Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable po…
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48 results for “toric Kähler geometry”
New structures on symplectic manifolds derived from convex functions and matrices.
problem Investigating new types of toric generalized Kaehler structures on compact manifolds.
method Characterizing structures by triples , proving canonical structures, and showing reversibility.
result Underlying each structure is a canonical toric Kähler structure with a symplectic potential given by .
Study of toric generalized Kähler structures with strong Hamiltonian torus actions.
problem Investigating a subclass of toric generalized Kähler manifolds.
method Introduced a generalized Delzant construction to produce non-abelian examples of strong Hamiltonian actions.
result Found a third canonical complex structure making the manifold toric Kähler.
For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The dimensional residue circle action on it admitting a hyperk…
K{ä}hler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functionalmath.DG
Develops Kähler geometry on new varieties for canonical metrics.
problem No specific problem stated; focuses on new varieties.
method Introduces new varieties, develops Kähler geometry, associates convex functions with metrics.
result Provides expression for Mabuchi functional and combinatorial sufficient condition of properness.
The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.
problem Understanding twins in Kähler and Sasaki geometry.
method Introducing weighted extremal Kähler twins and extremal Sasaki twins, studying their properties.
result Many twins appear in the Kähler setting and more than one extremal ray in the Sasaki cone.
We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …
Toric Pluripotential Theorymath.CV
Study of finite energy quasiplurisubharmonic functions on toric Kähler manifolds.
problem Characterizing and understanding finite energy quasiplurisubharmonic functions on toric Kähler manifolds.
method Characterization through convex functions and integrability properties of Legendre transforms.
result Log-Lipschitz convex functions on Delzant polytopes correspond to toric quasiplurisubharmonic functions with exponential integrability.
Study cohomology of quaternionic foliations and orbifolds.
problem Understanding cohomology of quaternionic foliations and orbifolds.
method Definition and proof of foliated versions of classical results for quaternionic Kähler manifolds.
result Formulation and proof of foliated versions of classical results for quaternionic Kähler manifolds.
The paper explores metric reduction in generalized geometry and constructs holomorphic bundles.
problem Metric reduction in generalized geometry and constructing holomorphic bundles.
method Investigation of Bismut connections and construction of metric generalized principal bundles.
result Construction of a family of generalized holomorphic line bundles over .
Toric actions in cosymplectic geometry linked to symplectomorphisms.
problem Understanding toric actions in cosymplectic geometry.
method Showed that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.
result Compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.
New SKT manifolds created using toric geometry.
problem Creating SKT manifolds.
method Using toric geometry and -construction.
result Infinite families of SKT manifolds produced.
Toric Quasifoldsmath.SG
Toric quasifolds extend toric geometry to non-rational polytopes.
problem Extending toric geometry to non-rational polytopes.
method Introduced toric quasifolds to solve symplectic extension problem.
result Illustrated toric quasifolds and their atlases.
Examines nonrational polytopes and fans in toric geometry.
problem Understanding nonrational convex polytopes and fans in toric geometry.
method Discussion and interrelation of recent developments.
result Exploration of nonrational polytopes and fans in toric geometry.
The paper introduces toric separable geometries and finds new extremal metrics.
problem Finding explicit extremal Kähler metrics on toric manifolds.
method Introducing toric separable geometries and analyzing their moduli space.
result Explicit computation of scalar curvature and derivation of necessary conditions for extremality.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
Algebraic Toric Quasifoldsmath.AG
Paper generalizes toric concepts to nonrational settings.
problem Nonrational toric structures.
method Algebraic geometry perspective.
result Reframed symplectic and complex toric quasifolds.
Study global geometry of toric nearly Kähler manifolds using multi-moment maps.
problem Global properties of toric nearly Kähler manifolds.
method Description using multi-moment maps, investigation of polynomial and radial solutions.
result Description of global geometry of toric nearly Kähler manifolds.
Geometrically connects toric varieties to normed spaces.
problem Connecting algebraic geometry with convex analysis.
method Establishes a 1-1 correspondence using topological models.
result Toric varieties correspond to horofunction compactifications of polyhedral norms.
Introduces symplectic reduction in nonrational toric geometry.
problem Symplectic reduction in nonrational toric geometry.
method Specializes to nonrational toric geometry and rational case for symplectic reduction.
result Symplectic reduction for nonrational Lie subgroups.
Toric Kato manifoldsmath.AG
Study of special Kato manifolds derived from toric geometry.
problem Characterize and study properties of Kato manifolds.
method Construction from toric geometry, topological and analytical properties, combinatorial data, flat degenerations, Hermitian geometry.
result No Kato manifold supports balanced or pluriclosed metrics.
The paper studies a generalized Pythagorean theorem on dually flat spaces via toric geometry.
problem Understanding the geometry of dually flat spaces and their toric Kähler manifolds.
method Introducing a dually flat structure and Bregman divergence on the boundary of toric Kähler manifolds.
result A continuity and generalized Pythagorean theorem for the divergence on the boundary.
Generalized Laurent monomials for nonrational spaces.
problem Handling singular spaces in toric geometry.
method Extending Laurent monomials to nonrational toric quasifolds.
result Generalized Laurent monomials defined for nonrational toric quasifolds.
Toric contact manifolds defined in any dimension.
problem No specific problem stated; abstract focuses on definition.
method Description via labelled polytope in grassmannian.
result Toric contact manifolds in arbitrary codimension.
This study connects ReLU neural networks to toric geometry to analyze function realization.
problem Determining which continuous piecewise linear functions can be realized by ReLU neural networks.
method Established a connection between toric geometry and ReLU neural networks, defining key structures like the ReLU fan, toric variety, and Cartier divisor.
result Proved a criterion for functions realizable by unbiased shallow ReLU networks using intersection numbers.
Study Hessian geometry of multi-Taub-NUT metrics and their phase changes.
problem Understanding phase changes in multi-Taub-NUT metrics.
method Analysis via moment maps of Hessian geometry.
result Generalization of earlier work on toric Gibbons-Hawking metrics.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of -extremal metrics on toric manifolds.
The article explores toric spaces of regular polyhedra, highlighting rational and non-rational cases.
problem Exploring toric spaces associated with regular convex polyhedra.
method Symplectic and complex toric spaces associated with five regular convex polyhedra.
result The regular dodecahedron and icosahedron cannot be treated via standard toric geometry.
Toric Hyperkahler Varietiesmath.AG
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawren…
Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.
problem Properties of para-Kähler manifolds with conformal Einstein soliton metrics.
method Investigated curvature properties of para-Kähler manifolds admitting conformal Einstein soliton.
result Certain curvature properties of para-Kähler manifolds were studied.
Study of multi-toric geometries with larger compact symmetry groups.
problem Exploring manifolds with special holonomy and toric structures.
method Analyzing manifolds with multi-moment maps and connected non-Abelian symmetry groups.
result Only structures with cohomogeneity-two action of for -manifolds.
The paper develops quaternionic toric geometry and classifies local actions.
problem Classifying local quaternionic torus actions on manifolds.
method Develops local -actions, introduces invariants, and studies tetraplectic structures.
result Classifies local quaternionic torus actions up to homeomorphism.
This paper classifies instantons with closed reductions and provides examples of non-closed reductions.
problem Understanding the geometry of toric Kähler instantons with and without closed reductions.
method Sharp geometric criteria and examples of instantons with different reduction types.
result Established geometric criteria for closed reductions and classified asymptotic geometries.
The paper connects fibrations to generalized complex structures in semi-toric geometry.
problem Understanding the relationship between fibrations and generalized complex structures.
method Using moment maps in semi-toric geometry and Gompf--Thurston methods for Lie algebroids.
result Constructs self-crossing stable generalized complex four-manifolds and proves compatibility with connected sums.
Classifies all toric Kahler surfaces with twistor 2-forms.
problem Classifying smooth 4-dimensional Kahler geometries with specific properties.
method Complete classification through algebraic and geometric analysis.
result Found six geometrically distinct families of toric Kahler surfaces.
The aim of this thesis is to construct new examples of compact orbifolds which admit a self dual Einstein (SDE) metric of positive scalar curvature , with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galick…
Study -manifolds from symplectic -manifolds with -symmetry.
problem Understanding -manifolds from symplectic -manifolds.
method Cohomological lifting of multi-toric graphs.
result Compact part of -moment graph can be obtained cohomologically from the base.
The paper explores Hamiltonian stationary Lagrangian fibrations in various geometries.
problem Understanding HSLAG submanifolds and fibrations in different geometries.
method Deforming toric Kähler metrics into non-toric almost Kähler metrics to find HSLAG submanifolds.
result A large class of Hamiltonian stationary Lagrangian fibrations are found.
Extends cutting and blowing up to nonrational symplectic settings.
problem Nonrational symplectic toric structures.
method Cutting and blowing up in nonrational directions.
result Extension to symplectic toric manifolds and orbifolds.
Local equivalence found between certain solitons and generalized Kähler-Ricci solitons.
problem Understanding and constructing generalized Kähler-Ricci solitons.
method Establishing local equivalence and extending to complete GKRS under natural conditions.
result Local classification and construction of new examples in all dimensions, especially in four dimensions.
The paper examines stability conditions for toric manifolds using algebraic geometry.
problem Investigating stability conditions for toric manifolds in algebraic geometry.
method Using criteria for relative Chow and K-stability, the paper applies the Hibert-Mumford criterion and considers maximal torus actions and -actions.
result The paper provides a criterion for relative K-stability and instability of toric Fano manifolds and presents counter-examples of relative K-stable but asymptotically Chow unstable manifolds.
Book teaches how Lagrangian torus fibration base geometry can be read off.
problem Understanding geometry of Lagrangian torus fibrations.
method Integral affine structure on fibration base for total space geometry.
result Read off interesting geometry of total space from base.
In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how …
The paper studies CR geometry and introduces a new quotient method.
problem CR geometry in arbitrary codimension.
method Levi-Kahler quotient for constructing Kahler metrics from CR structures.
result Explicit descriptions and characterizations of Levi-Kahler quotients of toric CR manifolds, especially products of odd dimensional spheres.
Quaternionic toric manifoldsmath.DG
New quaternionic toric manifolds introduced, studied with properties.
problem Developing a new type of toric manifold in quaternionic geometry.
method Construction from Delzant polytopes, -plectic structure, generalized moment map.
result Quaternionic toric manifolds are a large class for testing new quaternionic geometry results.
Lecture notes on Kähler geometry in toric varieties.
problem Existence of extremal Kähler metrics in toric manifolds.
method Using the Delzant polytope and stability condition.
result Existence of extremal Kähler metrics in terms of stability condition.
The paper studies Ricci curvature on Kähler-Ricci flow.
problem Analyzing Ricci curvature on Kähler-Ricci flow.
method Examining n-dimensional compact Kähler manifolds with semi-ample canonical line bundles under Kähler Ricci Flow.
result Ricci curvature converges to negative of generalized Kähler Einstein metric locally away from singular set.
Toric Vaisman Manifoldsmath.DG
Vaisman manifolds are strongly related to Kähler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if…