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.
Survey on two non-Kähler geometry conjectures.
problem Constant holomorphic sectional curvature and Fino-Vezzoni conjectures in non-Kähler geometry.
method Survey and discussion of historical and recent developments.
result Discussion of conjectures without new results.
Constructs special Lagrangian 3-spheres in non-Kähler compact threefolds.
problem Understanding transitions between Kähler and non-Kähler geometries.
method Analyzes topological transitions of Calabi-Yau threefolds to construct special Lagrangian cycles.
result Special Lagrangian 3-spheres emerge from non-Kähler geometries, exchanging holomorphic 2-cycles for 3-cycles.
Study shows continuity of non-Kähler Calabi-Yau conifold transitions.
problem Understanding the geometry of Calabi-Yau conifold transitions.
method Use of balanced and Hermitian-Yang-Mills metrics to analyze conifold transitions.
result The conifold transition is continuous in the Gromov-Hausdorff topology.
Classifies non-Kahler surfaces and proves local conformal Kahler property.
problem Classifying non-Kahler surfaces and understanding their geometric properties.
method Proof based on Buchdahl-Lamari theorem.
result All non-Kahler surfaces that are not of class VII are locally conformally Kahler.
In recent years, Streets and Tian introduced a series of curvature flows to study non-Kähler geometry. In this paper, we study how to construct second order curvature flows in a uniform way, under some natural assumptions which holds in Streets and Tian's works. As a result, by classifying the lower order tensors, we c…
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 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. Survey on Strominger system and Ricci flow in non-Kähler geometry.
problem Non-Kähler geometry on Calabi-Yau threefolds.
method Discussion of various geometric flows and equations.
result Exploration of balanced metrics and solutions to the Strominger system.
Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.
problem Understanding non-Kähler complex threefolds and conifold transitions.
method Review of basics, description of topological features, survey of recent developments.
result Survey of recent developments on the geometrization of conifold transitions.
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.
New solutions found for complex geometry problem.
problem Solving the Hull-Strominger system on non-Kähler manifolds.
method Constructing solutions with a specific connection ansatz and using moduli spaces of sheaves.
result First T-dual solutions on compact non-Kähler manifolds with different topology.
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.
Flow of conformally balanced metrics converges on Kahler manifolds.
problem Non-Kahler geometry and its relation to string theory.
method Anomaly flow, convergence proof on Kahler manifolds.
result Flow converges on Kahler manifolds for suitable initial data.
Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.
problem Classical Euclidean Black Hole Uniqueness conjecture
method Analyzing the Chen-Teo gravitational instanton
result The Chen-Teo instanton is Hermitian and non-Kähler
Describes Teichmüller stack structure near Kähler points.
problem Understanding Teichmüller spaces/stacks near Kähler points.
method Analyzes local structure of Teichmüller stack near Kähler points.
result Affirmative answer to Catanese Kur=Teich question at generic Kähler points.
New non-Kähler 3-folds constructed via log conifold transitions.
problem Constructing new non-Kähler 3-folds from Fano threefold pairs.
method Defining log conifold transitions and studying their deformation theory.
result Local smoothings of nodes can be lifted to global first-order deformations.
In arXiv:1008.1018 it is shown that a given stable vector bundle V on a Calabi-Yau threefold X which satisfies c2(X)=c2(V) can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explic…
Study of complex 3-folds with vanishing Bismut Ricci form via special Kähler geometry.
problem Characterizing non-Kähler BHE 3-folds with vanishing Bismut Ricci form.
method Dimensional reduction to special Kähler geometry, solving 6th order PDE, momentum map interpretation.
result Characterization of Samelson locally homogeneous BHE 3-folds and construction of new non-Kähler BHE structures.
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.
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 explores non-Kähler SYZ mirrors for solvmanifolds, proving cohomological properties and constructing new mirror pairs.
problem Understanding non-Kähler SYZ mirrors for solvmanifolds and their cohomological properties.
method Investigates geometric and cohomological properties, proving relationships and constructing mirror pairs.
result Proves the Fourier-Mukai transform exchanges type-A and type-B cycles, and provides criteria for non-Kähler SYZ mirror pairs.
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.
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.
The paper shows bounds for a geometric flow related to Type IIB string theory.
problem Establishing derivative bounds for a geometric flow in non-Kähler geometry.
method Unified formulation of the flow with Ricci flow, proving bounds from metric and torsion 1-form uniform bounds.
result Derivative bounds follow from uniform metric and torsion 1-form bounds.
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.
We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
Unified Schwarz lemma in Kähler and Hermitian geometry.
problem Various forms of the Schwarz lemma in Kähler and Hermitian geometry.
method Introducing new curvatures to refine and elucidate the real bisectional curvature.
result Unified Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas.
On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler setting, one can seek to understand extensions of these characterizations to genera…
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.
We revisit the backgrounds of type IIB on manifolds with SU(4)-structure and discuss two sets of solutions arising from internal geometries that are complex and symplectic respectively. Both can be realized in terms of generalized complex geometry. We identify a map which relates the complex and symplectic supersymme…
The study shows how certain geometries can be mapped to simpler structures.
problem Understanding the structure of geometries with specific properties.
method Reframing known submersion results and applying them to new contexts.
result Holonomy decompositions and holomorphic submersions are derived for specific geometries.
New approach to Mirror Symmetry for non-Kähler manifolds, proving Iwasawa manifold is its own mirror.
problem Applying Mirror Symmetry to non-Kähler compact complex manifolds, particularly Iwasawa manifold.
method Developed a new approach to Mirror Symmetry, proving Iwasawa manifold is its own mirror.
result Iwasawa manifold is its own mirror dual, with clear geometric meaning for essential deformations.
The paper describes Hermitian non-Kähler structures on complex flag manifolds.
problem Understanding Hermitian non-Kähler structures on products of principal S¹-bundles.
method Using representation theory of complex simple Lie algebras and Cartan-Ehresmann connections.
result Explicit description of Hermitian non-Kähler manifolds and families of complex structures.
Proves Liouville theorem for non-Kähler manifolds.
problem Extending Liouville theorem to non-Kähler manifolds.
method Proves Liouville theorem for a specific class of complete Gauduchon manifolds.
result Generalizes Yau's result to non-Kähler manifolds.
Constructs non-Kähler Calabi-Yau 3-folds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau 3-folds with high Betti numbers.
method Smoothing normal crossing varieties with trivial dualizing sheaves.
result Constructs non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers.
Paper solves open problem in complex Finsler geometry.
problem Existence of non-Kähler weakly Kähler Finsler metrics.
method Constructs a family of weakly Kähler Finsler metrics.
result Proves uniformization theorem for unitary invariant complex Randers metrics.
The connection between Hitchin's stable forms and vector cross products is observed. Using this correspondence, we construct new examples of non-Kahler Calabi-Yau 3-folds and manifolds with G2-structure of class W3. We also generalize and refine results of Calabi and Gray in the paracomplex setting.
Study examines flow behavior on non-Kähler manifolds.
problem Asymptotic behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Analysis of Hermitian-Yang-Mills flow over compact non-Kähler manifolds with specific conditions.
result Characterized the limiting behavior of the flow.
The study examines cohomological aspects on complex and symplectic manifolds.
problem Understanding qualitative properties through quantitative cohomological information.
method Analysis of Bott-Chern and Aeppli cohomology groups.
result Established comparisons among cohomology group dimensions and proved the ∂∂-lemma and Hard-Lefschetz condition. Study connects Kähler and non-Kähler hyperbolicity.
problem Understanding hyperbolicity in Kähler and non-Kähler settings.
method Establishes a connection between SKT and balanced hyperbolicity.
result Highlights relationship with Kähler hyperbolicity.
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.
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.
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.
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.
Paper finds infinitely many smooth solutions to complex geometry problem.
problem Finding smooth solutions to the Strominger system on specific geometric manifolds.
method Constructing solutions on generalized Calabi-Gray manifolds.
result Explicit construction of infinitely many solutions.
Paper generalizes ddc-condition to non-Kähler complex manifolds.
problem Characterizing complex manifolds beyond Kähler geometry.
method Introduces and studies a new ddc-type condition for complex manifolds.
result The new condition is open under deformations and applies to various manifolds.