaces with flat metrics.
method Explicit expressions for determinants of self-adjoint extensions of Laplacians using Bergman tau-function and theta-constants.
result An explicit expression for the determinant of the Szegö extension and comparison formulas for different extensions.
Study special circle bundles over moduli spaces of quadratic differentials.
problem Analyse of circle bundles over specific moduli spaces of quadratic differentials.
method Formalism of the Bergman tau-function applied to combinatorial models.
result Sections of circle bundles over moduli spaces are given by the argument of the modular discriminant.
Paper proves BGW tau-function can be represented as Q-polynomials.
problem Enumerative geometric interpretations of BGW tau-function.
method Proves BGW tau-functions are hypergeometric tau functions of BKP hierarchy.
result Original BGW tau-function can be represented as a linear combination of Schur Q-polynomials.
Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.
problem Proving the Kontsevich-Witten tau-function formula.
method Directly shows Q-polynomial expansion satisfies Virasoro constraints.
result Direct proof of the formula without matrix model.
In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivia…
There is a general method for constructing a soliton hierarchy from a splitting of a loop group as a positive and a negative sub-groups together with a commuting linearly independent sequence in the positive Lie subalgebra. Many known soliton hierarchies can be constructed this way. The formal inverse scattering associ…
We prove that the extended Toda hierarchy of \cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m≥−1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau fu…
Study on hypermaps and KP hierarchy, proving tau function and enumerative meaning.
problem Understanding the partition function of meromorphic functions on the Riemann sphere.
method Analysis of Hurwitz Dubrovin--Frobenius manifold structure and rational reductions of the KP hierarchy.
result The all genera partition function is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy.
For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies are related by a Legendre type transformation. We also consider relationships b…
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 simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.
Study on Bergman kernels near analytic hypersurfaces with exponential decay.
problem Analyzing the behavior of Bergman kernels near analytic hypersurfaces.
method Uniform estimates and asymptotic analysis of partial Bergman kernels for sections vanishing along hypersurfaces.
result Uniform estimate and asymptotic behavior of Bergman kernels near vanishing locus.
Solves constant pre-factor problem for tt*-Toda equations using asymptotic data and symplectic structures.
problem Constant pre-factor problem for the tt*-Toda equations.
method Explicit evaluation using asymptotic data and introduction of symplectic structures.
result Preservation of symplectic structures by Riemann-Hilbert correspondence for wider class of solutions.
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.
Topological recursion recovers a specific partition function for colored knots.
problem Recovering the extended Ooguri-Vafa partition function for colored HOMFLY-PT polynomials of torus knots.
method Applying topological recursion to the spectral curve of colored HOMFLY-PT polynomials of torus knots.
result Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.
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.
Estimates Bergman kernel of punctured disk with improved results.
problem Estimating Bergman kernel of punctured disk.
method Techniques from \cite{SunSun} applied to punctured disk with Poincaré metric.
result Improved results on Bergman kernels of punctured Riemann surfaces near singularities.
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.
The paper studies Bergman kernels on surfaces with punctures.
problem Analyzing Bergman kernels on surfaces with singularities.
method Localized analysis of Bergman kernels near punctures, using the standard Poincaré metric.
result Optimal uniform estimate of the supremum norm of the Bergman kernel.
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.
Continuing the previous work, we propose a further extension of the structure equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro algebra of non-local symmetries. Via a canonical dressing transformation, we first define a wave function for the CMC hierarchy. This leads to a pair of additiona…
Study estimates Bergman kernel on hyperbolic surfaces.
problem Estimating Bergman kernel on hyperbolic surfaces.
method Derive off-diagonal estimates for tensor-products of cotangent line bundles.
result Derived off-diagonal estimates for Bergman kernel.
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. 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.
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.
Study on symplectic manifolds finds asymptotic expansions for Bergman kernels.
problem Analyzing asymptotic expansions of Bergman kernels on symplectic manifolds.
method Established asymptotic expansions for renormalized Bochner Laplacians on high tensor powers of positive line bundles.
result Full off-diagonal asymptotic expansion for generalized Bergman kernels.