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 asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.
problem Asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces with ADE singularities.
method Investigates a function on the unit disc defined by fiber integrals of the forms with a smooth test function, showing a lower bound of Hölder exponent at the origin for both cscK-metrics and Ricci-flat metrics.
result Shows bounds of Hölder exponent for both cscK-metrics and Ricci-flat metrics.
The paper constructs flat metrics on orbifolds and resolutions.
problem Finding flat metrics on orbifolds and their resolutions.
method Gluing construction and analysis of singularities.
result All crepant resolutions of non-Kähler Calabi-Yau orbifolds with Chern-Ricci flat balanced metrics admit such metrics.
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.
Blowing up flat metrics yields balanced ones with constant curvature.
problem Constructing balanced metrics with constant curvature on orbifolds.
method Blowing up a compact orbifold with balanced Chern-Ricci flat metrics.
result Blown-up orbifolds admit balanced metrics with constant Chern scalar curvature.
Uniform estimates for complex Monge-Ampère equations on hermitian varieties.
problem Uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
method Proving uniform a priori estimates for degenerate complex Monge-Ampère equations.
result Generalization of a theorem to hermitian contexts.
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. New metrics solve complex equations on special 3D shapes.
problem Finding metrics on complex 3D shapes.
method Gluing construction to solve equations.
result Solves dilatino equation on small resolutions.
This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the (1,1) curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. …
Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.
problem Characterizing Bismut Einstein metrics on compact complex manifolds.
method Observing the (2,0)-part of Bismut Ricci form and using it to prove properties of the metrics.
result Bismut Einstein metrics with non-zero Einstein constant are Kähler Einstein, and those with zero are Bismut Ricci flat.
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.
In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic.
In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the (1,1)- component of the curvature 2-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Her…
Study on Calabi-Yau locally conformally Kähler manifolds proving they are Vaisman.
problem Characterizing Calabi-Yau locally conformally Kähler manifolds.
method Analyzing properties of Calabi-Yau locally conformally Kähler manifolds, proving they are Vaisman.
result Calabi-Yau locally conformally Kähler manifolds are Vaisman.
We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi-Civita connection. The formula uses the intrinsic torsion of an underlying SL(n,R)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As…
Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a balanced metric which is hermitian Ricci--flat. This can be viewed as a differential f…
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.
Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by co-adjoint orbits. Conversely, compact isotropy co-adjoint orbits of …
The paper proves inequalities for orbifold second Chern classes in Fujiki's class.
problem Inequalities for orbifold second Chern classes of compact normal analytic varieties.
method Generic nefness theorems for tangent and cotangent sheaves, and an orbifold Bogomolov--Gieseker inequality for mixed polarizations.
result Semipositivity of the orbifold second Chern class for varieties with nef anti-canonical divisor.
We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Che…
Compact Kähler spaces with zero first Chern class have special geometric properties.
problem Characterizing Kähler spaces with zero first Chern class.
method Analyzing holomorphic tensors, decomposing tangent sheaf, and using Bochner principle.
result Spaces with zero first Chern class split off a complex torus and have specific holonomy representations.
Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.
problem Behavior of harmonic 2-forms on K3 surfaces with Ricci-flat metrics.
method Analysis of convergence of harmonic forms to flat 4D orbifold.
result Decomposition of harmonic 2-forms into converging subspaces.
Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat Kähler metric with cone singularities along the lines asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expr…
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
Study shows Kähler-Einstein metric singularities linked to curvature.
problem Understanding singularities of Kähler-Einstein metrics.
method Relates singularities to holomorphic sectional curvature of conical geometry.
result Provides second-order estimates with explicit constants.
The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.
problem Conditions for minimal compact Kähler manifolds with vanishing second Chern class.
method Study of the abundance conjecture and associated Iitaka fibrations.
result For a minimal compact Kähler manifold, the second Chern class vanishes if and only if the cotangent bundle is nef and the canonical bundle has numerical dimension 0 or 1.
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
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.
Constructs Ricci-flat K3 metrics using D-geometry.
problem Computing the BPS index of a heterotic string theory.
method Hyper-Kähler quotient and D-geometry technology.
result Contains solution to BPS state counting problem.
We construct the first and second Chern-Ricci functions on negatively curved minimal surfaces in R3 using Gauss curvature and angle functions, and establish that they become harmonic functions on the minimal surfaces. We prove that a minimal surface has constant first Chern-Ricci function if and only if…
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.
This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.
problem Characterize all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces.
method Unified construction using axi-symmetric harmonic functions and methods from scalar-flat Kähler metrics.
result All such metrics are ALF and belong to known families.
This paper calculates Dijkgraaf-Witten invariants from Chern classes.
problem Computing Dijkgraaf-Witten invariants from specific 3-cocycles.
method Defines and computes invariants from second Chern classes of representations.
result Clarifies topological interpretation under certain conditions.
Constructs Einstein metrics on manifolds with specific orbits.
problem Finding Einstein metrics on manifolds with given orbits.
method Continuous families of metrics constructed using vector bundles and R4m+4. result Recovery of Spin(7) metrics A8 and B8. Study finds criteria for surfaces with specific curvature properties.
problem Understanding Kählerian or projective structures on surfaces with non-positive curvature.
method Established a criterion for compact Hermitian surfaces with non-positive second Chern-Ricci curvature.
result Found conditions for Kählerian or projective structures on surfaces with non-positive curvature.
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.
We provide a characterization of quotients of three-dimensional complex tori by finite groups that act freely in codimension one via a vanishing condition on the first and second orbifold Chern class. We also treat the case of actions free in codimension two, using instead the "birational" second Chern class, as we cal…
Maps are shown to be Riemannian products with Ricci-flat fibers.
problem Understanding maps between manifolds and their geometric properties.
method Spin geometry and representation theory of curvature operators.
result Scalar-rigid maps are essentially Riemannian products of base and Ricci-flat fibers.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
We present a simplification of Neumann's formula for the universal Cheeger-Chern-Simons class of the second Chern polynomial. Our approach is completely algebraic, and the final formula can be applied directly on a homology class in the bar complex.
We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters k=2 and k=3 naturally show up in the conformal rescaling that takes a representative metric in Nurowski's conformal class associated to a maximally symmetric (2,3,5)-distribution (desc…
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.
D3-brane solutions derived from Ricci-flat metrics on Kähler-Einstein surfaces.
problem Constructing D3-brane solutions in supergravity.
method Classical ansatz involving harmonic warp factor and Ricci-flat metrics.
result Existence of Kähler-Einstein metrics and Ricci-flat metrics on canonical bundles.
Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.
problem Understanding the gap between de Rham and symplectic-Bott-Chern harmonic forms on specific almost-Kähler manifolds.
method Analyzing the space of de Rham harmonic forms and symplectic-Bott-Chern harmonic forms on closed almost-Kähler manifolds.
result The second non-HLC degree measures the gap between de Rham and symplectic-Bott-Chern harmonic forms.
Paper proves a conjecture about minimal hypersurfaces in spheres.
problem Proving a conjecture about the second gap of minimal hypersurfaces with constant scalar curvature.
method Analyzing the squared norm of the second fundamental form of minimal hypersurfaces in spheres.
result Proves the Chern conjecture about the second gap of minimal hypersurfaces in spheres.
In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimensio…
We prove that the conical Kähler-Ricci flows introduced in \cite{CYW} exist for all time t∈[0,+∞). These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class C1,β is negative or zero, the corresponding conical Kähler-Ricci flows co…
Constructs Chern-Weil classes for Cartan geometries.
problem Defines characteristic classes for Cartan geometries.
method Defines a subalgebra of polynomials on the Atiyah algebroid of Q and a characteristic map. result Recover classical Chern-Weil map for specific cases.