Study cohomological and metric properties of non-Kähler complex manifolds.
problem Understanding cohomological invariants and metrics of non-Kähler complex manifolds.
method Partial account of problems through cohomological and metric properties.
result Partial insights into cohomological and metric properties of non-Kähler manifolds.
The abstract discusses how generalized Calabi-Gray manifolds help solve non-Kähler geometry questions.
problem Non-Kähler complex manifolds with explicit geometry.
method Demonstrates the use of generalized Calabi-Gray manifolds to address non-Kähler geometry questions.
result Generalized Calabi-Gray manifolds provide a framework to answer specific non-Kähler geometry problems.
Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.
problem Applying Kobayashi-Hitchin correspondence to non-Kähler manifolds.
method Continuity method for vortex equation, Kobayashi-Hitchin correspondence for holomorphic pairs.
result Proved solvability of vortex equation on holomorphic vector bundles over compact Hermitian manifolds.
Survey on metrics on non-Kähler complex manifolds.
problem Existence and properties of Hermitian metrics.
method Analytic study of Chern connection and related flows.
result Generalizations of Kähler-Einstein condition.
Identifies special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
problem Determining special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
method Algebraic equations, invariant distributions, deformation theory, SYZ mirror symmetry.
result Existence of topologically distinct SLags and non-Kähler SYZ mirrors.
Study on integrals on non-Kähler manifolds, focusing on special metrics.
problem Understanding Morse-type integrals on non-Kähler manifolds.
method Conjecture and proof for semipositive classes and special metrics.
result The conjecture holds for semipositive classes and certain special metrics.
Explains old and new non-Kähler metrics on compact manifolds.
problem Coexistence and behavior of non-Kähler metrics on compact complex manifolds.
method Focus on specific types of non-Kähler metrics, describe constructions and properties.
result Mechanism for constructing nilmanifolds with desired metrics.
Study on complex variation of Hodge structures for non-Kähler manifolds.
problem Understanding complex variation of Hodge structures for non-Kähler manifolds.
method Analyzes holomorphic families of compact complex manifolds with specific cohomology properties.
result Period map is holomorphic and transversal under given conditions.
The paper solves equations for Higgs bundles on non-Kähler manifolds.
problem Analytically stable Higgs bundles on non-Kähler manifolds.
method Solving the Hermitian-Einstein equation on analytically stable Higgs bundles under specific conditions.
result Solutions to the Hermitian-Einstein equation for analytically stable Higgs bundles on non-Kähler manifolds.
Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau manifolds with high Betti numbers.
method Smoothing normal crossing varieties to create K3 fibrations over smooth projective varieties.
result Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers.
Study on existence of balanced metrics on non-Kähler manifolds.
problem Existence of balanced metrics on non-Kähler complex manifolds.
method Analyzes obstructions and constructs examples, focusing on compact quotients of Lie groups.
result Proves non-existence on certain non-Kähler complex parallelizable manifolds and solvmanifolds.
Study non-Kahler symplectic manifolds, proving deformation and Torelli theorems.
problem Topology and deformation theory of non-Kahler holomorphically symplectic manifolds.
method Investigation of topology and deformation theory, proving local Torelli theorem and Fujiki formula.
result Holomorphically symplectic deformations of BG-manifolds are unobstructed, and the period map is locally a diffeomorphism.
The paper computes non-trivial triple Massey products on specific non-Kähler solvmanifolds.
problem Computing non-trivial triple Massey products on non-Kähler solvmanifolds.
method Explicit computations on specific non-Kähler solvmanifolds.
result Non-Kähler solvmanifolds have non-vanishing triple ABC-Massey products. Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
problem Understanding the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Monotonicity of eigenvalues of mean curvature, convergence to geometric invariants.
result Eigenvalues of mean curvature converge to geometric invariants in the Gauduchon case.
New non-Kähler manifolds constructed with specific properties.
problem Creating non-Kähler manifolds with cohomological properties.
method Four constructions using quotient singularities and spectral sequences.
result Disproved a conjecture by Popovici.
In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kähler manifolds to the complete non-Kähler case.
New non-Kähler examples of generalized Kähler manifolds constructed via mapping tori.
problem Constructing new non-Kähler generalized Kähler manifolds.
method Starting from a 3-torus and a compact Kähler manifold, constructing via mapping tori.
result Obtained new non-Kähler examples and recovered known examples of generalized Kähler solvmanifolds.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Constructing Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Using t-Gauduchon metrics on principal torus bundles over rational homogeneous varieties. result Examples of new metrics on non-Kähler Calabi-Yau manifolds.
We consider a complex Plateau problem for strongly pseudoconvex contours in non Kähler manifolds. A positive solution in the case of manifolds carrying a pluriclosed Hermitian metric forms is given. For the general case we propose a conjecture.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Finding Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Constructing new metrics on specific types of non-Kähler Calabi-Yau manifolds.
result Examples of Levi-Civita Ricci-flat metrics on various non-Kähler Calabi-Yau manifolds.
New complex non-Kähler manifolds with specific properties are constructed.
problem Constructing complex non-Kähler manifolds with special properties.
method Using families of compact solvmanifolds and properties of the ∂∂ˉ-Lemma. result Provided families of compact (n+1)-dimensional complex non-Kähler manifolds with specific properties. Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.
problem Computing Dolbeault cohomology for a new class of non-Kähler manifolds.
method Computed Dolbeault cohomology using the Hodge decomposition.
result Endo-Pajitnov manifolds satisfy the Hodge decomposition at the level of dimensions.
We produce new examples of non-Kähler complete expanding gradient Ricci solitons on trivial vector bundles over a product of Einstein manifolds with positive scalar curvature.
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were constructed independently by M. Atiyah, A. Blanchard and E. Calabi in the 1950's. In the 1980's V. Tsanov gav…
Study on Hermitian metrics and curvature properties of complex manifolds.
problem Analyzing curvature properties of Hermitian metrics on complex manifolds.
method Derivation of formulae and proofs for Chern-Ricci curvatures and holomorphic sectional curvatures.
result Examples of metrics with specific curvature properties.
New proof of Aubin-Yau theorem for complex non-Kähler manifolds.
problem Transversally Kähler foliations as a generalization of Kähler manifolds.
method Adapting classical Aubin-Yau methods to transversally Kähler case under homological orientability condition.
result New, simpler proof of Vaisman Aubin-Yau theorem.
In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kähler manifold (X,g) with the Hermitian metric g satisfying the Gauduchon and Astheno-Kähler condition.
The paper classifies 4D Ricci-flat ALE manifolds and their properties.
problem Characterizing 4D Ricci-flat ALE manifolds and their properties.
method Analyzing the correspondence between ALE gravitational instantons and Kähler orbifolds, and proving properties of these structures.
result There is a one-to-one correspondence between Hermitian non-Kähler ALE gravitational instantons and Bach-flat Kähler orbifolds in 2D.
The paper finds conditions for smooth curves of balanced metrics in Hermitian non-Kähler settings.
problem Balanced metrics under deformations of complex structures.
method Necessary conditions for smooth curves of balanced metrics.
result Existence of smooth curves of balanced metrics starting from a fixed balanced metric.
New metrics found on non-Kähler complex manifolds.
problem Finding metrics on non-Kähler complex manifolds.
method Reinterpretation of Bismut Hermitian-Einstein condition and associated holomorphic Courant algebroid.
result Infinitely many non-Kähler manifolds without Bismut Hermitian-Einstein metrics.
In this thesis, we study cohomological properties of non-Kähler manifolds. In particular, we are concerned in investigating the cohomology of compact (almost-)complex manifolds, and of manifolds endowed with special structures, e.g., symplectic structures, D-complex structures in the sense of F. R. Harvey and H. B. Law…
Study properties of holomorphic p-contact manifolds, including non-Kähler hyperbolicity and deformations.
problem Characterize the geometric and algebraic properties of holomorphic p-contact manifolds. method Explores non-Kähler hyperbolicity, differential calculus, and p-contact deformations, proving unobstructedness theorems. result Proves a Bogomolov-Tian-Todorov-type unobstructedness theorem for p-contact deformations up to order two. We study complex non-Kähler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kähler metrics, too.
Study of t-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.
problem Investigating the t-Gauduchon Ricci-flat condition on non-Kähler manifolds. method Chern-Ricci flow approach, examples of non-Kähler Calabi-Yau manifolds, and geometric flow analysis.
result Examples of Chern-Ricci flow on non-Kähler Calabi-Yau manifolds that do not preserve the t-Gauduchon Ricci-flat condition. The paper studies non-Kähler LVMB manifolds and their metrics.
problem Existence and properties of special metrics on LVMB manifolds.
method Analyzes LVMB manifolds with holomorphic bundle structures over toric bases, using combinatorial data.
result Constructs a new example of a balanced metric on an LVMB manifold.
In this note we present, for every n≥4, a non-Kähler compact complex manifold X of complex dimension n admitting a balanced metric and an astheno-Kähler metric which is in addition k-th Gauduchon for any 1≤k≤n−1.
Study classifies gravitational instantons based on their asymptotic geometry.
problem Classifying gravitational instantons based on their asymptotic properties.
method Investigation of asymptotic geometry of Hermitian non-Kähler Ricci-flat metrics.
result All Hermitian non-Kähler gravitational instantons can be compactified to log del Pezzo surfaces.
Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.
problem Understanding special non-Kähler metrics on complex nilmanifolds.
method Analyzing locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced metrics on compact complex manifolds. result Compact complex nilmanifolds with balanced or k-Gauduchon metrics are tori, extending previous results. The article finds non-Abelian DT-instantons on non-Kähler manifolds.
problem Finding solutions to DT-instanton equations on non-Kähler manifolds.
method Constructing examples of DT-instantons for homogeneous almost Hermitian structures on the manifold of full flags in C^3.
result Explicit classification and phenomena of reducibility and disappearance of DT-instantons.
Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.
problem Investigate positivity of Bismut curvature in non-Kähler manifolds.
method Analyze Bismut-Griffiths-positivity under Hermitian curvature flows.
result Identify HCFs that do not preserve Bismut-Griffiths-positivity.
We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern cla…
Study Hermitian curvature flow on complex surfaces, finding singularities and limits.
problem Characterize and analyze Hermitian curvature flow on complex surfaces.
method Case-by-case analysis of flow on each complex model geometry.
result First example of a compact complex non-Kähler manifold with finite time singularity.
We propose a new construction of compact non-Kähler Calabi-Yau manifolds with balanced metrics and study the Strominger system on them. In particular, we obtain explicit solutions to the Strominger system with degeneracies on Σg×T4, where Σg is an immersed minimal surface of genus g≥3 in T3.
Study on new hyperbolicity notions for non-Kähler manifolds and their deformations.
problem Analyzing new hyperbolicity notions for non-Kähler complex manifolds.
method Introducing and analyzing two new notions of hyperbolicity for compact complex non-Kähler manifolds, and studying their behavior under smooth modifications.
result Established openness results for p-HS hyperbolicity and p-Kähler hyperbolicity under holomorphic deformations. We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold M=G2/T. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus G2/T turns out to be the first …
Study on non-Kähler Calabi-Yau geometries on 3-folds with constraints.
problem Characterizing non-Kähler Calabi-Yau geometries on threefolds.
method Symmetry reduction and scalar PDE for Kähler structure.
result Equality of Bott-Chern number hBC1,1≥2 with specific conditions. This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.
problem Calculating characteristic class relations in Dolbeault cohomology.
method Representation theory of structure group, without metric or connection.
result First results on Chern-Simons invariants of Cartan geometries.