Study the intersection of positive closed currents using tangent currents and King's residue formula.
problem Investigate the intersection of positive closed currents in complex manifolds.
method Employ tangent currents and King's residue formula to establish a natural condition for intersection.
result Derive an integral representation of the intersection of positive closed currents.
Extends Lelong number theory to positive plurisubharmonic currents.
problem Lack of in-depth exploration of Lelong number theory for positive plurisubharmonic currents.
method Introduces generalized Lelong numbers and studies their properties using Lelong-Jensen formulas for the normal bundle.
result Shows the top degree Lelong number of a positive plurisubharmonic current is totally intrinsic.
The paper studies m-positive currents and line bundles on complex manifolds.
problem Understanding m-positive currents and their properties on complex manifolds. method Introducing m-plurisubharmonic functions, proving vanishing theorems, and regularisation theorems using viscosity solutions. result Global and local regularisation theorems for m-semi-positive currents. Conic Kähler-Einstein metrics glued from fibers of a map are positive.
problem Constructing Kähler-Einstein metrics on complex manifolds.
method Gluing fiberwise conic Kähler-Einstein metrics on the regular locus of a fibration.
result The glued current is positive and extends to a positive current on the manifold.
We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…
Decomposes geodesic currents on surfaces into measured laminations or positive systole components.
problem Decomposing geodesic currents on surfaces of finite type.
method Topological decomposition and analysis of intersection functions.
result Currents with positive systole are bilipschitz equivalent to length functions under hyperbolic metrics.
Kähler-Ricci flow smooths out positive closed currents with divisorial singularities
problem Geometric regularization of positive closed currents
method Kähler-Ricci flow
result Gradual replacement of divisorial singularities by Poincaré type ones
The paper proves a theorem for generalized p-Kähler manifolds.
problem Characterization of compact generalized p-Kähler manifolds.
method Proof based on duality between closed and exact positive forms and currents.
result Complete unified proof of Characterization Theorem for compact generalized p-Kähler manifolds.
The Kähler-Ricci flow smooths positive currents on Kähler manifolds.
problem Geometric regularization of positive closed currents on Kähler manifolds.
method Kähler-Ricci flow on compact Kähler manifolds.
result Local Arnold multiplicities linearly decrease to zero under the flow.
Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.
problem Analyzing Lelong numbers for currents in weakly hyperbolic foliations.
method Local and global analysis of directed positive harmonic currents and currents directed by foliations.
result Lelong numbers of currents at the singularity vanish.
Let L be a holomorphic line bundle over a compact Kähler manifold X endowed with a singular Hermitian metric h with curvature current c1(L,h)≥0. In certain cases when the wedge product c1(L,h)k is a well defined current for some positive integer k≤dimX, we prove that c1(L,h)k can be approxima…
Let X be a compact Kähler manifold. We prove that the Kähler-Ricci flow starting from arbitrary closed positive (1,1)-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has positive Lelong numbers for short time if the initial current does. We also prove that th…
Stability of positive mass theorem for hyperbolic graphs proven.
problem Proving stability of positive mass theorem for asymptotically hyperbolic graphs.
method Adapting ideas from previous work on asymptotically flat graphs to hyperbolic graphs.
result Stability of positive mass theorem for a class of n-dimensional asymptotically hyperbolic graphs.
The paper uses SVAR modeling to analyze how demographic changes affect the current account and economic growth.
problem The impacts of demographic changes on the current account and economic growth.
method SVAR modeling to track dynamic impacts between population growth, current account, and economic growth.
result The long-run net impact on economic growth of the domestic working population growth and demand labor for emigrants is positive.
New representations on surfaces with positive cross ratios.
problem Understanding representations of surfaces with specific geometric properties.
method Using geodesic currents and Anosov representations, proving systolic inequalities.
result Systolic inequalities hold for all positively ratioed representations.
We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
Maximal representations are studied using tree embeddings and geodesic currents.
problem Maximal representations of surface groups in symplectic groups.
method Metric properties, geodesic currents, and tree embeddings.
result Translation length can be computed as intersection with a geodesic current.
Maps preserving mass and injective on boundary are isometries.
problem Stability of mass-preserving maps in integral current spaces.
method Proving rigidity of mass-preserving 1-Lipschitz maps.
result Maps preserving mass and injective on boundary are isometries.
The note proves positive currents induced by VKE with mixed singularities.
problem Variation of Kahler-Einstein metrics with mixed singularities.
method Fiberation between compact Kahler manifolds with generic smooth log canonical pairs.
result Current induced by VKE with mixed cone and Poincare singularities is positive.
To a tropical p-cycle VT in Rn, we naturally associate a normal closed and (p,p)-dimensional current on (C∗)n denoted by Tnp(VT). Such a "tropical current" Tnp(VT) will not be an integration current along any analytic set, si…
Geodesic currents in strongly hyperbolic spaces are dense.
problem Characterizing geodesic currents with strongly hyperbolic dual pseudometrics.
method Combining finite-cover argument and boundary data characterization.
result Dense subset of geodesic currents with strongly hyperbolic dual pseudometrics.
Study b-divisors on Kähler manifolds linking them to currents.
problem Intersection theory of b-divisors on Kähler manifolds.
method Established correspondence between closed positive currents and nef b-divisors.
result Intersection theory of nef b-divisors answered.
If (X,J) is an almost complex manifold, then a function u is said to be plurisubharmonic on X if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the fact that the (1,1)-cur…
New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.
problem Understanding Kähler-Einstein currents on singular spaces.
method Analyzing currents' properties and applying them to compact spaces with log terminal singularities.
result Kähler-Einstein currents dominate Kähler forms on compact spaces with log terminal singularities.
We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown t…
Constructs currents and heights on K3 surfaces.
problem Understanding the geometry and arithmetic of K3 surfaces.
method Constructs canonical positive currents and heights on K3 surfaces, equivariant for automorphism group.
result Continuous family of currents and heights defined over an enlarged boundary of the ample cone.
Study real rectifiable currents, generalize King's theorem, simplify proof, relate to Hodge conjecture.
problem Characterize currents defined by positive real holomorphic chains.
method Use Siu's semicontinuity theorem to simplify King's proof.
result Sufficient condition for the Hodge conjecture.
We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be prod…
Computes mapping classes for currents on compact surfaces.
problem Counting mapping classes on compact surfaces.
method Analyzes currents and mapping classes on compact surfaces.
result Proves a lattice counting theorem for Teichmüller space.
Paper shows regularizing flow for conical Kähler-Ricci equations.
problem Regularizing property of conical Kähler-Ricci flow.
method Regularizing property of the twisted conical Kähler-Ricci flow from a positive closed current with zero Lelong number.
result Extends regularizing property to conical singularity case.
This paper connects real closed fields to Hitchin representations and their properties.
problem Understanding representations of surface groups over real closed fields.
method Tarski-Seidenberg transfer principle and multiplicative Bonahon-Dreyer coordinates.
result Hitchin representations correspond to F-positive representations over real closed fields. Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,φ). In particular, on X there exist φ-plurisubharmonic functions, φ-convex domains, φ-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sens…
In this note we give an overview of some applications of the Calabi-Yau theorem to the construction of singular positive (1,1) currents on compact complex manifolds. We show how recent developments allow us to give streamlined proofs of existing results, as well as new ones.
Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.
problem Study of unlikely intersections for automorphisms of Markov surfaces with positive entropy.
method Arithmetic equidistribution for adelic line bundles, theory of laminar currents, quasi-Fuchsian representation theory.
result Two automorphisms with positive entropy share a Zariski dense set of periodic points if and only if they share a common iterate.
Establishes Kobayashi-Hitchin correspondence for nef and big classes.
problem Analyzing stability and positivity in algebraic geometry.
method Introducing adapted currents and metrics to establish correspondence.
result Equality cases of Bogomolov-Gieseker and Miyaoka-Yau inequalities.
A model for grid cells using vectors and matrices for position and motion.
problem Representing self-position and motion in a high-dimensional space.
method Vector-matrix multiplication, magnified local isometry, and global adjacency kernel.
result The model can learn hexagon patterns and correct errors.
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.
Study geodesic currents on surfaces, proving new properties of character varieties.
problem Characterizing discontinuity sets in Hitchin and maximal character varieties.
method Decompose geodesic currents into measured laminations or positive systole components.
result Bi-Lipschitz equivalence of intersection functions to length functions for currents with positive systole.
Proves stronger curvature condition for known nonnegative curvature manifolds.
problem Curvature conditions for manifolds with nonnegative sectional curvature.
method Modifies curvature operator with a 4-form to achieve positive-semidefiniteness.
result All known nonnegative curvature manifolds satisfy a stronger condition.
The paper introduces generalized Lelong numbers for currents and their applications in intersection theory.
problem Defining and studying generalized Lelong numbers for currents in intersection theory.
method Formulating generalized Lelong numbers for closed smooth (j,j)-forms, defining horizontal dimension, and establishing properties and formulas.
result Effective sufficient conditions for defining and continuity of intersections of positive closed currents.
We show that the non pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated to big cohomology classes are also bimeromorphic invariants. We compare the weighted energy functionals of currents with respect to different cohomolog…
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
Study complex manifolds with negative curvature, finding either a current or Kähler property.
problem Characterizing compact complex manifolds with negative curvature operators.
method Proves properties using metrics and the pluriclosed flow.
result Classification of complex surfaces with negative curvature operators.
Study rigid classes on hyperkahler manifolds, showing general ones are rigid.
problem Characterize rigid classes on compact hyperkahler manifolds.
method Analyze eigenvectors of hyperbolic automorphisms and use BBF form.
result General parabolic classes on hyperkahler manifolds are rigid.
New method learns from either positive or negative feedback alone.
problem Limited applicability of existing preference optimization methods in scenarios with only unpaired feedback.
method Decouples learning from positive and negative feedback, using expectation-maximization (EM) to optimize probability of positive outcomes and explicitly incorporate negative examples.
result Stable learning from negative feedback alone demonstrated.
Extends curve functions to geodesic currents with a simple criterion.
problem Continuous extension of curve functions to geodesic currents.
method Simple criterion based on smoothing property.
result Extends known curve functions and introduces new examples.
Study volumes of Bott-Chern classes on complex manifolds.
problem Understanding volumes of transcendental Bott-Chern classes.
method Extending non-pluripolar products to quasi-positive currents, establishing quasi-monotonicity of Monge-Ampère masses, and solving degenerate complex Monge-Ampère equations.
result Positive answer to Demailly-Păun-Boucksom conjecture regarding bounded mass property.
We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furth…