arXiv research

Study the intersection of positive closed currents using tangent currents and King's residue formula.

Extends Lelong number theory to positive plurisubharmonic currents.

The paper studies $m$-positive currents and line bundles on complex manifolds.

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…

Kähler-Ricci flow smooths out positive closed currents with divisorial singularities

The Kähler-Ricci flow smooths positive currents on Kähler manifolds.

Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.

Let $L$ be a holomorphic line bundle over a compact Kähler manifold $X$ endowed with a singular Hermitian metric $h$ with curvature current $c_1(L,h)\geq0$. In certain cases when the wedge product $c_1(L,h)^k$ is a well defined current for some positive integer $k\leq\dim X$, we prove that $c_1(L,h)^k$ 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…

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.

Maps preserving mass and injective on boundary are isometries.

The paper aims to explore the impacts of bi-demographic structure on the current account and growth. Using a SVAR modeling, we track the dynamic impacts between these underlying variables. New insights have been developed about the dynamic interrelation between population growth, current account and economic growth. Th…

The note proves positive currents induced by VKE with mixed singularities.

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the in…

To a tropical $p$-cycle $V_{\mathbb{T}}$ in $\mathbb{R}^n$, we naturally associate a normal closed and $(p,p)$-dimensional current on $(\mathbb{C}^*)^n$ denoted by $\mathscr{T}_n^p(V_{\mathbb{T}})$. Such a "tropical current" $\mathscr{T}_n^p(V_{\mathbb{T}})$ will not be an integration current along any analytic set, si…

Geodesic currents in strongly hyperbolic spaces are dense.

Study b-divisors on Kähler manifolds linking them to currents.

This paper is devoted, first of all, to give a complete unified proof of the Characterization Theorem for compact generalized $p-$Kähler manifolds (Theorem 3.2). The proof is based on the classical duality between "closed" positive forms and "exact" positive currents. In the last part of the paper we approach the gener…

Let $p:X\to Y$ be an holomorphic surjective map between compact Kähler manifolds and let $D$ be an effective divisor on $X$ with generically simple normal crossings support and coefficients in $(0,1)$. Provided that the adjoint canonical bundle $K_{X_y}+D_y$ of the generic fiber is ample, we show that the current obtai…

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.

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.

We study some fundamental properties of real rectifiable currents and give a generalization of King's theorem in characterizing currents defined by positive real holomorphic chains. Our proof uses Siu's semicontinuity theorem and largely simplifies King's proof. A consequence of this result is a sufficient condition fo…

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…

Paper shows regularizing flow for conical Kähler-Ricci equations.

This paper connects real closed fields to Hitchin representations and their properties.

For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\overlineΣ$, and for every compactly supported filling current $α$, we compute as $L \to \infty$, the number of mapping classes $φ$ so that $f(φ(α))\leq L$. As an application, when the surface in question is close…

The positive mass theorem states that the total mass of a complete asymptotically flat manifold with non-negative scalar curvature is non-negative; moreover, the total mass equals zero if and only if the manifold is isometric to the Euclidean space. Huang and Lee [2015] proved the stability of the Positive Mass Theorem…

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…

This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The mode…

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.

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.

The paper introduces generalized Lelong numbers for currents and their applications in intersection theory.

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.

Study rigid classes on hyperkahler manifolds, showing general ones are rigid.

New method learns from either positive or negative feedback alone.

We study the Weyl chamber length boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising…

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain …

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite.

Extends curve functions to geodesic currents with a simple criterion.

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…

We show that on a Riemann surface lamination locally embedded in $\mathbb{C}^2$, $C^1$ functions (in the sense of the $C^1$ structure of the lamination) are uniform limits of ambient $C^1$ functions, with $L^p$ control on the derivatives along the leaves. This implies that locally in $C^2$, a (1,1) positive closed curr…

Study volumes of Bott-Chern classes on complex manifolds.