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.
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.
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.
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. 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…
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.…
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. 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 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…
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.
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.
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
A (J2=±1)-metric manifold has an almost complex or almost product structure J and a compatible metric g. We show that there exists a canonical involution in the set of connections on such a manifold, which allows to define a projection over the set of connections adapted to J. This projection sends the Le…
We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces ad…
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying algebraic Ricci solitons on three-dimensional Lorentzian Lie groups.
method Computed canonical and Kobayashi-Nomizu connections and their curvatures; defined algebraic Ricci solitons.
result Classified algebraic Ricci solitons on specific Lorentzian Lie groups.
New normalization condition for sub-Riemannian connections.
problem Normalizing connections on sub-Riemannian manifolds.
method Formulated in terms of Cartan connections, depends on curvature's first degree of homogeneity.
result A compatible partial affine connection can be uniquely extended to a full affine connection and a grading of the tangent bundle.
Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
The paper proves monotonicity formulas for minimal connections and their applications.
problem Understanding critical points of volume functionals in Riemannian geometry.
method Developed monotonicity formulas for minimal connections under specific conditions.
result Established vanishing theorems for minimal connections on Euclidean spaces and dDT connections on G2-manifolds.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case …
Study of multiplicative connections in Lie groupoids.
problem Defining and understanding multiplicative connections in Lie groupoids.
method Definition and study of multiplicative connections satisfying compatibility with the groupoid structure.
result Identification of the obstruction to the existence of a multiplicative connection.
In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connect…
Develops torsion dual connections for statistical manifolds.
problem Defining statistical manifolds using dual connections.
method Introduces torsion dual connections and proves their properties.
result Curvature tensor of torsion dual connections has specific divergence.
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
Defines a new natural connection on Riemannian Π-manifolds.
problem Characterizing natural connections on Riemannian Π-manifolds.
method Introducing and analyzing the first natural connection with torsion.
result Relations between the first natural connection and Levi-Civita connection are established.
In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.
problem Determining the global meromorphic connection based on specified local behavior at singular points.
method Expository discussion of various problems related to meromorphic connections with specified local behavior, including Deligne-Simpson and rigidity problems.
result The existence and nonemptiness of moduli spaces of meromorphic connections with specified local behavior.
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the g…