We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.
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48 results for “polarized”
Uniform K-stability holds for close polarizations if original is canonical or anti-canonical.
problem Conditions for uniform K-stability of projective varieties.
method Analysis of stability conditions for close polarizations.
result Uniform K-stability is preserved for close polarizations if original is canonical or anti-canonical.
Study polar actions on Damek-Ricci spaces, proving existence and finding examples.
problem Characterize polar actions on Damek-Ricci spaces.
method Prove criteria for isometric actions to be polar, find examples, and classify actions.
result Non-trivial polar actions exist on all Damek-Ricci spaces.
The paper studies how Kähler polarizations degenerate to mixed polarizations on toric varieties.
problem Degeneration of Kähler polarizations to mixed polarizations on toric varieties.
method Constructing polarizations by Hamiltonian actions, finding one-parameter families of Kähler polarizations, and analyzing convergence of spaces of holomorphic sections.
result Kähler polarizations degenerate to mixed polarizations as increases, with specific convergence results for one-parameter families.
Polarized and -polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\Cal F$ (given by the action of a Lie group in the -polarized case) and a transverse CR distribution . Polarized means that is roughly speaking invariant by $\Cal F$. Both structures ar…
A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebr…
The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, w…
Paper uses ResUNet-CMB to reconstruct cosmic polarization rotation from CMB data.
problem Reconstructing anisotropic cosmic polarization rotation from CMB data.
method Extended ResUNet-CMB to handle gravitational lensing and patchy reionization.
result ResUNet-CMB outperforms standard quadratic estimator in reconstructing all three effects.
New method polarizes anisotropic Heisenberg groups.
problem Polarizing anisotropic Heisenberg groups.
method Implementing a technique to polarize anisotropic Heisenberg groups.
result New class of polarizable Carnot groups expanded.
The study introduces polarization of generalized Nijenhuis torsions and their relevance in operator fields.
problem Characterization of Haantjes -modules of operator fields.
method Introducing polarization of generalized Nijenhuis torsions and proving algebraic identities.
result Polarizations of generalized Nijenhuis torsions are relevant in the characterization of Haantjes -modules of operator fields.
The study classifies Riemannian homogeneous spaces with polar isotropy actions.
problem Characterizing Riemannian homogeneous spaces with polar isotropy actions.
method Analyzing simply connected Riemannian homogeneous spaces of compact semisimple Lie groups and various non-compact spaces.
result Classification and non-polar isotropy actions for specific spaces.
Classifies polar foliations on symmetric spaces.
problem Classifying polar foliations on symmetric spaces.
method Orbit equivalence and classification up to codimension two.
result Foliations are either hyperpolar or extensions of rank one foliations.
Paper examines trading polarity to predict market crashes.
problem Insufficient investigation of trading imbalance at high frequency.
method Investigates trading polarity from Shenzhen Stock Exchange data.
result Trading polarity correlates with market crashes and returns.
The paper studies polar normalizations of skew ruled surfaces in 3D space.
problem Understanding the properties and invariants of polar normalized skew ruled surfaces.
method Determination of invariants and analysis of Tchebychev and support vector fields.
result Special polar normalizations lead to degenerate curves.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
The main result of this paper is that a polar action on a compact irreducible homogeneous Kaehler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.
POLAR framework interprets word embeddings using polar opposites.
problem Lack of interpretability in pre-trained word embeddings.
method Adopt semantic differentials and polar opposites to transform embeddings.
result Interpretable word embeddings maintain performance comparable to original embeddings.
Study empty polar varieties' impact on singular function-germs.
problem Topology of singular function-germs with nonisolated singularities.
method Analysis of empty polar varieties.
result Topology implications of nonempty polar varieties.
Polar Homologymath.AG
For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue …
Sections of polar actionsmath.DG
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifol…
Carnot groups can be polarized if they have specific coordinate systems.
problem Understanding when Carnot groups can be polarized.
method Proving Carnot groups with certain coordinate systems are polarizable.
result Carnot groups with suitable horizontal polar coordinates are polarizable.
Polar manifolds are Riemannian G-manifolds admitting a "section", i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariantly diffeomorphic classification in dimensions 5 or less. As an application, we determine wh…
The paper bounds abnormal and Goh-abnormal sets for metabelian Lie groups with polarizations.
problem Bounding abnormal and Goh-abnormal sets for metabelian Lie groups.
method Analyzing rank 2 polarizations and sub-Riemannian structures on metabelian Lie groups.
result Metabelian Lie groups with polarizations satisfy the minimizing Sard property.
In this paper, we generalize the polar transforms of spacelike isothermic surfaces in to n-dimensional pseudo-Riemannian space forms . We show that there exist polar spacelike isothermic surfaces derived from a spacelike isothermic surface in , which are into , …
Classifies polar actions on 3D homogeneous spaces.
problem Classifying polar isometric actions on 3D homogeneous spaces.
method Orbit equivalence classification and study of cohomogeneity one actions.
result Classification of extrinsically homogeneous surfaces and orbit foliations.
Study shows horizontal diameter of unit spheres is π for specific foliations.
problem Understanding horizontal diameter in unit spheres with specific foliations.
method Analyzing singular Riemannian foliations on unit spheres.
result Horizontal diameter of unit spheres is π for polar and infinitesimally polar actions.
The paper introduces polarizations in symplectic and orthogonal settings.
problem Understanding polarizations in symplectic and orthogonal contexts.
method Exposition of symplectic and orthogonal polarizations, emphasizing symmetry.
result Grassmannians of polarizations in symplectic and orthogonal settings.
We give a simple sufficient condition for K-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kahler metric in the Kahler class corresponding to the polarization.
We classify infinitesimally polar actions on compact Riemannian symmetric spaces of rank one. We also prove that every polar action on one of those spaces has the same orbits as an asystatic action.
Investigates polar tangential angles of curves and their monotonicity.
problem Behavior of polar tangential angles of plane curves.
method Proof of monotonicity for certain curves of monotone curvature.
result Nonexistence results for an obstacle problem involving free elasticae.
Proved inequality for anti-self-polar polytopes.
problem Proving an inequality for anti-self-polar polytopes.
method Using Kalai's combinatorial inequality based on Whiteley's result.
result Proved a conjecture by Katz about anti-self-polar polytopes.
New method shows unitarity in quantization for toric manifolds.
problem Unitarity in quantization commutes with reduction for toric manifolds.
method Generalized coherent state transform (gCST) and geodesic rays of toric Kähler polarizations.
result Quantization commutes unitarily with reduction for the new mixed polarization.
Classifies foliations on specific symmetric spaces.
problem Classifying foliations on symmetric spaces of rank one.
method Orbit equivalence classification.
result Polar homogeneous foliations classified.
In this paper, assuming that a polarized algebraic manifold is strongly K-stable, we shall show that the polarization class admits a constant scalar curvature Kaehler metric.
Introduces Polar Depth for analyzing multivariate heavy-tailed data extremes.
problem Analyzing the behavior of extremes from multivariate heavy-tailed distributions.
method Introduces Polar Depth, a novel statistical depth function expressed in polar coordinates.
result The polar depth of the largest observations converges to the polar depth of the limiting distribution as the threshold increases.
The paper introduces a method to make neural networks more robust to adversarial attacks.
problem Vulnerability of deep neural networks to small, adversarially designed perturbations.
method A bottom-up strategy using a nonlinear front end that polarizes and quantizes data.
result The approach can completely eliminate adversarial perturbations on MNIST and Fashion MNIST datasets.
Paper uses polar field data to improve solar flare prediction accuracy.
problem Improving solar flare prediction accuracy using machine learning.
method Incorporates polar field data into machine learning models for solar flare classification.
result Improves solar flare prediction performance by up to 10.1% using a novel probabilistic mixture of experts model.
We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classificatio…
We relate the Lipschitz-Killing measures of a definable set in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of , such results were established by Langevin and Shifrin.Then we give infinitesimal versions of these results. As a corollary, we…
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributiona…
Projective varieties remain stable under close polarizations, extending to Kähler cones.
problem Maintaining stability of projective varieties under close polarizations.
method Uniformly valuative stability definition and extension to Kähler cones.
result Openness of uniformly valuative stability on the Kähler cone of projective manifolds.
Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)"math.DG
Clarifies definition of polarized canonical radius in Kahler Ricci flow.
problem Unclear definition of polarized canonical radius in Kahler Ricci flow.
method Clarification of the definition.
result Clarified definition of polarized canonical radius.
Totally geodesic dual leaves on curved manifolds are also curved.
problem Characterizing dual leaves of nonnegatively curved polar manifolds.
method Proving dual leaves are totally geodesic and closed, and inducing a Riemannian submersion.
result Dual leaves of nonnegatively curved polar manifolds are themselves nonnegatively curved and totally geodesic.
Developed concentrated liquidity in n-dimensional AMM with polar coordinates in Rust.
problem Risk of stacking too many stablecoin pools.
method Building concentrated liquidity positions with ticks in polar coordinates in Rust.
result Hedging risk of stacking stablecoin pools.
The paper defines generalized s-manifolds and explores their polars and antipodal sets.
problem Understanding polars and antipodal sets in generalized s-manifolds.
method Introduced generalized s-manifolds and provided a method to construct them. Studied polars and antipodal sets.
result Extended results on compact symmetric spaces to generalized s-manifolds.