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.
Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.
problem Smooth Chern-Ricci-flat metrics on Hermitian manifolds.
method An analogue of the Calabi flow for compact Hermitian manifolds with vanishing first Bott-Chern class.
result The flow converges to the unique Chern-Ricci-flat metric under certain conditions.
Chern's conjecture on affine manifolds is proven to be true.
problem The Euler characteristic of closed affine manifolds equals zero.
method Proof by mathematical analysis of affine manifolds.
result The conjecture is confirmed to be correct.
Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.
problem Characterizing Vaisman manifolds with vanishing first Chern class.
method Categorization into three types based on Bott-Chern class sign, showing canonical metrics, quasi-regularity, stability, and automorphism group behavior.
result Vaisman manifolds with non-positive Bott-Chern class admit canonical metrics and are stable under deformations.
The paper establishes inequalities for Chern classes and numbers on polarized manifolds and nef vector bundles.
problem Chern class and number inequalities on polarized manifolds and nef vector bundles.
method Sharp inequalities derived from polarized pairs and nef vector bundles.
result Bounding Chern numbers of nef vector bundles and classifying compact Kähler 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.
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
Hermitian metrics with zero second Chern Ricci curvature are rigid and exist on specific manifolds.
problem Characterizing Hermitian metrics with vanishing second Chern Ricci curvature.
method Analyzing the rigidity of the second Chern Ricci curvature on compact complex manifolds.
result Characterization of second Chern Ricci-flat Hermitian metrics and non-existence results.
Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.
problem Characterize Bott-Chern cohomology of Vaisman manifolds.
method Explicit description via basic cohomology, infer relationships between cohomology groups, show invariants are unbounded, cohomological characterization of formality.
result Bott-Chern and Dolbeault numbers determine each other for Vaisman manifolds, and cohomological invariants are unbounded.
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
Survey on Chern-Ricci flow for complex manifolds.
problem Understanding and solving open problems in Chern-Ricci flow.
method Parabolic flow of Hermitian metrics on complex manifolds.
result Open problems and new directions in Chern-Ricci flow highlighted.
Paper explores prescribing Chern scalar curvatures on noncompact Hermitian manifolds.
problem Prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds.
method Generalizes Aviles-McOwen's existence results to higher-dimensional Hermitian manifolds.
result Existence results for Chern scalar curvatures on Hermitian manifolds.
Defines Chern-Robinson connection on Robinson manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines Chern-Robinson connection on complexified tangent bundle of Robinson manifolds.
result Various Bianchi identities are derived and applied to geometry.
Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
The paper examines inequalities for Chern numbers on specific 4D Kähler manifolds.
problem Investigating Chern number inequalities on 4D Kähler manifolds with deformed Hermitian-Yang-Mills metrics.
method Analyzing 4D Kähler manifolds with deformed Hermitian-Yang-Mills metrics under the condition θ^∈(π,2π). result Established Chern number inequalities for the specified manifolds.
New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.
problem Conjectures about the reciprocity of Chern-Simons invariants of 3-manifolds.
method Supporting evidence through Galois descent of a K3-group. result The conjectures hold under the condition of Galois descent of a K3-group. In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.
Regularities and stability shown for a specific type of complex parallelizable manifolds.
problem Stability and regularity of Chern-flat metrics on complex parallelizable manifolds.
method Study of Hermitian metrics governed by the second Chern-Ricci form on compact complex manifolds.
result Chern-flat metrics are dynamically stable on compact complex parallelizable manifolds.
In this paper we study almost complex manifolds admitting a quasi-Kähler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-Kähler …
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying ∂∂-lemma but with non-vanishing Aeppli-Bott-Chern-Massey product. Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak W1,2 Bott-Chern and Dolbeault decompositions. result Strict relation between W1,2 Bott-Chern harmonic forms and the W1,2 Bott-Chern decomposition. The paper extends Bott-Chern Laplacian definition and explores its properties on almost Hermitian manifolds.
problem Exploring the properties of Bott-Chern Laplacian on almost Hermitian manifolds.
method Extending the definition of Bott-Chern Laplacian, proving ellipticity, and analyzing kernels on different types of manifolds.
result The dimensions of Bott-Chern and Dolbeault harmonic forms differ on almost complex 4-manifolds with specific metrics.
Study L2 Hilbert complexes on complex manifolds.
problem Analyse L2 Hilbert complexes on complex manifolds. method Define and study L2 Aeppli-Bott-Chern Hilbert complex; examine properties on various manifolds; use self-adjoint extensions of differential operators. result Kernels of operators on compact Hermitian manifolds are isomorphic to Aeppli or Bott-Chern cohomology.
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.
On a closed balanced manifold, we show that if the Chern scalar curvature is small enough in a certain Sobolev norm then a slightly modified version of the Chern-Yamabe flow~\cite{Angella:2015aa} converges to a solution of the Chern-Yamabe problem. We also prove that if the Chern scalar curvature, on closed almost-Herm…
Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.
problem Prescribing Chern scalar curvatures on noncompact manifolds.
method Solving a Kazdan-Warner type equation on noncompact non-Kähler manifolds with an analytic condition.
result Established existence results and provided a new proof of multiplicity theorem.
The study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.
problem Establishing bounds and ordering for Chern numbers on vector bundles.
method Combinatorial ideas to study Chern numbers on ample and numerically effective vector bundles.
result An effective lower bound for Chern numbers of ample vector bundles and reverse dominance ordering for nef vector bundles.
Study bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.
problem Formality and higher Aeppli-Bott-Chern-Massey products on complex manifolds.
method Introduce and study bigraded formality and Aeppli-Bott-Chern-Massey products, showing non-trivial pullbacks on blow-ups.
result Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to blow-ups under certain conditions.
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-…
New findings on Chern flat metrics and their criticality.
problem Understanding critical Hermitian metrics on Chern flat manifolds.
method Analyzing Chern flat manifolds as compact quotients of complex Lie groups and studying their criticality.
result Chern flat metrics on semi-simple Lie groups are torsion-critical and vice versa.
We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
problem Calculating the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
method Renormalization of the Chern-Simons invariant using asymptotics along an equidistance foliation.
result The leading coefficient introduces a complex-valued quantity consisting of mean curvature and torsion 2-form.
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.
From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey.
In this paper, we give a quantum interpretation of the Bismut-Chern character form (the loop space lifting of the Chern character form) as well as the Chern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by …
We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …
Paper proves ratios of Chern numbers differ for complex hyperbolic branched covers.
problem Proving differences in Chern number ratios for complex hyperbolic branched covers.
method Analyzes Chern numbers of complex hyperbolic branched covers and compares them to those of complex hyperbolic manifolds.
result Ratio of Chern numbers for branched covers are not all equal to those of complex hyperbolic manifolds.
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. Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function f. result Flow converges to a conformal Hermitian metric with specified curvature.
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…
New L∞ liftings derived from Chern-Simons classes for coherent sheaves.
problem Liftings of semiregularity maps for coherent sheaves on complex manifolds.
method Introducing Chern-Simons classes for curved DG-pairs and proving canonical liftings.
result Canonical L∞ liftings of Buchweitz-Flenner semiregularity maps. The paper explores positivity conditions for χy-genus and their implications on Chern numbers and symplectic manifolds.
problem Optimizing Chern number inequalities for almost-complex manifolds.
method Introducing and analyzing positivity conditions for the modified χy-genus and applying them to Chern numbers and symplectic manifolds. result Optimal Chern number inequalities hold for many important Kähler and symplectic manifolds.
Develops equivariant Chern characters for coherent sheaves with group actions.
problem Computing Chern characters for coherent sheaves on manifolds with group actions.
method Introduces equivariant Chern characters and proves Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.
result Establishes a Riemann-Roch-Grothendieck theorem for coherent sheaves with finite group actions.
In this paper, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds.
Short proof given for a complex formula in spin manifold geometry.
problem Localization formula for loop space Chern character in spin manifolds.
method Using rescaled spinor bundle on tangent groupoid.
result A concise proof of the localization formula.
Compact Kähler manifolds with positive curvature are projective and rationally connected.
problem Characterizing compact Kähler manifolds with positive curvature.
method Proving properties of compact Kähler manifolds with quasi-positive second Chern-Ricci curvature.
result Compact Kähler manifolds with quasi-positive second Chern-Ricci curvature are projective and rationally connected.