The paper examines positivity properties of singular Hermitian metrics.
problem Investigating positivity for singular Hermitian metrics.
method Exploring Griffiths, ω-trace, and RC positivity.
result These positivity notions imply cohomology vanishing and rational connectedness.
We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type v…
Classifies 4D toric Hermitian ALF metrics with conical singularities.
problem Classifying specific types of 4D Riemannian metrics.
method Explicit formulas provided for classification.
result Examples of metrics with conical singularities have infinitely many distinct topologies.
Gauduchon's theorem extended to singular spaces with smoothing.
problem Extending Gauduchon's theorem to singular spaces.
method Using smoothing techniques for singular spaces.
result Existence of conformally equivalent metrics on singular spaces.
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.
Alternative metric defined on vector bundles, proving vanishing theorem.
problem Defining singular Hermitian metrics on vector bundles.
method Alternative definition of singular Hermitian metric, discussing Griffiths and Nakano positivities.
result Generalised Griffiths' vanishing theorem proved.
Paper develops techniques for singular metrics on vector bundles.
problem Developing techniques for singular metrics on vector bundles.
method Introducing non-pluripolar products and defining I-good singularities. result Derives a Chern--Weil type formula for Hermitian vector bundles with I-good singularities. We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pair…
We show that normalized currents of integration along the common zeros of random m-tuples of sections of powers of m singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with sing…
We give a new variant of L2-extension theorem for the jets of holomorphic sections and discuss the relation between the extension problem of singular Hermitian metrics with semipositive curvature.
Study on a metric on Hermitian metrics space, proving diffeomorphisms and completeness.
problem Metric on space of Hermitian metrics on complex vector bundles.
method Compute metric spray, geodesics, curvature, and use Nash-Moser theorem.
result Metric completion of Hermitian metrics space is L2 integrable singular Hermitian metrics.
The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.
problem Analyzing vector bundles with singular Hermitian metrics and positivity.
method Develops harmonic theory and extends results from complex geometry.
result Extends Nakano's vanishing theorem to vector bundles with singular metrics.
An F-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metri…
Solves Lempert's question on Nakano semi-positivity preservation.
problem Preserving Nakano semi-positivity under limits of metrics.
method Establishes L2 extension theorem and uses multiplier submodule sheaves. result Affirmative solution to Lempert's question on Nakano semi-positivity.
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. Study Hermitian curvature flow on complex surfaces, finding singularities and limits.
problem Characterize and analyze Hermitian curvature flow on complex surfaces.
method Case-by-case analysis of flow on each complex model geometry.
result First example of a compact complex non-Kähler manifold with finite time singularity.
Tian's theorem applies to Moishezon spaces with singular metrics.
problem Distribution of currents on Moishezon spaces with singular metrics.
method Proving asymptotic distribution of Fubini-Study currents.
result Curvature currents of metrics on singular Hermitian line bundles.
Investigates admissible metrics on compact Kähler varieties and their stability.
problem Existence of admissible metrics on compact Kähler varieties and their stability.
method Analyzes admissible Hermitian metrics and Hermitian-Yang-Mills metrics on slope stable coherent sheaves.
result Existence of admissible metrics and Hermitian-Yang-Mills metrics under certain conditions.
The paper studies metrics on vector bundles with singularities and their associated forms.
problem Analyzing singular Hermitian metrics on vector bundles and their associated forms.
method Defines and analyzes the Segre and Chern forms of singular metrics, proving properties of their Lelong numbers.
result Lelong numbers of the associated forms are integers if singularities are integral.
Study of Fubini-Study forms on surfaces with punctures.
problem Analyzing Fubini-Study forms on surfaces with punctures.
method Using Hermitian metrics, holomorphic line bundles, and Kodaira maps.
result Fubini-Study forms grow polynomially near punctures.
We introduce and study a notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and P{ă}un. We define what it means for such a metric to be curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the cuvature Θh as a matrix of currents. We …
The paper classifies surfaces with pseudo-effective tangent bundles.
problem Understanding projective manifolds with pseudo-effective tangent bundles.
method Developed singular hermitian metrics on vector bundles and applied to projective manifolds.
result Projective manifolds with pseudo-effective tangent bundles admit a smooth fibration to a flat projective manifold.
Survey on complex Monge-Ampère equations in Hermitian contexts.
problem Complex Monge-Ampère equations in Hermitian settings.
method Survey of results from many authors over 15 years.
result Overview of known results on complex Monge-Ampère equations.
The paper proves estimates for Hermitian metrics and shows curvature blow-up on complex manifolds.
problem Estimating curvature blow-up in Hermitian metrics.
method Local Calabi and higher order estimates for continuity equations.
result Chern scalar curvature blows up at a finite-time singularity on compact complex manifolds.
We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence o…
Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.
problem Existence of Hermitian-Yang-Mills metrics for Kähler vector bundles.
method Analyzes slope polystability and uses Kähler currents with singularities.
result Validates existence of Hermitian-Yang-Mills metrics for stable bundles and nef-big currents.
Compactifies moduli spaces of Hermitian-Yang-Mills connections on balanced manifolds.
problem Analyzing Ω-Yang-Mills connections on Riemannian manifolds. method Extending known results on Yang-Mills connections to Ω-Yang-Mills connections, proving weak compactness and removable singularity theorems. result Compactification of moduli spaces of smooth Hermitian-Yang-Mills connections on unitary bundles over balanced manifolds.
We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the h…
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.
Holomorphic families yield metrics with explicit curvature formulas.
problem Positivity of direct images with Poincaré type singularities.
method Analyzes holomorphic families and line bundles with Poincaré type singular metrics.
result Explicit formula for curvature of direct image metrics.
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…
Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.
problem Existence of balanced metrics and Gieseker stability of vector bundles.
method Quot-scheme limit of Fubini-Study metrics and Bergman 1-parameter subgroups.
result Existence of balanced metrics equivalent to Gieseker stability for singular cases.
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.
We consider the Toda systems of VHS type with singular sources and provide a criterion for the existence of solutions with prescribed asymptotic behaviour near singularities. We also prove the uniqueness of solution. Our approach uses Simpson's theory of constructing Higgs-Hermitian-Yang-Mills metrics from stability.
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.
Local examples of singular connections with bounded energy.
problem Constructing singular Hermitian Yang-Mills connections with bounded energy.
method Local construction of singular connections over B1⊂C3. result Number of essential singular points can be arbitrarily large.
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the h…
We prove an analogue of the Kobayashi-Hitchin correspondence oncompact connected 3-folds that is fibered on orbifold Riemann surfaces and satisfy an integrability condition, which contains compact connected Sasakian 3-folds. We define mini-holomorphic bundles on such 3-folds and the algebraic Dirac-type singularities o…
Study shows stability of tangent bundle through conifold transitions.
problem Stability of tangent bundle through conifold transitions.
method Hermitian-Yang-Mills metric and conformally balanced metrics.
result Tangent bundle T1,0Xt admits a Hermitian-Yang-Mills metric Ht near vanishing cycles of Xt. We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If E is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a μ-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Ya…
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 main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with a c^-iterated edge metric on its regular part q-parabolic. Moreover, besides stratified pseudomanifolds, the q-parabolicity of other classes of singular spaces, such as compac…
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. Abstract: Study of metrics on line bundles over complex varieties.
problem Chern-Weil and Hilbert-Samuel formulae for singular metrics.
method Theory of b-divisors and multiplier ideal volume function.
result Generalization of a result for elliptic curves to higher degrees.
Kaehler-Einstein metrics on orbifolds derived from Einstein sequences.
problem Desingularizing Einstein orbifolds with Kaehler-Einstein metrics.
method Analyzing sequences of smooth compact Einstein 4-manifolds converging to orbifolds.
result The limit orbifold is Kaehler-Einstein and one of the classified orbifold limits.
Compactness results for Hermitian manifolds help understand Type IIB flow.
problem Understanding singularities in the Type IIB flow.
method Formulated convergence criteria for Hermitian manifolds and applied them to the Type IIB flow.
result Existence of singularity models for the Type IIB flow.
Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformatio…
New exponential decay estimate for Hermitian Yang-Mills metrics near branch points.
problem Understanding the behavior of Hermitian Yang-Mills metrics near branch points.
method Local radial solutions, global gluing construction, exponential estimate near branch points.
result Exponential decay estimate for local radial solutions near branch points.