Geometric equation defines canonical metrics on vector bundle families.
problem Finding canonical metrics on families of holomorphic vector bundles.
method Introducing a geometric partial differential equation for families of holomorphic vector bundles.
result Construction of Hermite--Einstein metrics in adiabatic classes on product manifolds and proof of the existence of a unique solution for the Dirichlet problem.
New stability criteria for vector bundles linked to Hermite-Einstein geometry.
problem Stability of higher-rank vector bundles and their moduli spaces.
method Introducing m-positivity and a smooth function for coherent subbundles, linking to Hermite-Einstein geometry. result Hermite-Einstein bundles are uniformly semi-stable, and new stability conditions are established.
On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of…
Establishes Hermite-Einstein metrics on complex spaces with singularities.
problem Existence of Hermite-Einstein metrics on complex spaces with singularities.
method Established existence of estimable Hermite-Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces with klt singularities.
result Obtained precise results for varieties with klt singularities.
Proves conditions for weighted Hermite-Einstein metrics on vector bundles.
problem Conditions for existence of weighted Hermite-Einstein metrics.
method Introduces weighted Hermite-Einstein equation, stability notions, and proves existence.
result Existence of weighted Hermite-Einstein metrics if and only if slope polystable.
The article describes canonical metrics on holomorphic fibre bundles.
problem Existence of canonical metrics on isotrivial Kähler fibrations.
method Induced from Hermite--Einstein connections on holomorphic principal bundles.
result Existence of optimal symplectic connections when principal bundles are polystable.
The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.
problem Existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles.
method Using Cartan's highest weight theory, the paper establishes an algebraic criterion for topological splitting and decouples the prescribed mean curvature equation.
result A sufficient algebraic condition for realizing an L2-function as the mean curvature of a singular Hermitian structure on an irreducible homogeneous bundle. Existence of metrics on non-Kähler varieties, generalizing previous work.
problem Existence of metrics on non-Kähler varieties.
method Definition of slope stability and existence of singular Hermite-Einstein metrics.
result Existence and uniqueness of singular Hermite-Einstein metrics for slope-stable sheaves.
Extends classical stability results to new geometric settings.
problem Stability of holomorphic vector bundles on complex manifolds.
method Introduces (ω,Ω)-Hermite-Einstein and (ω,Ω)-stable conditions. result Generalised Hermite-Einstein condition implies (ω,Ω)-semi-stability. We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and on…
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.
We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle F over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on F coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson.…
Proves stability of certain vector bundles on Kähler surfaces.
problem Stability of rank 2 holomorphic vector bundles on Kähler surfaces.
method Proves existence of Z-positive and Z-critical metrics leading to bundle stability. result Proves stability results for deformed Hermitian Yang-Mills and almost Hermite-Einstein equations for rank 2 bundles.
We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable …
In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fun…
The paper studies curvature properties of sheaves of twisted holomorphic forms on families of compact Kähler manifolds.
problem Investigating curvature properties of sheaves of twisted holomorphic forms on families of compact Kähler manifolds.
method Derives a general curvature formula and explores special cases.
result Provides insight into geometric and analytical properties of curvature in various contexts.
The thesis explores stability conditions and metrics in differential geometry.
problem Understanding extremal objects in differential geometry.
method Introduces and analyzes Z-critical metrics and optimal symplectic connections. result Proves a correspondence between existence of metrics and stability conditions.
The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, w…
Solves a long-standing problem in Kähler geometry.
problem Existence of constant scalar curvature Kähler metrics on projectivized vector bundles.
method Introduces adiabatic slope stability, a weaker version of K-stability, and uses test configurations from subsheaves.
result Equivalence between adiabatic slope stability and existence of cscK metrics for simple vector bundles.
Unique optimal symplectic connections found for submersions.
problem Finding unique optimal symplectic connections for submersions.
method Analytic results and geometric partial differential equations.
result Optimal symplectic connections are unique up to automorphism group.
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
We give a notion of compatibility between a Riemannian structure and a Jacobi structure. We prove that in case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, …
We give a notion of compatibility between a Riemannian metric and a Jacobi structure. We prove that in case of Poisson structures, contact structures and locally conformally symplectic structures, fundamental examples of Jacobi structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, $\fr…
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.
Study on G2∗ structures and almost para-contact structures in 7D.
problem Understanding the relation between G2∗ structures and almost para-contact structures. method Calculating projections using properties of G2∗ structures. result Determined the class of almost para-contact structures induced by G2∗ structures. Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Rie…
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of S2-family of generalized complex structures and study of twistor spaces. result Existence of generalized hypercomplex structures on 4n-dimensional tori with non-maximal types. Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
Extends corner structure study to general case, constructs normal Trans-Sasakian structures.
problem Extending corner structure study to general case without conditions.
method Extends corner structure to general case, constructs Trans-Sasakian structures from non-normal corner structures.
result Constructs normal Trans-Sasakian structures from non-normal corner structures.
New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…
Study equivalence between Hessian and Born structures on tangent bundles.
problem Equivalence between Hessian and Born structures on tangent bundles.
method Analyzing conditions for Hessian structures and integrability of induced almost Born structures.
result Conditions for equivalence between Hessian and Born structures are established.
Introduces compatibility between Dirac structures and Nijenhuis tensors.
problem No specific problem stated; focuses on extending Poisson-Nijenhuis structures.
method Introduces compatibility between Dirac structures and (1,1)-tensor fields.
result Properties of Dirac-Nijenhuis structures studied, including connections and integrations.
We introduce generalized almost contact structures which admit the B-field transformations on odd dimensional manifolds. We provide definition of generalized Sasakain structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which…
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
Introduces VB-structures for geometric objects on manifolds.
problem Properties of higher tangent lifts of geometric structures.
method Introduces weighted structures for various geometric objects on a manifold with a homogeneity structure.
result Proves interesting properties of various weighted structures.
The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined H-nilpotent structures for Lie subgroups of SO(2n,2n) related to neutral hyperKähler structures. result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an H-nilpotent structure. Spin-structures on real Bott manifolds with Kähler structure are characterized.
problem Existence of spin-structures on real Bott manifolds with Kähler structure.
method Ishida characterization and techniques from \cite{PS16} using characteristic classes.
result Necessary and sufficient condition for the existence of spin-structures on M. Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
Exotic hypercomplex structures on a torus are proven to not exist.
problem Existence of exotic hypercomplex structures on a torus.
method Classification of complete flat affine structures on real tori using the Obata connection.
result Exotic hypercomplex structures on a torus do not exist.
Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current ar…
In this paper, we show the existence of (co-oriented) contact structures on certain classes of G2-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G2-structure) admits an almost contact structure. We…
Two Kähler structures are PCR equivalent in the Siegel domain.
problem Equivalence of two Kähler structures in the Siegel domain.
method Construction of complex hyperbolic and Kähler structures from Sasakian structure.
result PCR Kähler equivalent structures in Siegel domain.
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, …
Introduces holed cone structures to generalize cone structures on 3-manifolds.
problem Generalizing cone structures to 3-manifolds with irreducible holonomy representations.
method Introduces holed cone structures and considers their deformation space.
result The deformation space of holed cone structures is a covering space of the character variety.
The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.
problem Classifying and constructing complex symplectic structures on 4D Lie algebras.
method Interpreting complex symplectic and pseudo-Kähler structures, developing a method for constructing hypersymplectic structures.
result Obtained an example of a hypersymplectic structure on a 4-step nilmanifold.