Paper proves contact structures overtwisted if small plastikstufe with toric core exists.
problem Contact structures and overtwistedness in higher-dimensional contact manifolds.
method Proves overtwistedness via existence of a small plastikstufe with toric core.
result Contact structures are overtwisted if and only if a small plastikstufe with toric core exists.
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…
We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…
Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
problem Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
method Filtration approach to prove the conjecture.
result Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
Constructs scalar-flat Kähler metrics on toric symplectic manifolds.
problem Creating scalar-flat Kähler metrics on toric symplectic manifolds.
method Explicit construction and alternative construction with conical singularity.
result Explicit construction of scalar-flat Kähler metrics on toric symplectic manifolds.
The paper explores toric Vaisman manifolds and their connections to Sasaki and Kähler geometry.
problem Understanding the geometric relationships between Vaisman, Sasaki, and Kähler manifolds in the toric context.
method Introducing and analyzing toric Vaisman structures, showing relationships between minimal coverings and associated Sasaki manifolds, and proving conditions for toricity.
result Toric Vaisman manifolds have a close relationship with toric Sasaki manifolds, and vice versa, under specific conditions.
Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.
problem Proving uniqueness of Kähler Ricci shrinkers on toric orbifolds.
method Extending results from toric manifolds to toric orbifolds.
result Uniqueness of Kähler Ricci shrinkers on toric orbifolds established.
Study on mean Euler characteristic of Gorenstein toric contact manifolds.
problem Calculating the mean Euler characteristic of Gorenstein toric contact manifolds.
method Using the relationship between mean Euler characteristic and the normalized volume of the toric diagram, and applying results from Batyrev and Dais.
result Twice the mean Euler characteristic of a Gorenstein toric contact manifold equals the Euler characteristic of any crepant toric symplectic filling.
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.
This paper provides a new method to construct b-symplectic toric manifolds from toric manifolds.
problem Classifying and constructing b-symplectic toric manifolds. method A new method to construct b-symplectic toric manifolds from toric manifolds. result This new method allows for the decomposition of b-symplectic toric manifolds into toric manifolds. 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.
Cylindrical contact homology linked to Ehrhart polynomials and Chen-Ruan cohomology.
problem Contact invariants of Q-Gorenstein toric contact manifolds.
method Relationships between cylindrical contact homology and Ehrhart polynomials, Chen-Ruan cohomology.
result Cylindrical contact homology invariants linked to Ehrhart polynomials and Chen-Ruan cohomology.
Classifies toric fibers in S2imesS2.
problem Identify Hamiltonian isotopy of toric fibers.
method Comprehensive classification of toric fibers in FOOO's construction.
result Determines Hamiltonian isotopy of toric fibers.
Study characterizes toric LCK manifolds, proving conjecture and showing differences from symplectic case.
problem Characterizing compact toric locally conformally Kähler manifolds.
method Proves conjecture about toric LCK manifolds, constructs examples to show differences from symplectic case.
result Proves a conjecture about toric LCK manifolds and shows differences from symplectic case.
Paper generalizes toric concepts to nonrational settings.
problem Nonrational toric structures.
method Algebraic geometry perspective.
result Reframed symplectic and complex toric quasifolds.
We introduce the fibred toric varieties as equivariant CPr bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these…
New SKT manifolds created using toric geometry.
problem Creating SKT manifolds.
method Using toric geometry and J-construction. result Infinite families of SKT manifolds produced.
Defines and classifies toric co-Higgs bundles on projective toric varieties.
problem Classifying co-Higgs bundles on toric varieties.
method Using Klyachko's fan filtration and studying the co-Higgs bundle fiber at a closed point.
result Provides a Lie-theoretic classification of toric co-Higgs bundles.
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.
Study on compact toric locally conformally Kähler manifolds, finding specific properties.
problem Characterizing properties of compact toric locally conformally Kähler manifolds.
method Analyzing Kodaira dimension, using specific examples and mappings.
result Kodaira dimension is -∞ for underlying complex manifolds, and specific properties for surfaces and Vaisman manifolds.
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…
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.
Moment polytope of toric exponential families is a projection of a simplex.
problem Understanding the geometry of exponential families in finite sample spaces.
method Toric torification and projection of higher-dimensional simplices.
result Moment polytope is a projection of a higher-dimensional simplex.
Study shows certain toric arrangements have minimal topological complements.
problem Understanding the topology of toric arrangements.
method Associated a matrix to toric arrangements and analyzed those with maximal rank.
result Complement manifolds of these toric arrangements are diffeomorphic to centered ones.
We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the s…
New measure for nonrationality of toric quasifolds.
problem Measuring nonrationality of toric quasifolds.
method Defined nonrationality degree using quasilattice and quasitorus.
result Reframed Gordan's lemma in the nonrational setting.
A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. T…
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.
Study differential inequalities on polytopes and toric bundles.
problem Prove differential inequalities for homogeneous toric bundles.
method Study generalized Abreu Equation in polytopes.
result Prove differential inequalities for homogeneous toric bundles.
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 C∗-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.
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 derives a formula for Chow weights of toric blow-ups.
problem Chow weights of toric blow-ups.
method Combinatorial formula derived from toric manifold and Delzant polytope.
result Explicit formula for Chow weights of blow-ups.
Existence of Kähler-Einstein metrics on toric varieties proven.
problem Existence of Kähler-Einstein metrics on toric varieties.
method Characterization of K-stability using log Cox ring and universal orbifold cover.
result Every Q-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric.
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.
Study co-Higgs sheaves on toric varieties, finding explicit examples.
problem Characterizing and understanding co-Higgs sheaves on toric varieties.
method Characterization and explicit computation of examples.
result Explicit examples of co-Higgs sheaves on toric varieties computed.
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.
Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.
problem Finding metrics for toric Kähler cones with conical singularities.
method Parametrized family of Calabi-Yau cone metrics with conical singularities.
result Any toric Calabi-Yau cone metric with conical singularities belongs to this optimal family.
New toric Fano manifolds found without extremal Kähler metrics.
problem Finding toric Fano manifolds without extremal Kähler metrics.
method Constructing specific toric Fano manifolds of dimensions 10 and n (n≥11) that do not admit extremal Kähler metrics.
result Existence of toric Fano manifolds of dimension 10 and higher that do not admit extremal Kähler metrics.
Researchers describe ECC of concave lens spaces and compute symplectic capacities.
problem Computing symplectic capacities of concave toric domains.
method Combinatorial description of ECC, ECH capacities computation, packing of symplectic manifolds.
result Computed ECH capacities of certain concave toric domains.
We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with c…
Solves a general problem for toric manifolds in Kaehler-Ricci solitons.
problem General problem stated by authors for toric manifolds in Kaehler-Ricci solitons.
method Proves that a Calabi extremal Kaehler-Ricci soliton on a compact toric Kaehler manifold is Einstein.
result Solves for the class of toric manifolds a general problem stated by the authors.
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.
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.
Proves Witten genus vanishes for certain string intersections.
problem Proving the Witten genus vanishes for specific string intersections.
method Using equivariant localization formula of toric varieties.
result Witten genus vanishes for some string complete intersections.
The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T∗Q∖Q. If the geodesic flow is toric integrable, the cosphere bundle admit…
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.
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.
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.