arXiv research

The paper extends positivity results from vector bundles to Kobayashi positive ones.

In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex proj…

Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.

Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.

Study visibility properties of Kobayashi distance on unbounded domains.

New examples of hyperbolic Riemannian manifolds proven.

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated …

Classifies Ricci solitons on specific Lorentzian Lie groups.

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $C^\infty$ boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also …

New proof of Kobayashi's properness criterion using metric geometry.

In this paper we define Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. We prove that the pseudodistance induced by this pseudonorm coincides with the Kobayashi pseudodistance defined for the almost complex c…

Regularity properties of intrinsic objects for a large class of Stein Manifolds, namely of Monge-Ampère exhaustions and Kobayashi distance, is interpreted in terms of modular data. The results lead to a construction of an infinite dimensional family of convex domains with squared Kobayashi distance of prescribed regula…

Classifies hyperbolic manifolds with specific automorphism groups.

Constructs a convex Finsler metric on vector bundles under specific conditions.

A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is…

The Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper mainly consists of three parts. Firstly, we give the explicit expression o…

Survey on recent breakthrough linking curvature and Kobayashi hyperbolicity.

The paper proves conditions for vector bundles to be Kobayashi and Griffiths positive.

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk i…

We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.

We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familliar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold and finish with some examples.

Starting from the description of Segre forms as direct images of (powers of) the first Chern form of the (anti)tautological line bundle on the projectivized bundle of a holomorphic hermitian vector bundle, we derive a version of the pointwise Kobayashi-Lübke inequality.

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 study proves Gromov hyperbolicity for certain complex domains.

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-3$. This result complements existing classifications for automorphism group dimension $n^2-2$ (which is in some sense critical) and greater.

We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps.

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption.

In this paper we extend the notion of the Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. The main theorem on coincidence of the pseudodistance induced by this pseudonorm with the Kobayashi pseudodistance for…

Proves non-hyperbolicity of symplectic varieties with specific properties.

Classifies Ricci collineations on specific 3D Lorentzian Lie groups.

The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.

Rapid progress has been made recently on symmetry breaking operators for real reductive groups. Based on Program A-C for branching problems (T.Kobayashi [Progr.Math.2015]), we illustrate a scheme of the classification of (local and nonlocal) symmetry breaking operators by an example of conformal representations on diff…

The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.

This article continues a line of research aimed at solving an important problem of T. Kobayashi of the existence of compact Clifford-Klein forms of reductive homogeneous spaces. We contribute to this topic by showing that almost all symmetric spaces and 3-symmetric spaces do not admit solvable compact CliffordfKlein fo…

Compact Kähler orbifolds with non-negative Ricci curvature are simply connected.

Study proves correspondence for special bundles on complex surfaces.

Characterizes visibility and geodesic loops in complex domains.

A simple trick invoking objective B-fields is employed to refine the concept of characteristic classes for twisted bundles. Then the objective stability and objective Einstein metrics are introduced and a new Hitchin-Kobayashi correspondence is established between them. As an application the SO(3)-instanton moduli spac…

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 4$ whose group of holomorphic automorphisms has dimension either $n^2-4$, or $n^2-5$, or $n^2-6$. This paper continues a series of articles that achieve classifications for automorphism group dimension $n^2-3$ and greater.

It is proved that all invariant functions of a complex Finsler manifold can be totally recovered from the torsion and curvature of the connection introduced by Kobayashi for holomorphic vector bundles with complex Finsler structures. Equations of the geodesics and Jacobi fields of a generic complex Finsler manifold, ex…

Study of pseudometric properties on domains in Nagano spaces.

We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pair…

In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,φ)$ over a compact generalized Kähler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(α,σ,τ)$-polystable if and only if $\ma…

Let $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, …

We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi di…