Formula identifies boundary flux for Kähler manifolds under parallel deformation.
problem Identifying boundary flux for Kähler manifolds under parallel deformation.
method One-sided Hadamard formula for normalized Monge-Ampère energy.
result Identifies boundary component as negative outward Anzellotti trace of divergence-measure flux current.
Study Higgs sections and flat sections for nonlinear harmonic bundles.
problem Equivalence of Higgs and flat sections for nonlinear harmonic bundles.
method Analyze harmonic vector bundles, generalize to sub-fibrations and morphisms.
result Vanishing of a degree obstruction for general nonlinear harmonic bundles.
The paper studies the distribution of random degeneracy sets on complex manifolds.
problem Distribution of random degeneracy sets on compact Kähler manifolds.
method Asymptotic expansion of induced Grassmannian Chern forms, meromorphic transforms, and Wishart distribution.
result Normalized currents converge to curvature forms with quantitative estimates.
Criterion for solvability of complex 2-Hessian equation on compact Kähler manifolds.
problem Solvability of complex 2-Hessian equation on compact Kähler manifolds.
method Nakai--Moishezon-type criterion associated with the complex 2-Hessian equation.
result Criterion equivalent to existence of a smooth 2-admissible representative in complex dimension three.
The study sets limits on heat equation solutions' Hessians on curved spaces.
problem Bounding Hessians of positive solutions to heat equations on Kähler manifolds.
method Global and local upper bounds for Hessian matrices under curvature constraints.
result Improved bounds on Hessians for Riemannian manifolds with lower sectional curvature.
Introduces Poisson K-stability for Kähler manifolds and proves existence of constant scalar curvature structures.
problem Stability conditions for Poisson structures on Kähler manifolds.
method Infinite-dimensional momentum map techniques.
result Existence of constant scalar curvature symplectic generalized Kähler structures on Kähler-Einstein Fano manifolds.
Iterative method finds Hermitian-Einstein metrics on stable bundles.
problem Finding Hermitian-Einstein metrics on stable bundles over Kähler or Gauduchon manifolds.
method Iterative construction using a specific metric update formula.
result Smooth convergence to a Hermitian-Einstein metric from any initial metric.
Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
problem Bottom of the spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
method Survey on Kähler hyperbolic manifolds and bounded symmetric domains
result Proposed several open problems
Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds
problem Establishing an equivalence between diffeological substacks of Higgs and flat bundles
method Using diffeological moduli stacks
result Shows equivalence of categories between semistable Higgs bundles and flat bundles
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy Study third order Einstein deformations for Kähler-Einstein metrics on compact manifolds.
problem Existence of non-trivial Einstein deformations of Kähler metrics.
method Explicitly determined the obstruction to third order Einstein deformation and formulated it in terms of polynomial identities.
result Third order integrability for the Einstein equation is equivalent to Maurer-Cartan type equations and polynomial identities.
Sharp inequalities for Kähler manifolds' systolic invariants are established.
problem Understanding systolic invariants of Kähler manifolds.
method Analyzing metrics with positive scalar curvature on Kähler manifolds and their products.
result Bounds for systolic invariants attain equality for specific manifolds.
Study small eigenvalues on Kähler manifolds degenerating with induced metrics.
problem Analyzing the asymptotic rates of small eigenvalues on degenerate Kähler manifolds.
method Combining Li's uniform Skoda inequality with Monge-Ampère equations.
result Established exact asymptotic rates for small eigenvalues.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
The paper examines ellipticity of specific equations on vector bundles.
problem Investigating ellipticity of vector bundle versions of Monge-Ampère equations.
method Analyzing continuity paths and preserving ellipticity of equations.
result Not all equations preserve ellipticity along continuity paths, but σ2 does. Study solves complex equation on specific types of manifolds.
problem Solving complex Monge-Ampère equation on Kähler manifolds.
method Flow-based arguments to establish existence of smooth solutions.
result Existence of smooth solutions under decreasing right-hand side.
The paper proves rational connectedness for certain Kähler manifolds.
problem Rational connectedness of compact Kähler manifolds.
method Uniform weak RC-positivity of the tangent bundle.
result Compact Kähler manifolds with uniformly weakly RC-positive tangent bundles are projective and rationally connected.
Study b-divisors on Kähler manifolds linking them to currents.
problem Intersection theory of b-divisors on Kähler manifolds.
method Established correspondence between closed positive currents and nef b-divisors.
result Intersection theory of nef b-divisors answered.
Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.
problem Existence and uniqueness of bounded solutions to complex Monge-Ampère flows.
method Proved existence and uniqueness of bounded solutions with specific conditions on the right-hand side.
result Existence and uniqueness of bounded solutions to the complex Monge-Ampère flow on compact Kähler manifolds.
Paper solves tensor problem for holomorphic vector bundles.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors.
method New comparison theorem for Hermitian-Yang-Mills tensors.
result Existence of unique smooth Hermitian metric for any positive-definite tensor.
New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.
problem Proving Sobolev inequalities on Kähler manifolds.
method Using a classical Bidaut-Véron and Véron approach.
result Proved new Sobolev inequalities on compact Kähler manifolds with positive Ricci curvature.
Extends Higgs fields theory to complex fiber bundles.
problem Characterize nonlinear flat connections on complex fiber bundles.
method Representation of extension class by curvature, nonlinear Higgs bundles, and nonabelian Hodge structure.
result Established a faithful functor from nonlinear flat bundles to nonlinear Higgs bundles.
The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.
problem Conditions for Kähler and Riemannian manifolds to be simply connected.
method Spectral positivity assumptions for Kähler manifolds and a specific spectral positivity assumption for Riemannian manifolds.
result Compact Kähler manifolds and Riemannian manifolds under the specified spectral positivity assumptions are simply connected.
The L2-∂∂-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.
problem Extending the L2-∂∂-Lemma to non-compact Kähler manifolds. method Proving the L2-∂∂-Lemma on complete Kähler manifolds with a gap in the spectrum. result The L2-∂∂-Lemma is generalized to complete Kähler manifolds. Paper bounds Kähler manifolds' diameter using Orlicz spaces and complex Monge-Ampère equations.
problem Establishing diameter bounds for Kähler manifolds in Orlicz spaces.
method Proving a priori estimates for solutions of complex Monge-Ampère equations in Orlicz spaces using Kołodziej's and Guo-Phong-Tong-Wang's approaches.
result Uniform estimates for Green's function and its gradient for Kähler metrics.
The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
problem Diameter rigidity of Kähler manifolds with positive holomorphic sectional curvature.
method Establishing diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature.
result Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
Formula found for energy slope in complex geometry.
problem Calculating the asymptotic slope of a K-energy.
method Established a formula for the asymptotic slope.
result Found a formula for the asymptotic slope of α-K-energy.
Uniform RC-positivity results for direct image bundles.
problem Understanding the relation between rational connectedness and RC-positivity.
method Analyzing vector bundles and their direct images, using weak RC-positivity as a starting point.
result Uniform RC-positivity of direct image bundles under weak RC-positivity conditions.
Study Clairaut semi-slant/hemi-slant maps to Kähler manifolds.
problem Understanding Riemannian maps between specific types of manifolds.
method Analyzing Clairaut semi-slant/hemi-slant maps to Kähler manifolds.
result New insights into the properties of these maps.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
The paper studies the twisted Calabi flow on Kähler manifolds.
problem Analyzing the behavior of the twisted Calabi flow on compact Kähler manifolds.
method Establishing convexity, proving short-time existence, and demonstrating stability of the flow.
result The stability of the twisted Calabi flow near twisted constant scalar curvature Kähler metrics.
New linking numbers link complex cycles to Calabi-Yau 3-folds.
problem Understanding complex analytic cycles and their Massey products.
method Relating ABC Massey products to holomorphic linking numbers.
result Constructed a family of Calabi-Yau 3-folds with non-vanishing Massey products.
The study computes Bergman kernels and point process asymptotics on Kähler manifolds.
problem Computing asymptotics of Bergman kernels and point process distributions on Kähler manifolds.
method Equivariant and partial Bergman kernels, determinantal point processes, asymptotic analysis.
result The distribution of linear statistics converges to a centered normal variable with specific variances.
The paper quantizes Kähler manifolds using differential operators.
problem Quantizing classical observables on Kähler manifolds as differential operators.
method Constructing higher-order differential operators using Fedosov-type constructions and proving asymptotic equivalence to Berezin-Toeplitz operators.
result Holomorphic differential operators are precisely those that arise as Berezin-Toeplitz operators for quantizable functions.
Study the intersection form on Kähler manifolds of dimension 4 and above.
problem Understand the intersection form on higher-dimensional Kähler manifolds.
method Investigate fundamental properties and applications to birational geometry.
result Present open problems in the relationship between birational invariants and topological invariants.
SU(n)-structures derived from Kähler manifolds with torus actions.
problem Understanding SU(n)-structures on quotient spaces.
method Using Kähler manifolds with Hamiltonian actions of tori.
result Symplectic quotients inherit SU(n)-structures under certain conditions.
New curvature condition helps characterize Kähler manifolds.
problem Characterize compact Kähler manifolds with specific curvature properties.
method Introduce and utilize 2−positive bisectional curvature condition. result Deduce characterization theorem for manifolds with 2−positive bisectional curvature. Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.
problem Analyzing properties of Kähler manifolds with nonnegative bisectional curvature.
method Established precise relations among minimal degree, volume growth, and scalar curvature decay.
result Unified understanding of Kähler-Ricci flow through polynomial growth holomorphic functions.
Study on m-positivity in Kähler manifolds with new Monge-Ampère-type equation.
problem Exploring m-positivity in Kähler manifolds and its geometric applications. method Generalizing pseudo-effective and big Bott-Chern cohomology classes, proposing a new Monge-Ampère-type equation.
result Proof of a form of uniqueness for solutions of the Monge-Ampère-type equation.
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.
Strong formal properties for toric and homogeneous Kähler manifolds.
problem Understanding formal properties of Kähler manifolds.
method Analyzing rationally and strongly formal properties of toric and homogeneous Kähler manifolds.
result Toric and homogeneous Kähler manifolds are both rationally and strongly formal.
Optimizes dimension estimate for holomorphic functions on Kähler manifolds.
problem Determining the optimal dimension for holomorphic functions with polynomial growth.
method Analyzes Kähler manifolds with non-negative holomorphic bisectional curvature.
result Identifies the specific gap and optimal dimension for maximal volume growth.
Calabi flow extended with bounded curvature integrals.
problem Extending Calabi flow on compact Kähler manifolds.
method Using bounded Lp scalar curvature integrals. result Calabi flow can be extended under certain curvature conditions.
Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.
problem Rigidity of Kähler manifolds with nonnegative Ricci curvature.
method Analysis of Kähler manifolds with specific properties.
result Complete noncompact Kähler surface with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay is biholomorphic to the resolution of an affine algebraic variety.
New theorems compare Laplacian on Kähler manifolds.
problem Comparing Laplacian on Kähler manifolds.
method New curvature notions between Ricci and holomorphic bisectional curvatures.
result Established Laplacian comparison theorems and rigidity theorems.
New algebraic structure derived from Kähler manifolds.
problem Understanding algebraic structures on differential forms.
method Introducing L∞[1] R-algebras and proving linearization theorems. result Induced L∞[1] R-algebra structures on Γ(L) are linearizable under certain conditions. Researchers introduce new energies to study constant scalar curvature metrics.
problem Understanding constant scalar curvature metrics on compact Kähler manifolds.
method Introduced a family of Kβ energies using Berman's quantization and intersection theory. Combined with non-Archimedean techniques, provided a uniform Yau-Tian-Donaldson correspondence. result Uniform Yau-Tian-Donaldson correspondence characterizes the existence of a unique constant scalar curvature Kähler metric.
The paper proves a unique cscK metric for uniformly K-stable Kähler manifolds.
problem Finding a unique cscK metric for uniformly K-stable Kähler manifolds.
method Developed non-Archimedean pluripotential theory, used valuative criterion, and extended Calabi-Yau Theorem.
result Proved existence and uniqueness of cscK metrics for uniformly K-stable Kähler manifolds.
Extends Hodge theory to nearly Kähler manifolds of arbitrary dimensions.
problem Generalize Hodge-theoretic results to nearly Kähler manifolds of arbitrary dimensions.
method Apply Hodge theory to nearly Kähler manifolds of arbitrary dimensions, relating Hodge numbers to Betti numbers.
result Hodge numbers of compact nearly Kähler manifolds are related to Betti numbers in the same way as on a compact Kähler manifold.
New formulas with quadratic curvature terms on Kähler manifolds for Hodge number estimates.
problem Estimating Hodge numbers under weak curvature conditions.
method Established new Bochner-Kodaira formulas with quadratic curvature terms.
result Derivation of Weitzenböck-Bochner-Kodaira formulas with quadratic curvature terms on compact Kähler manifolds.
Two remarks on curvature properties of Kähler manifolds.
problem Curvature properties of Kähler manifolds.
method Analyzing semi-positive holomorphic sectional curvature and quasi-negative k-Ricci curvature. result For semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration equals the number of non-truly-flat directions. For quasi-negative k-Ricci curvature, the canonical bundle is ample. This paper improves Green's function estimates for compact Kähler manifolds.
problem Estimating Green's function norms for compact Kähler manifolds without curvature bounds.
method Proves an improved integral estimate for Green's function under volume density condition.
result Improved global geometric estimates, including eigenvalue bounds for Laplacian.
The paper studies curvature properties of direct image bundles.
problem Investigating curvature properties of direct image bundles.
method Using subharmonic metrics and mean curvature analysis.
result Direct image bundles carry metrics with positive mean curvature.