arXiv research

The paper proves a theorem for generalized p-Kähler manifolds.

New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.

Let $(M,I,J,K)$ be a hyperkaehler manifold, $\dim_\R M =4n$. We study positive, Dolbeault-closed $(2p,0)$-forms on $(M,I)$. These forms are quaternionic analogues of the positive $(p,p)$-forms. We construct an injective homomorphism mapping Dolbeault-closed $(2p,0)$-forms to closed $(n+p,n+p)$-forms, and positive $(2p,…

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…

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

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.

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…

Degenerate twistor deformations of Kähler manifolds are also Kähler.

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

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic equation to show that this metric is strictly positive on $\mathcal X$, unless the fam…

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

Holomorphic family of strongly pseudoconvex domains in Kähler manifolds are studied.

New system tracks musical performances in raw sheet images without preprocessing.

Extends Lelong number theory to positive plurisubharmonic currents.

K-energy is not strictly convex on certain complex manifolds, but under specific conditions, it is.

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

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

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…

The paper proves a function extension on Kähler 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…

The paper studies the distribution of random degeneracy sets on complex manifolds.

Extends classical stability results to new geometric settings.

Dynamics of four-dimensional massless fields of all spins is formulated in the Siegel space of complex $4\times 4$ symmetric matrices. It is shown that the unfolded equations of free massless fields, that have a form of multidimensional Schrodinger equations, naturally distinguish between positive- and negative-frequen…

Decomposes geodesic currents on surfaces into measured laminations or positive systole components.

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

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>σ_{n}K_{\max}$, where $σ_n\in (\frac{1}{4},1)$ is an explicit positive constan…

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…

Stability of positive mass theorem for hyperbolic graphs proven.

The paper uses SVAR modeling to analyze how demographic changes affect the current account and economic growth.

The abstract discusses conjectures about metrics on complex manifolds.

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 note proves positive currents induced by VKE with mixed singularities.

It has been shown that for each Killing-Yano (KY)-form accepted by an $n$-dimensional (pseudo)Riemannian manifold of arbitrary signature, two basic gravitational currents can be defined. Conservation of the currents are explicitly proved by showing co-exactness of the one and co-closedness of the other. Some general ge…

Geodesic currents in strongly hyperbolic spaces are dense.

Currents on Lie groups form a Hopf algebra structure.

The study defines differential forms and currents on orbifolds with corners.

The Riemannian hemisphere has a lower bound for its mass.

The paper examines convergence of currents and forms under smooth diffeomorphisms.

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

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…

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…

The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.

The paper defines new types of positivity and proves properties of Schur forms for vector bundles.