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.
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.
Holomorphic tensors on Vaisman manifolds are invariant under the Lee field.
problem Characterizing holomorphic tensors on Vaisman manifolds.
method Using the parallelism of the Lee form and properties of the Lee field.
result The Kodaira dimension of Vaisman manifolds is invariant under certain quotients.
We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions …
We study two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold. First we study compact l.c.K. manifolds with parallel Lee form by means of the existence of a holomorphic l.c.K. flow. Next, we introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms pres…
Calabi-Yau theorem extended to Vaisman manifolds.
problem Uniqueness of Vaisman metrics and their characterization.
method Analyzing the Lee form and Lee class properties.
result Vaisman metrics uniquely determined by volume and Lee class.
Study on Lee classes of complex surfaces, proving connectedness and bounds.
problem Understanding Lee classes of complex surfaces with LCS structures.
method Analyzing deRham classes of Lee 1-forms and using properties of PSH functions.
result Connectedness of Lee deRham classes and explicit negative upper bound on hyperbolic Kato surfaces.
We characterize compact locally conformally Kähler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee form. We introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffe…
We obtain conditions on the Lee form under which a holomorphic map between almost Hermitian manifolds is a harmonic map or morphism. Then we discuss under what conditions (i) the image of a holomorphic map from a cosymplectic manifold is also cosymplectic, (ii) a holomophic map with Hermitian image defines a Hermitian …
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to Cn. We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) mani…
The moduli space of Hermitian-Einstein connections on certain manifolds has a strong Kähler with torsion structure.
problem Characterizing the moduli space of Hermitian-Einstein connections on manifolds with a dilaton field.
method Demonstrates the existence of a strong Kähler with torsion structure on the moduli space under specific conditions on the Lee form and dilaton field.
result The moduli space admits an induced holomorphic and Killing vector field when the manifold has a holomorphic and Killing vector field invariant under the dilaton.
Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.
problem Analyzing properties of Kähler manifolds with nonnegative bisectional curvature.
method Established precise relations among minimal degree, volume growth, and scalar curvature decay.
result Unified understanding of Kähler-Ricci flow through polynomial growth holomorphic functions.
Study Einstein-Yang-Mills fields on specific manifolds, proving field deformations.
problem Deforming Einstein-Yang-Mills fields over conformally compact manifolds.
method Deformation theory using 0-calculus of Mazzeo and Melrose. result Any small perturbation of boundary data can be realized as an Einstein-Yang-Mills field.
Article finds a geometric connection between structures on manifolds.
problem Understanding geometric structures on manifolds.
method Investigates locally conformally Spin(7) manifolds with 2-vector fields.
result Establishes a relationship between nearly Kähler structures and Lee forms.
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.
The paper studies LCAK metrics on complex manifolds and their properties.
problem Characterizing and understanding LCAK metrics on complex manifolds.
method Analyzes the geometric structures induced by LCAK metrics and their properties.
result Pluricanonical LCAK metrics have parallel Lee form on compact manifolds.
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.
New invariants from divisibility of Lee classes for slice-torus.
problem Determining slice-torus knots using Lee class divisibility.
method Defined new invariants from divisibility of reduced Lee class invariants.
result New invariants coincide with Rasmussen invariant for certain cases.
Study BV operators on holomorphic polyvector fields on toric varieties.
problem Existence of BV operators in Gerstenhaber algebras.
method Analyzing BV operators on holomorphic polyvector fields on smooth compact toric varieties.
result Necessary and sufficient condition for BV operators existence.
We show that on an HKT manifold the holonomy of the Obata connection is contained in SL(n,H) if and only if the Lee form is an exact one form. As an application, we show compact HKT manifolds with holomorphically trivial canonical bundle which are not balanced. A simple criterion for non-existence of HKT metric on hype…
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.
New concept of V-minimality applied to Kaehler and non-Kaehler manifolds.
problem Understanding minimality in Kaehler and non-Kaehler manifolds.
method Introducing V-minimality and proving properties for various manifolds. result Complex submanifolds in non-Kaehler l.c.K manifolds are V-minimal for a specific vector field. 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…
We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant und…
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.
Killing fields on compact pseudo-Kähler manifolds are holomorphic.
problem Characterizing Killing fields on compact pseudo-Kähler manifolds.
method Detailed explanation and argumentation of why a previous claim was incomplete.
result Killing fields on compact pseudo-Kähler manifolds are holomorphic.
The paper proves UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
problem Proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
method Rigorously proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories on Rd′imesCd. result Proves vanishing anomalies for hybrid topological-holomorphic field theories, allowing for the definition of a factorization algebra structure for quantum observables.
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.
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.
The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
problem Conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
method Investigation of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
result Identification and determination of weight functions with real holomorphic gradient fields on specific metrics.
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…
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.
For a representation of a finite group G on a complex vector space V we determine when a holomorphic (qp)-tensor field on the principle stratum of the orbit space V/G can be lifted to a holomorphic G-invariant tensor field on V. This extends also to connections. As a consequence we determine those h…
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. The paper quantizes hybrid topological-holomorphic field theories on RmimesCn.
problem Quantizing hybrid topological-holomorphic field theories rigorously.
method Constructing perturbative, one-loop quantizations on RmimesCn. result The one-loop obstruction to quantization vanishes when m≥1. Study curvature in holomorphic fibration fields.
problem Curvature operator in Bergman spaces.
method Careful study of curvature operator in Kähler manifolds.
result Detailed analysis of curvature in smoothly bounded pseudoconvex domains.
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.
The paper counts ends of differential forms on surfaces.
problem Counting ends of meromorphic 1-forms on Riemann surfaces.
method Degeneration techniques and moduli space construction.
result Enumeration of ends for meromorphic 1-forms.
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.
The study examines Lee metrics on groups and their properties.
problem Characterizing groups that admit Lee metrics.
method Analyzing conditions for groups to have or not have Lee metrics, studying specific families of groups, and providing tables for groups of order ≤ 31.
result Conditions for groups to have Lee metrics, including specific families and non-cyclic groups.
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 paper explores mixed curvature for Hermitian manifolds and its implications.
problem Investigating the properties of mixed curvature for Hermitian manifolds.
method Analyzing convex combinations of first Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant mixed curvature are Kähler unless specific conditions are met.
Extends Ho-Lee model for time-dependent parameters.
problem Fixes limitations of the Ho-Lee model.
method Introduces a more flexible no-arbitrage condition.
result Resolves a drawback of the Ho-Lee model.
Vanishing theorem for certain tensor fields on compact Hermitian manifolds.
problem Vanishing theorem for holomorphic tensor fields on compact Hermitian manifolds.
method Inspired by X. Yang and L. Ni-F. Zheng's ideas, the proof uses the definiteness of holomorphic sectional curvature.
result Spaces of certain holomorphic tensor fields are trivial under the definiteness of holomorphic sectional curvature.
In this paper an analytic proof of a generalization of a theorem of Bismut ([Bis1, Theorem 5.1]) is given, which says that, when v is a transversal holomorphic vector field on a compact complex manifold X with a zero point set Y, the embedding j:Y→X induces a natural isomorphism between the holomorphic equiv…
Lee spectral sequence bounds unknotting number, proving Knight Move Conjecture for small knots.
problem Determining the unknotting number of knots.
method Using the Lee spectral sequence to collapse at specific pages.
result For knots with unknotting number less than 3, the Lee spectral sequence collapses at the E_2 page, proving the Knight Move Conjecture.
New theorem on Lee classes for LCK manifolds with potential.
problem Determining Lee classes on LCK manifolds with potential.
method Analyzing cohomology classes of Lee forms and proving the result for Vaisman manifolds.
result The set of Lee classes on LCK manifolds with potential forms an open half-space in H1(M,R).