The paper classifies Hermitian manifolds with specific connection properties.
problem Classifying Hermitian manifolds with specific connection properties.
method Algebraic consideration of holonomy systems, structure theorems, and classification theorems.
result The universal cover of such Hermitian manifolds is the product of a complex Lie group and Hermitian symmetric spaces.
Study on hermitian Yang-Mills connections on blown-up manifolds.
problem Analyzing hermitian Yang-Mills connections on blown-up Kähler manifolds.
method Investigates connections for pullback vector bundles under specific conditions.
result Provides numerical criterion for convergence of hermitian Yang-Mills connections.
Study of Hermitian and Gauduchon connections on Lie groups with almost Hermitian structures.
problem Characterizing connections on Lie groups with almost Hermitian structures.
method Analyzing left-invariant Hermitian and Gauduchon connections on Lie groups equipped with almost Hermitian structures.
result Explicit formulas for torsion components and curvature of Gauduchon connections on Lie groups.
Study on special Hermitian manifolds with specific connection properties.
problem Compact Hermitian manifolds with a particular Chern connection.
method Proved structure theorems for manifolds with parallel torsion and curvature.
result Structure theorems for manifolds with these properties.
The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.
problem Understanding the relationship between Chern and Riemann sectional curvatures on Hermitian manifolds.
method Derivation of Chern sectional curvature expressions and subsequent results on Ricci and scalar curvatures.
result A Hermitian metric is Kähler if and only if its Riemann sectional curvature equals its Chern sectional curvature.
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
In this paper, following the constructions of N. R. O'Brian, J. H. Rawnsley and I. Vaisman, we define four almost Hermitian structures (up to conjugation) on the twistor space of a Hermitian surface by using canonical connections, including the Lichnerowicz connection and the Chern connection. We also study the relatio…
Characterizes tangent cones for specific connections on reflexive sheaves.
problem Analyzing tangent cones of admissible Hermitian-Yang-Mills connections over reflexive sheaves.
method Algebro-geometric characterization of analytic tangent cones.
result Complete characterization of tangent cones for admissible Hermitian-Yang-Mills connections over reflexive sheaves.
Study deforms Hermitian metrics with positive curvature.
problem Deforming Hermitian metrics with positive curvature.
method Adapted conformal perturbation method to Hermitian setting.
result Hermitian metrics with quasi-positive curvature can be deformed to positive curvature.
From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian ge…
A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conf…
Local examples of singular connections with bounded energy.
problem Constructing singular Hermitian Yang-Mills connections with bounded energy.
method Local construction of singular connections over B1⊂C3. result Number of essential singular points can be arbitrarily large.
Verify conjecture for special Hermitian manifolds.
problem Conjecture about space forms for canonical metric connections.
method Verify conjecture for complex nilmanifolds and Bismut torsion-parallel manifolds.
result Verify conjecture for two special types of Hermitian manifolds.
The paper studies quarter-symmetric connections on Hermitian and Kähler manifolds.
problem Examining quarter-symmetric connections on almost Hermitian and Kähler manifolds.
method Analyzing the curvature tensors and their properties with respect to quarter-symmetric connections.
result Constructed tensors that do not depend on the quarter-symmetric connection generator, including the Weyl projective curvature tensor.
On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manif…
Study hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
problem Investigate hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
method Obtain a criterion for the existence of hermitian Yang-Mills connections on pullback bundles, using intersection numbers on the base.
result Determine conditions under which pullback bundles of stable or unstable bundles remain stable or unstable for adiabatic classes.
Complete classification of Hermitian manifolds with flat Gauduchon connections.
problem Classifying compact Hermitian manifolds with flat Gauduchon connections.
method Analyzing properties of Hermitian manifolds and using Gauduchon connections.
result Established a conjecture about Kähler-like conditions and flatness.
Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.
problem Improving Schwarz lemma for holomorphic maps between Hermitian manifolds.
method Introducing new curvature constraints on source and target manifolds, controlling by holomorphic sectional curvature.
result Significant improvements on the Wu--Yau theorem and Schwarz lemma for Gauduchon connections.
Compactifies moduli spaces of Hermitian-Yang-Mills connections on balanced manifolds.
problem Analyzing Ω-Yang-Mills connections on Riemannian manifolds. method Extending known results on Yang-Mills connections to Ω-Yang-Mills connections, proving weak compactness and removable singularity theorems. result Compactification of moduli spaces of smooth Hermitian-Yang-Mills connections on unitary bundles over balanced manifolds.
The paper explores the geometry of real connections on Hermitian manifolds and derives Kähler-Einstein metrics.
problem Investigating the geometry of real connections on Hermitian manifolds.
method Analyzing the relationship between real connections and Hermitian connections, focusing on the real Chern connection.
result Derives Kähler-Einstein metrics using real Chern-Einstein metrics.
Study magnetic curves in Sasakian manifolds, classifying and parametrizing them.
problem Classify and parameterize pseudo-Hermitian magnetic curves in Sasakian manifolds.
method Define and classify pseudo-Hermitian magnetic curves, construct parametrizations.
result Complete classification theorem for pseudo-Hermitian magnetic curves in Sasakian manifolds.
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.
Surveying locally homogeneous almost-Hermitian spaces with formulas for curvature.
problem Understanding the geometry of locally homogeneous almost-Hermitian spaces.
method Using the framework of varying Lie brackets to compute curvature of Gauduchon connections.
result Explicit formulas and examples for curvature of Gauduchon connections on locally homogeneous almost-Hermitian spaces.
Fractional Laplacian inverse problem solved for connection Laplacians.
problem Determining structures from metric, bundle, and map knowledge.
method Local knowledge of metric, bundle, and map determines global structures.
result Global structures determined from local knowledge of metric, bundle, and map.
We pursuit the research line proposed in \cite{YZ-Gflat} about the classification of Hermitian manifolds whose s-Gauduchon connection ∇s=(1−2s)∇c+2s∇b is flat, where s∈R and ∇c and ∇b are the Chern and the Bismut connections, respectively. We foc…
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
Study pseudoholomorphic maps using canonical connection.
problem Characterize pseudoholomorphic maps between almost Hermitian manifolds.
method Use canonical connection and Bochner formulas to derive estimates and theorems.
result Obtained C2-estimate of canonical second fundamental form and Liouville type theorems. In this paper, we systematically compute the Bianchi identities for the canonical connection on an almost Hermitian manifold. Moreover, we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection. As applications of the c…
The paper studies deformations of Hermitian Yang-Mills and Donaldson-Thomas connections on G2-manifolds.
problem Deformation theory of connections on G2-manifolds. method Introducing new coclosed G2-structures and analyzing elliptic complexes. result Moduli spaces of connections are shown to be tori under certain conditions.
Study proves properties of compact Hermitian surfaces with specific curvature conditions.
problem Characterizing compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature.
method Analyzes surfaces with Gauduchon connections and Lichnerowicz holomorphic sectional curvature.
result Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature are either Kähler or isosceles Hopf surfaces.
New examples of deformed Hermitian-Yang-Mills connections found.
problem Constructing deformed Hermitian-Yang-Mills connections on manifolds.
method Constructed first higher rank, irreducible deformed Hermitian-Yang-Mills connections in both small and large radius regimes.
result Existence of solutions with any possible angle and ruling out some stability conditions.
Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.
problem Defining and analyzing volume functionals for Hermitian connections on manifolds.
method Introduces the line bundle mean curvature flow and relates it to deformed connections and special submanifolds.
result Proves the mirror equality for mSpin(7)-dDT connections and deduces their properties. Harmonic Hermitian structures found on specific Riemannian manifolds.
problem Finding conditions for harmonic Hermitian structures on Riemannian manifolds with skew-torsion.
method Geometric conditions on a four-dimensional Hermitian manifold with a metric connection of totally skew-symmetric torsion.
result The complex structure is a harmonic map into the twistor space under certain conditions.
This paper discusses partial answers and a proof for conjectures about Gauduchon connections on Hermitian manifolds.
problem Conjectures about Gauduchon connections and Hermitian metrics on compact manifolds.
method Analyzes partial answers to conjectures and provides a proof for a related conjecture, discovering a duality phenomenon.
result Proof of the second conjecture about two Kähler-like Gauduchon connections implying a Kähler metric.
Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.
problem Moduli space of non-Hermitian Yang--Mills connections over a compact Kähler manifold
method Using normalized harmonic metrics
result Near the Hermitian locus, the unobstructed locus carries an almost hypercomplex structure compatible with the associated Riemannian metric.
Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our p…
The Strominger conjecture is confirmed for compact Hermitian manifolds in 2D and special higher dimensions.
problem Determining conditions for a Hermitian metric to be Kähler based on the Strominger connection's curvature.
method Analyzing the Strominger connection's holomorphic sectional curvature in compact Hermitian manifolds.
result The Strominger conjecture is confirmed in 2D and special higher dimensions.
Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.
problem Geometry of perturbations in Drury-Arveson space.
method Analysis of smooth vector bundles with Hermitian connections and computation of parallel transport operators.
result Found natural Hermitian connections on perturbed submodules.
Vortices and coupled vortices arise from Yang-Mills-Higgs theories and can be viewed as generalizations or analogues to Yang-Mills connections and, in particular, Hermitian-Yang-Mills connections. We proved an analytic compactification of the moduli spaces of vortices and coupled vortices on hermitian vector bundles ov…
The paper characterizes curvature of quaternionic skew-Hermitian manifolds and constructs related geometric structures.
problem Characterizing the curvature of quaternionic skew-Hermitian manifolds.
method Holonomy theory of symplectic connections and bundle constructions.
result Existence and integrability of almost hypercomplex skew-Hermitian structures on Swann bundles.
The study examines geometric properties of complex Hermitian manifolds and their holonomy groups.
problem Understanding the geometric properties and restrictions of Hermitian manifolds and their holonomy groups.
method Analyzing the representation of restricted holonomy groups and their geometric consequences.
result Established criteria for when a Hermitian manifold is Kähler or projective based on its holonomy group.
The canonical connection on a Riemannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with non-integrable almost product structure. We construct and characterize an…
The paper proves unique Levi-Civita connections on noncommutative forms.
problem Existence and uniqueness of Levi-Civita connections on noncommutative differential forms.
method Combining Hilbert module and algebraic techniques, proving conditions for existence and uniqueness of Hermitian torsion-free connections.
result Existence and uniqueness of Levi-Civita connections on θ-deformations of compact Riemannian manifolds.
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.
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.
The paper investigates differential geometry of CR manifolds with new metrics.
problem Investigate differential geometry of CR manifolds with new metrics.
method Introduce a new Riemannian metric and canonical connection, derive curvature and torsion properties.
result Generalize known results in Riemannian geometry to the pseudo-Hermitian case.
Study of Hermitian structures on Lie groups with 2D commutator subgroups.
problem Classifying Hermitian structures on Lie groups with specific commutator subgroups.
method Explicit classification of Type I and Type II structures, computation of Bismut connections, and examples of Kahler structures.
result Classification of Kahler structures within Type I and Type II structures.
We characterize quasi Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic f…