The paper describes a specific type of Kähler surfaces.
problem Understanding QCH Kähler surfaces with quasi-constant holomorphic sectional curvature.
method Detailed description of generalized Calabi type QCH Kähler surfaces.
result Detailed description of QCH Kähler surfaces of generalized Calabi type.
The paper proves bounded curvature for Kähler Ricci shrinker surfaces.
problem Understanding the curvature of Kähler Ricci shrinker surfaces.
method Proved bounded sectional curvature using earlier work.
result Complete classification of all Kähler Ricci shrinker surfaces.
Weakly Einstein Kähler surfaces are characterized and classified.
problem Characterizing and classifying weakly Einstein Kähler surfaces.
method Several conditions and constructions to characterize and classify weakly Einstein Kähler surfaces.
result Classification of weakly Einstein Kähler surfaces with specific properties and construction of new examples.
We give a mostly self-contained proof of the classification of non-Kahler surfaces based on Buchdahl-Lamari theorem. We also prove that all non-Kahler surfaces which are not of class VII are locally conformally Kahler.
The paper proves properties of Kähler surfaces with zero scalar curvature.
problem Characterizing Kähler surfaces with zero scalar curvature.
method Analyzing families of generalized Taub-Nut Kähler surfaces and Burn's metric.
result Proves that certain Kähler surfaces are QCH and of specific types.
The paper describes a new type of Kähler surfaces and their properties.
problem Characterizing and understanding Kähler surfaces of generalized orthotoric type.
method Introducing a distinguished orthonormal frame and integrating structure equations.
result A new way to classify and understand Kähler surfaces, especially orthotoric ones.
We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature Kähler metric for every Kähler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rat…
Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.
problem Characterizing log del Pezzo surfaces with specific geometric properties.
method Examining two classes of non-toric log del Pezzo surfaces and analyzing their geometric properties.
result Examples found that admit Kähler-Ricci solitons but not Sasaki-Einstein cone links.
The paper proves the existence of a special Kähler metric on a minimal ruled surface.
problem Existence of higher extremal Kähler metrics on a minimal ruled surface.
method Proved the existence of a higher extremal Kähler metric by showing it satisfies a specific equation and computed the top Bando-Futaki invariant.
result Higher extremal Kähler metrics exist on a minimal ruled surface, but not higher constant scalar curvature Kähler metrics.
Study connects two types of metrics on complex surfaces.
problem Interrelations between bi-Hermitian and locally conformally Kähler metrics.
method Analysis of metrics on complex surfaces.
result Found interconnections between bi-Hermitian and locally conformally Kähler metrics.
Proves a new inequality for certain complex surfaces.
problem Analyzes compact Kähler surfaces with positive scalar curvature.
method Uses properties of holomorphic maps and ruled surfaces.
result Proves a 2-systolic inequality on these surfaces.
Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.
problem Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
method Established optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities.
result Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.
problem Determining when Hermitian surfaces are Kählener.
method Used explicit identities linking Strominger-Bismut Ricci curvatures to torsion, and Chern number identities.
result Proves several Kählerness criteria for compact Hermitian surfaces.
The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.
problem Characterizing normal subgroups of Kähler groups.
method Analyzing embeddings and conjugation actions of surface groups and one-ended hyperbolic groups.
result Restrictions on normal subgroups of Kähler groups, including virtual direct products and surface group properties.
Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.
problem Existence and uniqueness of solutions to a generalized Monge-Ampère equation.
method Analyzes the equation's dependence on almost Kähler structure and proves Donaldson's conjecture.
result Proves Donaldson's conjecture for tamed almost complex 4-manifolds.
The study examines non-Kähler threefolds with specific metrics and finds they are quasi-bundles over surfaces.
problem Understanding non-Kähler threefolds with algebraic dimension two.
method Examining compact complex non-Kähler threefolds with locally conformally Kähler metrics and proving they are quasi-bundles over projective surfaces under certain assumptions.
result They are blown-up quasi-bundles over a projective surface.
The abstract conjectures and proves conditions for scalar-flat Kähler surfaces with specific tensor properties.
problem Conditions for scalar-flat Kähler surfaces with special tensor properties.
method Conjecture and prove in three special cases.
result The conjecture is proven in three special cases.
Study extends Yang-Mills energy gap to Kähler surfaces.
problem Extend Yang-Mills energy gap to Kähler surfaces.
method Extend L2-energy gap to compact Kähler surfaces with generic Kähler metrics. result All ASD connections on principal bundles over Kähler surfaces are irreducible.
Study on line bundle flow on Kähler surfaces converging to a singular solution.
problem Analyzing the mean curvature flow on Kähler surfaces.
method Investigates the flow under hypercritical phase and semipositivity conditions.
result The flow converges to a singular solution away from curves of negative self-intersection.
Survey on rational curves on complex surfaces, highlighting different approaches.
problem Existence of rational curves on complex surfaces.
method Classification of complex surfaces and systematic study of rational curves in each class.
result Highlighting the different approaches to study rational curves on complex surfaces.
Sharp estimate for 2-systole on Kähler surfaces with positive scalar curvature.
problem Estimating the 2-systole on compact Kähler surfaces with positive scalar curvature.
method Combining classification of positive scalar curvature Kähler surfaces with Stern's level set method adapted to Kähler setting.
result Proved the sharp estimate minXS(ω)⋅sys2(ω)≤12π. Researchers compute c-projective symmetry algebras for Kähler surfaces.
problem Understanding symmetries in Kähler surfaces.
method Defined and analyzed c-projective vector fields and computed their symmetries.
result Computed c-projective symmetry algebras for Kähler surfaces with essential c-projective vector fields.
Study on compact Kähler surfaces for sign-changing curvatures.
problem Prescribing sign-changing Chern scalar curvatures on compact Kähler surfaces.
method Established a Chen-Li type existence theorem and provided an alternative proof.
result Alternative proof of Ding-Liu's theorem on sign-changing Gaussian curvatures.
We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow converges smoothly to a singular Kahler metric away from a finite number of curves of n…
Study on Kähler metrics on ruled surfaces, proving existence and non-existence.
problem Existence and non-existence of Kähler metrics on minimal ruled surfaces.
method Analysis of twisted and coupled constant scalar curvature Kähler metrics.
result Bound for Chen-Cheng invariant on ruled surfaces.
New classifications of totally real surfaces in nearly Kähler C⁴.
problem Classifying totally real surfaces in nearly Kähler C⁴.
method Investigation under various assumptions, including extrinsic homogeneity, minimality, total umbilicity, and Codazzi-like properties.
result Multiple new examples of totally real surfaces, including parallel and non-parallel Codazzi-like cases.
This paper studies symplectic critical surfaces in Hermite surfaces.
problem Generalizing results about Kähler angle to the general case.
method Focuses on symplectic critical surfaces in Hermite surfaces.
result Provides a definition of symplectic critical surfaces in Hermite surfaces.
Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
It is shown that if a minimal ruled surface admits a Kähler Yamabe minimizer, then this metric must be generalized Kähler-Einstein and the underlying holomorphic vector bundle of the ruled surface must be quasi-stable.
In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…
Study extends gauge-theoretic invariant to higher-dimensional Kahler surfaces and calculates homotopy groups.
problem Calculate higher homotopy groups of symplectic mapping spaces on modified Kahler surfaces.
method Apply deformation of complex objects and gauge-theoretic techniques to closed Kahler surfaces.
result Show that even-dimensional higher homotopy groups of symplectic mapping spaces are infinitely generated.
Study on properties of special Kähler metrics and their interplay.
problem Properties and interplay of Strong Kähler with torsion and astheno-Kähler metrics.
method Analyzes families of astheno-Kähler nilmanifolds, studies complex blowups, and investigates interplay between metrics.
result Existence of astheno-Kähler metrics is not preserved by blowup, and some metrics are geometrically Bott-Chern formal.
Yamabe invariants of certain non-Kähler surfaces are zero.
problem Determining the sign of Yamabe invariants for non-Kähler surfaces.
method Analyzing Inoue surfaces and Kodaira surfaces, their blowups, and applying Seiberg-Witten theory.
result Yamabe invariants of Inoue surfaces and their blowups are all zero.
We study Kahler surfaces with harmonic anti-selfdual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kahler metrics, including Kahler-Einstein metrics and conformally Einstein Kahler metrics. We also extend s…
We present a construction that produces infinite classes of Kähler groups that arise as fundamental groups of fibres of maps to higher dimensional tori. Following the work of Delzant and Gromov, there is great interest in knowing which subgroups of direct products of surface groups are Kähler. We apply our construction…
We show that on smooth minimal surfaces of general type, the Kähler-Ricci flow starting at any initial Kähler metric converges in the Gromov-Hausdorff sense to a Kähler-Einstein orbifold surface. In particular, the diameter of the evolving metrics is uniformly bounded for all time and the Kähler-Ricci flow contracts al…
The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.
problem Analyzing limits of flows on Kähler surfaces and their convergence to solutions of equations.
method Using a property of limits of viscosity subsolutions.
result Proves convergence of flows to weak solutions of the Monge-Ampère equation.
New metrics found on Hirzebruch surfaces solve complex geometry questions.
problem Finding new conformally Kähler, Einstein-Maxwell metrics on Hirzebruch surfaces.
method Used special Killing potentials found by Futaki and Ono.
result Proved existence of new conformally Kähler, Einstein-Maxwell metrics.
The study finds surfaces with specific curvature properties are essentially known manifolds.
problem Investigating curvature properties on Kähler manifolds.
method Analyzing the curvature operator of the second kind on closed Kähler surfaces.
result Closed Kähler surfaces with six-positive curvature operator of the second kind are biholomorphic to CP2. In this paper we give new examples of QCH Kahler surfaces whose opposite almost Hermitian strucure is Hermitian and not locally conformally Kahler. In this way we give also a large class of examples of Hermitian surfaces with J-invariant Ricci tensor which are not l.c.k.
In this paper we give classification of Kähler surfaces of generalized Calabi type.
Study on K3 surfaces' collapsing and special Kähler structures.
problem Understanding the structure of K3 surfaces' collapsing metrics.
method Analyzing M2 and establishing connections to SKSs and Jacobian elliptic K3 surfaces. result Established a bijection between integral singular SKSs on P1 and Jacobian elliptic K3 surfaces. Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.
problem Verifying a conjecture about Kähler-Einstein edge metrics on Hirzebruch surfaces.
method Using the Calabi ansatz, constructing a family of metrics and studying their angle deformation.
result Verification of a conjecture and finding a rigid singularity.
The aim of this paper is to describe complex foliations on Kahler surfaces.
Study classifies totally geodesic surfaces in nearly Kähler space.
problem Classifying totally geodesic surfaces in nearly Kähler space.
method Detailed description of nearly Kähler space and classification of surfaces.
result Classification of totally geodesic almost complex surfaces.
The paper examines conditions for Lagrangian surfaces in Kähler-Einstein manifolds.
problem Characterizing Hamiltonian stationary Lagrangian surfaces with non-negative Gaussian curvature.
method Simple conditions and characterization of surfaces in Kähler-Einstein manifolds.
result Conditions for surfaces to have Euclidean factors or be fiber bundles over circles.
Let M be a Kähler surface and Σ be a closed symplectic surface which is smoothly immersed in M. Let α be the Kähler angle of Σ in M. We first deduce the Euler-Lagrange equation of the functional L=∫Σcosα1dμ in the class of symplectic surfaces. It is cos3αH=(J(J∇cosα)⊤)⊥, wh…
Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an ap…