We study the J-invariant and J-anti-invariant cohomological subgroups of the de Rham cohomology of a compact manifold M endowed with an almost-Kähler structure (J, ω, g). In particular, almost-Kähler manifolds satisfying a Lefschetz type property, and solvmanifolds endowed with left-invariant almost-complex structures …
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48 results for “almost-Kahler structures”
The paper calculates the Chern-Ricci form for a twisted almost Kähler structure.
problem Calculating the Chern-Ricci form for a specific type of almost Kähler manifold.
method Using a twisted almost Kähler structure, the paper derives an explicit formula for the local connection 1-form and calculates the Chern-Ricci form.
result An explicit formula for the local connection 1-form and the Chern-Ricci form of a twisted almost Kähler structure are provided.
We construct an example of Ricci-flat almost-Kähler non-Kähler structure in four dimensions.
The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.
problem Analyzing harmonic forms and Hodge decomposition on almost Kähler manifolds.
method Using Hodge decomposition and the Hard Lefschetz Condition to study almost Kähler manifolds.
result The spaces of harmonic forms have the Hodge decomposition and the Hard Lefschetz Condition is satisfied.
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 finds conditions for the existence of extremal toric almost Kähler metrics.
problem Conditions for the existence of extremal toric almost Kähler metrics.
method Observation and application of recent results on K-stability and Abreu equation.
result Existence of extremal toric almost Kähler structures is equivalent to uniform K-stability.
Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler metrics compatible with t…
The study classifies Chern-Einstein homogeneous almost Kähler manifolds.
problem Classifying Chern-Einstein homogeneous almost Kähler manifolds.
method Description of homogeneous spaces with invariant almost Kähler structures and classification based on Chern-Einstein condition.
result Classification of Chern-Einstein homogeneous almost Kähler manifolds for specific Lie groups.
We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gr…
Formula derived for mass of almost Kähler manifolds, extending previous results.
problem Calculating mass in almost Kähler geometry.
method Spin adaptation of Witten's proof, extending previous complex-geometric methods.
result Explicit formula for ADM mass in terms of Hermitian scalar curvature and topological data.
The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups such that the Ricci tensor is -invariant.
An almost Kähler structure on a symplectic manifold consists of a Riemannian metric and an almost complex structure such that the symplectic form satisfies . Any symplectic manifold admits an almost Kähler structure and we refer to as an almost Käh…
Formulae for special almost-complex structures on Vogan diagrams.
problem Existence of special almost-complex structures on almost-Kähler manifolds.
method Combinatorics of Vogan diagrams for classical semisimple Lie groups.
result Explicit formulae for special almost-complex structures.
We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.
The paper explores geometric and topological properties of almost Kähler manifolds using harmonic theory.
problem Understanding the geometric and topological aspects of almost Kähler manifolds.
method Deduction of geometric and topological consequences from extended Kähler identities for compact almost Kähler manifolds.
result Generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition for -harmonic forms.
We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence Kähler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the con…
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 …
Abstract: Almost Kähler Hodge numbers vary with metric choices.
problem Variation of almost Kähler Hodge numbers with metrics.
method Analysis of almost complex Hodge numbers under different almost Kähler metrics.
result The almost Kähler Hodge number varies with metric choices.
On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of…
Study proves Hirzebruch genus inequality for almost Kähler manifolds with negative curvature.
problem Proving Hirzebruch genus inequality for almost Kähler manifolds with negative sectional curvature.
method Combining \(L^2\)-estimates for harmonic forms, refined vanishing theorem, and Atiyah's \(L^2\)-index theorem.
result Components of Hirzebruch genus satisfy inequality \((-1)^{n-p}χ_{p}(X) \geq 1\) for all \(p\).
Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.
problem Understanding Weyl curvature pinching on 4-manifolds.
method Analyzing harmonic and pinched self-dual Weyl curvature, proving anti-self-duality.
result Proves anti-self-duality for compact 4-manifolds with pinched self-dual Weyl curvature.
Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.
problem Classifying self-dual almost-Kähler four-manifolds.
method Using LeBrun's result and properties of Ricci tensor, the authors classify manifolds of different types.
result Any self-dual almost-Kähler metric on is the Fubini-Study metric up to rescaling.
Study of generalized almost-Kähler-Ricci solitons and their implications.
problem Existence of first-Chern-Einstein almost-Kähler metrics on compact symplectic Fano manifolds.
method Generalization of Kähler-Ricci solitons to almost-Kähler setting, study of moment map and Lie algebra of holomorphic vector fields.
result Existence of generalized almost-Kähler-Ricci solitons as obstructions and implications for symplectic Fano manifolds.
For a closed smooth manifold admitting a symplectic structure, we define a smooth topological invariant using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce depending on symplectic deformation equivalence class . We first prove tha…
Paper shows examples of almost Kähler manifolds satisfying Hard Lefschetz but not Betti-Hodge equality.
problem Understanding the Hard Lefschetz condition in almost Kähler manifolds.
method Examples and counterexamples of compact almost Kähler manifolds.
result The Hard Lefschetz condition does not imply the equality between Betti and Hodge numbers in almost Kähler manifolds.
Study on Kodaira dimension of almost Kähler manifolds and their curvature.
problem Understanding Kodaira dimension in almost Kähler manifolds.
method Explicit computation and analysis of curvature of the canonical connection.
result Ricci curvature vanishes for members of the family of almost Kähler manifolds.
The paper generalizes inequalities on almost Kähler manifolds.
problem Generalizing inequalities on almost Kähler manifolds.
method Considered Donaldson gauge functional and twisted Aubin functionals.
result Generalized inequality between Aubin functionals.
Decomposes harmonic forms on almost Kähler manifolds, revealing non-trivial structure.
problem Primitive decomposition of harmonic forms on compact almost Kähler manifolds.
method Primitive decomposition of , Bott-Chern and Aeppli-harmonic -forms.
result Primitive components of harmonic forms are constants multiples of .
The study finds conditions for almost-Kähler 4-manifolds to be Kähler.
problem Conditions for almost-Kähler 4-manifolds to be Kähler.
method Analyzes harmonic self-dual Weyl curvature and constant scalar curvature.
result Compact almost-Kähler 4-manifolds with harmonic self-dual Weyl curvature and constant scalar curvature are Kähler if .
The paper proves the existence of metrics with constant curvature on smoothings of orbifolds with singularities.
problem Existence of metrics with constant curvature on smoothings of orbifolds with singularities.
method Construction of almost-Kähler structures with constant Hermitian curvature on smoothings of constant scalar curvature Kähler orbifolds with singularities.
result Almost-Kähler smoothings of constant scalar curvature Kähler orbifolds with singularities admit almost-Kähler structures of constant Hermitian curvature.
Study extends Kähler-Ricci flow to symplectic manifolds.
problem Extend Kähler-Ricci flow to symplectic manifolds.
method Establish new formulas for canonical quantities and characterize fixed points.
result Extended characterization of fixed points.
Almost-Kähler metrics found on a specific 4-manifold.
problem Finding anti-self-dual metrics on a specific 4-manifold.
method Using twistor space to construct anti-self-dual classes, then showing existence of almost-Kähler representatives.
result Some anti-self-dual classes on have almost-Kähler representatives.
Study classifies special 4D shapes with certain curvature.
problem Classifying specific types of 4D shapes.
method Classifying compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
result Classified compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.
problem Understanding the gap between de Rham and symplectic-Bott-Chern harmonic forms on specific almost-Kähler manifolds.
method Analyzing the space of de Rham harmonic forms and symplectic-Bott-Chern harmonic forms on closed almost-Kähler manifolds.
result The second non-HLC degree measures the gap between de Rham and symplectic-Bott-Chern harmonic forms.
The paper classifies orbit closures of symplectic Lie algebras.
problem Classifying orbit closures of symplectic Lie algebras under the action of .
method Analyzing the natural action of on the set of 4-dimensional Lie algebras with symplectic structures.
result A complete classification of orbit closures of 4-dimensional symplectic Lie algebras.
Paper introduces a new operator and solves equations on higher-dimensional almost Kähler manifolds.
problem Investigate -problem and generalized Monge-Ampère equation on almost Kähler manifolds.
method Introduce operator and use it to solve equations.
result Established a uniqueness up to a constant and local existence theorem for the generalized Monge-Ampère equation.
The study constructs a Kähler structure on non-elliptic symplectic manifolds.
problem Defining a Kähler structure on non-elliptic symplectic manifolds.
method Using almost Kähler structures and Lipschitz Kähler flat metrics, the study globally deforms and decomposes the almost Kähler structure.
result The signed Euler characteristic satisfies for non-elliptic symplectic manifolds.
The study constructs symplectic solvmanifolds satisfying the hard-Lefschetz condition.
problem Developing an analogue of Hodge theory for symplectic manifolds.
method Analyzing specific Lie algebras and their associated Lie groups, exploiting connections with Kneser graphs.
result Examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition are constructed.
New metrics constructed dual to specific wave-like geometries.
problem Constructing metrics dual to general plane-fronted wave Lorentzian metrics.
method Explains construction of extremal and non-Kähler almost-Kähler metrics.
result Constructs canonical almost-Kähler metrics dual to general plane-fronted wave Lorentzian metrics.
In the spirit of [10,2], we study the Calabi-Yau equation on -bundles over endowed with an invariant non-Lagrangian almost-Kähler structure showing that for -invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space $M…
Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.
problem Computing invariants on six-dimensional solvmanifolds.
method Computed almost-complex and almost-Hermitian invariants on families of solvmanifolds.
result Provides obstructions to symplectic structures on compact almost-complex manifolds.
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.
problem Classifying natural almost Hermitian structures on Lie groups with minimal conformal leaves.
method Analyzing Lie groups with a 2-dimensional conformal foliation and classifying structures based on Lie algebra properties.
result 16 multi-dimensional almost Kähler families, 18 integrable families, and 11 Kähler families were constructed.
We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non-compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with non-negative -scalar curvature are cokähler…
Two applications of Aleksandrov-Bakelman-Pucci estimate to Calabi-Yau equation.
problem Solvability of Calabi-Yau equation on symplectic four-manifolds.
method Aleksandrov-Bakelman-Pucci estimate applied to Calabi-Yau equation.
result Extensions of solvability results on specific manifolds.
The paper proves a generalized Lefschetz duality for a specific type of manifold.
problem Proving the hard Lefschetz duality for a new class of manifolds.
method Generalizing Kähler identities to prove the duality for locally conformally almost Kähler manifolds.
result The hard Lefschetz duality is established for locally conformally almost Kähler manifolds.
New metrics link Kähler and pp-wave spacetimes.
problem Connecting Kähler and pp-wave spacetimes.
method Constructing families of complete almost Kähler metrics by deforming pp-waves.
result Established a one-to-one correspondence between almost Kähler metrics and pp-wave spacetimes.
We study almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly ). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this condition forces the torsion of this connection to be parallel in directions ortho…