arXiv research

Abstract: Almost Kähler Hodge numbers vary with metric choices.

Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.

Study of generalized almost-Kähler-Ricci solitons and their implications.

The paper generalizes inequalities on almost Kähler manifolds.

The study finds conditions for almost-Kähler 4-manifolds to be Kähler.

Study classifies special 4D shapes with certain curvature.

Paper introduces a new operator and solves equations on higher-dimensional almost Kähler manifolds.

The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concent…

Formula derived for mass of almost Kähler manifolds, extending previous results.

Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.

New metrics constructed dual to specific wave-like geometries.

The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.

Study primitive decompositions for harmonic forms on almost Kähler manifolds.

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…

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 …

The paper proves a generalized Lefschetz duality for a specific type of manifold.

The paper calculates the Chern-Ricci form for a twisted almost Kähler structure.

New metrics link Kähler and pp-wave spacetimes.

We construct an example of Ricci-flat almost-Kähler non-Kähler structure in four dimensions.

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kahler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermit…

We show the existence of strictly almost-Kahler anti-self-dual metrics on certain 4-manifolds by deforming scalar-flat Kahler metrics. On the other hand, we prove the non-existence of such metrics on certain other 4-manifolds by means of Seiberg-Witten theory. In the process, we provide a simple new proof of the fact t…

Researchers decompose harmonic forms on specific types of manifolds.

Among other results, a compact almost Kähler manifold is proved to be Kähler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order covariant derivatives are Hermitian. Moreover, it is shown that there are no compact a…

We prove that any $4$-dimensional almost-Kähler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is Kähler.

It is proved that if an almost Kähler manifold of dimension greater or equal to 8 is of pointwise constant antiholomorphic sectional curvature, then it is a complex space form.

The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups $(g, J, Ω)$ such that the Ricci tensor is $J$-invariant.

Study proves Hirzebruch genus inequality for almost Kähler manifolds with negative curvature.

Donaldson-Friedman constructed anti-self-dual classes on $K3\#3\overline{\mathbb{CP}_{2}}$ using twistor space. We show that some of these conformal classes have almost-Kähler representatives.

We survey some recent results and constructions of almost-Kähler manifolds whose curvature tensors have certain algebraic symmetries. This is an updated and corrected version of the (to be) published manuscript.

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…

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.

This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kähler manifolds obtained as smoothings of a constant scalar curvature Kähler orbifold, with $A_1$ singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we sho…

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…

The paper develops $L^2$-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.

Decomposes harmonic forms on almost Kähler manifolds, revealing non-trivial structure.

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 …

An almost Kähler structure on a symplectic manifold $(N, ω)$ consists of a Riemannian metric $g$ and an almost complex structure $J$ such that the symplectic form $ω$ satisfies $ω(\cdot, \cdot)=g(J(\cdot), \cdot)$. Any symplectic manifold admits an almost Kähler structure and we refer to $(N, ω, g, J)$ as an almost Käh…

Paper shows examples of almost Kähler manifolds satisfying Hard Lefschetz but not Betti-Hodge equality.

An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive …

We show that any non-Kahler, almost Kahler 4-manifold for which both the Ricci and the Weyl curvatures have the same algebraic symmetries as they have for a Kahler metric is locally isometric to the (only) proper 3-symmetric 4-dimensional space described by O. Kowalski.

The paper studies invariants of almost complex and almost Kähler manifolds, proving properties and relationships between them.

Formulae for special almost-complex structures on Vogan diagrams.

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard…

The paper revisits and analyzes the tmd-operator in almost Kähler manifolds.

Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.

Study $L^{2}$-harmonic forms on almost Kähler manifolds, extending vanishing theorems.

Given a non compact semisimple Lie group $G$ we describe all homogeneous spaces $G/L$ carrying an invariant almost Kähler structure $(ω,J)$. When $L$ is abelian and $G$ is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy $ρ= λω$ for some $λ\in\mathbb R$, where $ρ$ is the Ricci…

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M, [[ω]])$ depending on symplectic deformation equivalence class $[[ω]]$. We first prove tha…