Flat holomorphic connections on stable bundles over LVMB manifolds are always flat.
problem Characterizing LVMB manifolds and their holomorphic connections.
method Analyzing LVMB manifolds and their tangent bundles, deducing properties of holomorphic connections.
result Holomorphic connections on semi-stable bundles over LVMB manifolds are always flat.
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
In this article we study compact Kähler manifolds X admitting non-singular holomorphic vector fields with the aim of extending to this setting the classical birational classification of projective varieties with tangent vector fields. We prove that any such a Kähler manifold X admits an arbitrarily small deformatio…
Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.
problem Understanding properties of holomorphic tensor fields on Kähler manifolds.
method Established vanishing theorems for uniformly rational connected (RC) k-positive Hermitian holomorphic vector bundles. result Holomorphic tangent bundles of Kähler manifolds with positive k-Ricci curvature are uniformly RC k-positive. Study splitting submanifolds in specific homogeneous spaces.
problem Classify splitting submanifolds in rational homogeneous spaces of Picard number one.
method Use global holomorphic vector fields and projection maps to analyze submanifolds.
result Proves submanifolds in certain spaces are rational or Hermitian symmetric.
The paper connects two descriptions of Teichmüller space tangent spaces using harmonic vector fields.
problem Describing tangent spaces to Teichmüller space in two different ways.
method Using harmonic vector fields inspired by harmonic maps to connect the two descriptions.
result A harmonic vector field on the upper half plane describes a connection on the universal Teichmüller curve.
Affine vector fields on pseudo-Kähler manifolds are symplectic.
problem Characterize affine vector fields on compact pseudo-Kähler manifolds.
method Two proofs provided, showing affine vector fields are symplectic and discuss properties of Lie derivatives.
result Affine vector fields on compact pseudo-Kähler manifolds are symplectic.
The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two Kähler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tan…
Classifies 3D F-manifolds with or without Euler fields.
problem Local classification of 3D F-manifolds.
method Integrability condition on multiplication in holomorphic tangent bundle.
result Local classification of 3D F-manifolds.
Study describes lcK structures on Vaisman-type manifolds with holomorphic Lee vector field.
problem Characterizing lcK structures with holomorphic Lee vector field on Vaisman-type manifolds.
method Complete description through potential analysis and vector field properties.
result Examples of lcK structures with non-homothetic Lee vector field.
Paper describes holomorphic polyvector fields on toric varieties.
problem No specific problem stated; general description of fields.
method Explicit description of holomorphic polyvector fields on smooth compact toric varieties.
result Generalizes Demazure's result of holomorphic vector fields on toric varieties.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…
Compact lcK manifolds with holomorphic Lee field are Vaisman under certain conditions.
problem Characterizing compact locally conformally Kähler manifolds with holomorphic Lee fields.
method Analyzing conditions for a compact lcK manifold to be Vaisman when it has a holomorphic Lee vector field.
result Compact lcK manifolds with holomorphic Lee field are Vaisman if the Lee field has constant norm or the metric is Gauduchon.
The study classifies vector fields tangent to Seifert fiberings.
problem Classifying vector fields tangent to Seifert fiberings.
method Classification based on Seifert fiberings.
result A classification of vector fields tangent to Seifert fiberings.
The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.
problem Existence of complete holomorphic vector fields on complex manifolds with specific metrics.
method Method of potential scaling to find a potential function with constant length differential, then constructing a vector field from its gradient.
result A complete holomorphic vector field is constructed on a complex manifold with a Kähler-Einstein metric.
Study on special null submanifolds in indefinite Sasakian manifolds.
problem Characterizing null submanifolds in indefinite Sasakian manifolds.
method Proving properties of screen conformal null submanifolds and defining a new class.
result Existence of contact screen conformal r-null submanifolds in indefinite Sasakian space forms. Conformal vector fields on lcK manifolds are shown to be Killing or holomorphic.
problem Characterizing conformal vector fields on lcK manifolds.
method Analyzing properties of conformal vector fields on compact lcK manifolds.
result Conformal vector fields on compact lcK manifolds are either Killing or holomorphic.
New condition extends Diederich-Fornæss index for pseudoconvex domains.
problem Determine sufficient conditions for Diederich-Fornæss index to be close to 1.
method Derive sufficient condition on Levi-flat sets of the boundary.
result Diederich-Fornæss index is 1 if Levi-flat sets are transversal to holomorphic tangent vector fields.
Study on Ricci solitons on tangent and unit tangent bundles.
problem Characterizing Ricci solitons on tangent and unit tangent bundles.
method Analyzing pseudo-Riemannian g-natural metrics and their Ricci soliton properties. result Classification of conformal vector fields and existence of non-Einstein Ricci solitons.
Embeddings of 3-manifolds into complex 3-space with specific tangents.
problem Embedding closed 3-manifolds into C3 with specified complex tangents. method Constructing embeddings based on link and 2-plane field properties.
result Existence of embeddings with specified complex tangents and holomorphic tangent spaces.
The study proves rationality of complex projective varieties with holomorphic vector fields.
problem Rationality of complex projective varieties with holomorphic vector fields.
method Key technique by Harvey-Lawson on finite volume flows.
result Uniform upper bound on Betti numbers for varieties with holomorphic vector fields.
Classifies Kähler manifolds with special gradient vector fields.
problem Characterizing Kähler manifolds with geodesic holomorphic gradients.
method Classification based on specific vector fields and integrability conditions.
result All such manifolds are biholomorphic to bundles of complex projective spaces.
Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector Yi∈C to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can b…
The tangent bundle of a Riemannian manifold (M,g) with non-degenerated g-natural metric G that admits a Killing vector field is investigated. Using Taylor's formula (TM,G) is decomposed into four classes that are investigated separately. The equivalence of the existence of Killing vector field on M and TM is proved. Ke…
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…
The paper introduces new metrics on Finsler manifolds and characterizes associated vector fields.
problem Characterizing vector fields on Finsler manifolds with new metrics.
method Introducing F-natural metrics and characterizing conformal, homothetic, and Killing vector fields. result Characterization of vector fields on slit tangent bundles of Finsler manifolds.
Study on points where random spherical harmonic nodal set meets tangent vector field.
problem Distribution of points on nodal sets of random spherical harmonics.
method Analysis of expected counting function and eigenvalue asymptotics.
result Asymptotic behavior of counting function is independent of the vector field.
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
The paper establishes conditions for complex structures on manifolds with given vector fields.
problem Conditions for complex structures on manifolds with given vector fields.
method Intrinsic, diffeomorphic invariant conditions for vector fields to have desired regularity.
result Quantitative results for sub-Hermitian geometry and formally integrable elliptic structures.
Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometricians. Here we define a Riemannian or pseudo-Riemannian lift metric on TM, which is in some senses more…
Study on singularities of bundle homomorphisms induced by Morin maps.
problem Characterizing singular points of bundle homomorphisms.
method Analyzing conditions for singularities induced by Morin maps, using Hamilton vector fields for contact structures.
result Characterization of singularities in bundle homomorphisms induced by Morin maps.
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
Defines quaternionic k-vector fields on quaternionic Kähler manifolds.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on HPn. The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic characterizations the type: one in terms of Lie brackets of a complex tangent v…
Study Kähler-Ricci flow on manifolds with singularities.
problem Behavior of Kähler-Ricci flow on manifolds with finite-time singularities.
method Use of holomorphic vector fields to prove estimates.
result Proves estimates related to previous work on the flow.
Explains complex analytic invariants of vector fields and foliations.
problem Integrating theories of singular varieties and foliations.
method Expository discussion of invariants.
result Introduces connections between complex analytic singular varieties and foliations.
The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic las…
Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. The bienergy of a vector field on a Riemannian manifold (M,g) is defined to be the bienergy of the corresponding map (M,g) ---> (TM,g_S), where the tangent bundle TM is equipped with the Sasaki metric g_S. The constrained variational problem is studied, where variations are confined to vector fields, and the correspond…
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
Holomorphic vector bundles and sections on jet schemes calculated.
problem Calculating spaces of holomorphic sections on jet schemes.
method Constructing vector bundles and calculating holomorphic sections.
result Space of holomorphic vector fields on jet scheme Jm(X) calculated for specific manifolds. Study examines null vector fields on Lorentzian manifolds.
problem Understanding the structure of null vector fields on Lorentzian manifolds.
method Investigates the bundle structure and ternary product of nowhere vanishing null vector fields.
result Null tangent bundle is a non-polynomial graded bundle with a para-associative ternary product.
Applying concepts and tools from classical tangent bundle geometry and using the apparatus of the calculus along the tangent bundle projection ('pull-back formalism'), first we enrich the known lists of the characterizations of affine vector fields on a spray manifold and conformal vector fields on a Finsler manifold. …
In the paper we investigate submanifolds in a tangent bundle endowed with g-natural metric G, defined by a vector field on a base manifold. We give a sufficient condition for a vector field on M to defined totally geodesic submanifold in (TM,G). The parallel vector field is discussed in more detail.
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex 3-folds of the form P(T∗X) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such t…
The paper proves a splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds.
problem Understanding the structure of sheaves of holomorphic k-vectors on complex contact manifolds.
method Proving a splitting theorem using sheaves and cohomology.
result The sheaf of holomorphic k-vectors splits into two components.
Holomorphic residue formula for complex supermanifolds.
problem Residue localization on complex supermanifolds.
method Holomorphic residue localization formula for odd vector fields.
result Explicit local residue formula under isolated non-degeneracy hypotheses.
A novel method for parallel transport and geodesics on submanifolds.
problem Understanding parallel transport and geodesics on submanifolds.
method Rolling tangent space to visualize and analyze parallel transport and geodesics.
result Conditions for parallel transport and geodesics are simplified and visualized in the tangent space.