Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.
problem Establishing conditions for optimal L2 extension in complex geometry.
method Analyzing singular Nakano positivity and applying L2 extension theorem.
result Necessary condition for equality in optimal L2 extension theorem.
New dHYM connections found on complex vector bundles.
problem Existence of dHYM connections on higher rank vector bundles.
method Constructing explicit non-trivial examples and providing algebraic conditions.
result First explicit non-trivial dHYM connections on higher rank holomorphic vector bundles.
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
In this paper, we introduce the notions of α-Hermitian-Einstein metric and α-stability for I±-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for I±-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include…
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M, the projectivized vector bundle P(E) admits a Kähler metric with positive holomorphic sectional curvature.
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. Holomorphic Lie algebroid connections on Riemann surfaces are characterized.
problem Characterizing holomorphic Lie algebroid connections on Riemann surfaces.
method Analyzes conditions for holomorphic vector bundles to admit Lie algebroid connections based on Lie algebroid properties.
result Conditions for holomorphic vector bundles to admit holomorphic Lie algebroid connections are determined.
Atiyah reviewed holomorphic vector bundles and gauge theories.
problem Holomorphic vector bundles and gauge theories.
method Review of Atiyah's work from 1952-1990.
result Holomorphic vector bundles and gauge theories are interconnected.
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previ…
Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
problem Solving Monge-Ampère type equations for Nakano positive curvature tensors of holomorphic vector bundles.
method Solves the Monge-Ampère type equation in the conformal class of a Nakano positive Hermitian metric.
result Solves the Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
Finite vector bundles over complex manifolds are trivializable via finite covers.
problem Understanding when holomorphic vector bundles over compact complex manifolds are trivializable.
method Introducing finite bundles and using finite étale covers to trivialize holomorphic vector bundles.
result Holomorphic vector bundles over compact complex manifolds are finite if and only if they admit a flat holomorphic connection with finite monodromy.
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
Develops theory of d-holomorphic connections on Klein surfaces.
problem No specific problem stated; focuses on theory development.
method Constructs Atiyah exact sequence for d-holomorphic bundles and provides existence criterion.
result Established theory of d-holomorphic connections and existence criterion.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…
The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on D…
Criterion for Lie algebroid connections on compact Riemann surfaces.
problem Finding conditions for Lie algebroid connections on compact Riemann surfaces.
method Analyzing stable holomorphic vector bundles and their connections.
result Necessary and sufficient condition for Lie algebroid connections on compact Riemann surfaces.
The paper extends orthogonal decomposition results to hermitian Higgs bundles.
problem Classical propositions on holomorphic vector bundles do not always extend to Higgs bundles.
method The approach involves extending propositions on orthogonal decompositions and the second fundamental form to hermitian Higgs bundles.
result Extended propositions concerning orthogonal decompositions and the second fundamental form have applications in Higgs bundles.
The paper defines positivity for singular metrics on vector bundles and proves related theorems.
problem Positivity of singular Hermitian metrics for holomorphic vector bundles.
method The method of Berndtsson and Lempert, along with a Berndtsson-type positivity theorem for holomorphic vector bundles.
result Sharp L2 extension theorem for holomorphic vector bundles. Griffiths' first obstruction formula for vector bundles is derived.
problem Extending holomorphic vector bundles from submanifolds.
method Explicit formula using Atiyah class.
result Formula for the first obstruction.
Proves generic surjectivity of vector bundles via degeneration.
problem Understanding generic surjectivity of vector bundles.
method Uses degeneration argument, Berndtsson's theorem, and Lempert's proof.
result Generalizes previous work on L2 division theorem. For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between …
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 study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperkähler manifold M. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π:Tw(M)\rightarrow CP^1. It is sho…
We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern…
In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the…
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 paper finds many negatively curved Kähler metrics on complex manifolds.
problem Finding Kähler metrics with negative curvature on complex manifolds.
method Analyzes vector bundles and proves dimension estimates and Liouville theorems.
result Proves existence of complete Kähler metrics with negative curvature on certain total spaces.
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
The paper introduces new functionals and equations for complex vector bundles.
problem Complex vector bundle equations and their solutions.
method Introducing generalized Donaldson's functionals and discussing their Euler-Lagrange equations.
result Uniqueness of solutions to complex vector bundle equations.
We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds G/Γ, where G is a complex connected Lie group and Γ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles Eρ associated to any irreducible representa…
Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M) \geq 1 and X Hyper-Kaehler. To obtain invariants of Hyper-Kaehler X one finds that …
Established equivalence of Atiyah classes for generalized holomorphic vector bundles.
problem Defining and comparing Atiyah classes for generalized holomorphic vector bundles.
method Used three approaches: \(\check{C}\)ech cohomology, first jet short exact sequence, and Lie algebroid pairs.
result Equivalence of Atiyah classes defined by different methods.
Holomorphic connections on Calabi-Yau manifolds are flat.
problem Existence of holomorphic connections on Calabi-Yau manifolds.
method Proving the existence of flat holomorphic connections for holomorphic vector bundles.
result Holomorphic vector bundles over compact Kähler Calabi-Yau manifolds admit flat holomorphic connections.
The paper develops quantitative estimates for holomorphic sections over bounded domains.
problem Establishing precise inequalities for holomorphic sections over bounded domains.
method Develops Sobolev-type inequalities and applies them to holomorphic sections of Hermitian vector bundles.
result Quantitative Carleman-type estimates for holomorphic sections are derived, improving on previous non-quantitative results.
The paper explores linear generalised complex structures over vector bundles.
problem Understanding holomorphic vector bundles in a generalized geometry context.
method Adapted linear splitting and equivalence to C-multiplication and C-Lie algebroid structure. result Generalised complex Lie algebroids are expressed as complex conjugated Lie bialgebroids.
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable. Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.
problem Detecting stability of holomorphic vector bundles using Seiberg-Witten equations.
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple-spinors.
result Constructs a numerical invariant related to φ−stability of SU(n)−holomorphic vector bundles. In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in view of root systems.
Study Kähler-Ricci flow on manifolds with singularities.
problem Behavior of Kähler-Ricci flow on manifolds with finite-time singularities.
method Use of holomorphic vector fields to prove estimates.
result Proves estimates related to previous work on the flow.
Holomorphic vector bundles on Hopf manifolds admit flat connections.
problem Understanding flat connections on holomorphic vector bundles over Hopf manifolds.
method Defining resonant and non-resonant Mall bundles, proving the existence of flat connections on non-resonant bundles, and applying the Poincare-Dulac theorem.
result Non-resonant Hopf manifolds are linearizable, generalizing Kodaira's result.
In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in…
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
We investigate the analog of holomorphic vector bundles in the context of Sasakian manifolds.
Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.
problem Constructing local models for vector bundles on Kähler manifolds.
method Applying Geometric Invariant Theory to Kähler manifolds to construct analytic GIT-quotients.
result Existence of Weil-Petersson forms on parameter spaces for stable vector bundles.
New characterizations of curvature operators for specific forms via L2-estimates.
problem Characterizing semi-positive and semi-negative curvature operators for (n,q) and (p,n)-forms. method Using L2-estimates to characterize curvature operators for (n,q) and (p,n)-forms. result New characterizations of Nakano semi-positivity and semi-negativity.
A family of holomorphic vector bundles is constructed on a complex manifold X. The space of the holomorphic sections of these bundles are calculated in certain cases. As an application, if X is an N-dimensional compact Kähler manifold with holonomy group SU(N), the space of holomorphic vector fields on its jet …
It is known that given a stable holomorphic pair (E,φ), where E is a holomorphic vector bundle on a compact Kähler manifold X and φ is a holomorphic section of E, the vector bundle E admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(\E ,φ)$, where $\E$ is a reflexive shea…