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…
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48 results for “Kobayashi conjecture”
Solves Kobayashi's conjecture on homogeneous spaces.
problem Rank condition for homogeneous spaces not admitting compact Clifford-Klein forms.
method Cohomological obstruction and Sullivan model for reductive pairs.
result Affirmative proof of Kobayashi's rank conjecture.
The paper extends positivity results from vector bundles to Kobayashi positive ones.
problem Extending positivity results from vector bundles to Kobayashi positive ones.
method Using convexity of Kobayashi positive Finsler metrics and duality for convex Finsler metrics.
result The quotient and tensor product of Kobayashi positive vector bundles are also Kobayashi positive.
The paper studies Kobayashi pseudometric, complex automorphisms, and hyperkaehler manifolds.
problem Characterizing automorphisms and pseudodistances on complex manifolds.
method Defining Kobayashi quotient, proving isomorphisms, and analyzing pseudodistances.
result Proves Kobayashi's conjecture on hyperkaehler manifolds and automorphisms.
Study on Einstein manifolds with specific properties.
problem Identifying all locally homogeneous compact pseudo-Riemannian Einstein manifolds.
method Analyzing standard compact Clifford-Klein forms of simple non-compact Lie groups and conjecturing based on T. Kobayashi's work.
result Found at least one Einstein metric in standard compact Clifford-Klein forms and conjecturing these are the only possible ones.
Four-dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature math.DG
Classifies 4D spaces with compact Clifford-Klein forms.
problem Classifying 4D symmetric spaces with compact Clifford-Klein forms.
method Developed a method for 1-connected solvable symmetric spaces.
result Classification of 4D spaces with compact Clifford-Klein forms.
The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric.…
Survey on recent breakthrough linking curvature and Kobayashi hyperbolicity.
problem Connecting curvature properties to Kobayashi hyperbolicity in complex geometry.
method Detailed analysis of Wu-Yau theorem and its proof.
result A compact complex manifold with negative holomorphic sectional curvature admits a Kähler metric with negative Ricci curvature.
Proves non-hyperbolicity of symplectic varieties with specific properties.
problem Non-hyperbolicity of primitive symplectic varieties with certain conditions.
method Uses ergodicity, birational contractions, and cycle spaces.
result Vanishing Kobayashi pseudometric for symplectic varieties with Lagrangian fibrations.
Improved bounds for algebraic degeneracy and hyperbolicity of hypersurfaces.
problem Improving bounds for algebraic degeneracy and hyperbolicity of hypersurfaces in complex projective space.
method Combining techniques from Diverio-Merker-Rousseau, Bérczi, and Darondeau with computer explorations.
result New degree bounds for algebraic degeneracy and hyperbolicity, improving previous results.
Supports conjecture about harmonic maps from S³ to S².
problem Tackles conjecture about harmonic maps from S³ to S².
method Uses conditions on Hessian and singular values to prove validity.
result Obtains pinching theorem for minimal hypersurfaces in the sphere.
The paper establishes a correspondence between solutions of extended Bogomolny equations and Higgs bundles.
problem Solutions of the extended Bogomolny equations on $Σ imes \RP$ with specific singularities.
method Develops a Kobayashi-Hitchin type correspondence and partial correspondence for solutions with Nahm pole and knot singularities.
result Verifies a conjecture and proves existence and uniqueness of solutions with knot singularities.
We show that there exist knots K in S^3 with g(E(K))=2 and g(E(K#K#K))=6. Together with Theorem~1.5 of [1], this proves existence of counterexamples to Morimoto's Conjecture (Conjecture 1.5 of [2]). This is a special case of arxiv.org/abs/math.GT/0701765 [1] Tsuyoshi Kobayashi and Yo'av Rieck. On the growth rate of the…
Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.
problem Solving a 1975 problem posed by S. Kobayashi about complex Finsler vector bundles with positive Kobayashi curvature.
method Analyzes properties of complex Finsler vector bundles with positive Kobayashi curvature.
result Complex Finsler vector bundles with positive Kobayashi curvature are ample and biholomorphic to projective space.
This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over , and the generalized quiver representations) were included. The main goal is the construction of gauge th…
Kobayashi-Hitchin Correspondence for Saturated Reflexive Parabolic Sheaves on Kähler manifoldsmath.DG
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.
problem Applying Kobayashi-Hitchin correspondence to non-Kähler manifolds.
method Continuity method for vortex equation, Kobayashi-Hitchin correspondence for holomorphic pairs.
result Proved solvability of vortex equation on holomorphic vector bundles over compact Hermitian manifolds.
This paper calculates and analyzes the Kobayashi pseudometric for a specific domain.
problem Calculating the Kobayashi pseudometric for a complex domain.
method Explicit formulas for geodesics and pseudometric calculations.
result Explicit expressions and calculations of the Kobayashi pseudometric.
Study visibility properties of Kobayashi distance on unbounded domains.
problem Understanding visibility properties of Kobayashi distance on unbounded domains.
method Analyzing visibility properties in the context of Kobayashi hyperbolic domains, focusing on unbounded domains and their boundary behavior.
result Carathéodory-type extension theorem for biholomorphisms between planar domains, including infinitely-connected domains.
New examples of hyperbolic Riemannian manifolds proven.
problem Understanding hyperbolic Riemannian manifolds.
method Proving an analogue of the Brody lemma and presenting new examples.
result New examples of Kobayashi hyperbolic Riemannian manifolds.
The Kobayashi-Hitchin correspondence is proven for twisted vector bundles on Kähler manifolds.
problem Proving the Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles.
method Proved the correspondence and approximate correspondence for twisted holomorphic vector bundles on compact Kähler manifolds.
result A twisted holomorphic vector bundle is polystable if and only if it is Hermite-Einstein, and semistable if and only if it is approximate Hermite-Einstein.
Established a correspondence between Higgs bundles and Bogomolny equations.
problem Geometric Langlands program and S-duality in N=4 super Yang-Mills theory.
method Kobayashi-Hitchin correspondence between Higgs bundles and solutions of the extended Bogomolny equation.
result Proved the conjecture by Witten linking Higgs bundles and Bogomolny equations.
Canonical connections and algebraic Ricci solitons of three-dimensional Lorentzian Lie groupsmath.DG
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying algebraic Ricci solitons on three-dimensional Lorentzian Lie groups.
method Computed canonical and Kobayashi-Nomizu connections and their curvatures; defined algebraic Ricci solitons.
result Classified algebraic Ricci solitons on specific Lorentzian Lie groups.
The paper examines conditions for the Kobayashi metric to be Gromov hyperbolic on complex convex sets.
problem Conditions for the Kobayashi metric to be Gromov hyperbolic on complex convex sets.
method Develops necessary and sufficient conditions for Gromov hyperbolicity, extends results to -smooth boundaries, and provides counterexamples.
result Gromov hyperbolicity of Kobayashi metric on -convex sets with -smooth boundary, and counterexamples showing boundary regularity is necessary.
New examples of knots with special bridge positions found.
problem Understanding special types of bridge positions for knots.
method Derived examples of knots with unperturbed weakly reducible non-minimal bridge positions.
result Connected sum of unperturbed bridge positions is unperturbed (a new conjecture).
Classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on three-dimensional Lorentzian Lie groups.
method Examined canonical and perturbed canonical connections, as well as Kobayashi-Nomizu connections and perturbed Kobayashi-Nomizu connections.
result Classified affine Ricci solitons on three-dimensional Lorentzian Lie groups with product structure.
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.
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 boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also …
T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irr…
New proof of Kobayashi's properness criterion using metric geometry.
problem Properness of -action on homogeneous spaces.
method CAT(0) metric geometry on non-compact Riemannian symmetric spaces.
result Established a similar criterion for properness of -action on homogeneous spaces.
Given integers g_i > 1 (i=1,...,n) we prove that there exist infinitely may knots K_i in S^3 so that g(E(K_i)) = g_i and the Heegaard genus of the exterior of the connected sum of K_1,...,K_n is the sum the Heegaard genera of K_1,...,K_n, that is: g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). (Here, E() denotes the ex…
Study shows Calabi-Yau manifolds are non-hyperbolic via mirror symmetry.
problem Proving non-hyperbolicity of Calabi-Yau manifolds.
method Using mirror symmetry, entire curves are found on Calabi-Yau manifolds.
result Calabi-Yau manifolds are Kobayashi non-hyperbolic.
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…
Study finds all hyperbolic manifolds with specific automorphism group dimensions.
problem Classifying homogeneous Kobayashi-hyperbolic manifolds based on automorphism group size.
method Analyzing holomorphic automorphism groups of dimension for manifolds of dimension .
result All connected homogeneous Kobayashi-hyperbolic manifolds of dimension with automorphism group of dimension are identified.
Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphismsmath.CV
Study finds all specific hyperbolic manifolds with high automorphism groups.
problem Classifying hyperbolic manifolds with high-dimensional automorphism groups.
method Examined homogeneous Kobayashi-hyperbolic manifolds of dimension .
result Determined all such manifolds with holomorphic automorphism group dimension .
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…
Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear gr…
The Kobayashi-Hitchin correspondence is established for mini-holomorphic bundles on certain 3-folds.
problem Establishing a correspondence between mini-holomorphic bundles and HE-monopoles.
method Defining mini-holomorphic bundles, Dirac-type singularities, and admissible BHE-metrics.
result A special Hermitian metric (admissible BHE-metric) exists on slope stable mini-holomorphic bundles.
Classifies hyperbolic manifolds with specific automorphism groups.
problem Classifying Kobayashi-hyperbolic manifolds with high-dimensional automorphism groups.
method Analyzes manifolds of dimension with automorphism groups of dimensions or .
result Completes the classification for automorphism groups and .
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
This paper classifies specific types of complex manifolds with high automorphism groups.
problem Classifying homogeneous Kobayashi-hyperbolic manifolds with high-dimensional automorphism groups.
method Analyzing manifolds of dimension and determining their automorphism groups.
result All connected homogeneous Kobayashi-hyperbolic manifolds of dimension are classified for specific automorphism group dimensions.
The Hitchin-Kobayashi correspondence is extended to foliated Riemannian manifolds.
problem Proving a Hitchin-Kobayashi correspondence for foliated manifolds.
method Adapting Uhlenbeck and Yau's proof to the foliated setting, defining stability for foliated bundles, and relating to higher-dimensional instantons.
result Foliated Hermitian-Einstein connections correspond to polystability, with implications for higher-dimensional instantons.
The paper proves conditions for vector bundles to be Kobayashi and Griffiths positive.
problem Conditions for vector bundles to be Kobayashi and Griffiths positive.
method Comparing the curvature of and for large and using duality of convex Finsler metrics.
result Conditions for vector bundles to be Kobayashi and Griffiths positive.
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.
The coherent state representation of the Jacobi group is indexed with two parameters, , describing the part coming from the Heisenberg group, and , characterizing the positive discrete series representation of . The Ricci form, the scalar curvature and the geodesics of th…
The paper establishes a correspondence for quiver bundles over generalized Kähler manifolds.
problem Establishing a correspondence between stability and Hermitian-Einstein metrics for quiver bundles.
method Analyzing -holomorphic quiver bundles over compact generalized Kähler manifolds.
result A quiver bundle is -polystable if and only if it admits an -Hermitian-Einstein metric.
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.