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.
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.
Study of CR Yamabe problem on toric contact manifolds.
problem CR Yamabe problem on toric contact manifolds.
method Equivalence to boundary value problem for elliptic PDE.
result Many contact toric manifolds admit CR structures with distinct CR Yamabe invariants.
Study properties of contact structures on symplectic disk bundles with concave boundaries.
problem Understanding the geometric properties of contact structures on concave boundaries of symplectic disk bundles.
method Use tools from toric geometry and algebraic torsion measurements from embedded contact homology.
result All such contact manifolds have a global contact toric structure, and can be tight or overtwisted.
A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells…
Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric var…
The paper extends graph complexity calculations for certain 3-manifolds up to 14.
problem Calculating the minimum 4-colored graph complexity of compact 3-manifolds.
method Exact calculations and two-sided bounds for graph complexity.
result Exact value of graph complexity computed for an infinite family of tetrahedral manifolds.
We find all extremal Lagrangian tori in symplectic unit balls and some toric domains.
problem Characterize extremal Lagrangian tori in symplectic manifolds.
method Analyzing symplectic area and using geometric properties of toric domains.
result Every extremal Lagrangian torus in the unit ball is on the boundary.
We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donalds…
Graphs with 4 colors classify 3-manifolds with toric boundaries.
problem Classifying 3-manifolds with toric boundaries.
method Using 4-colored graphs to represent and classify manifolds.
result Complete catalog of 3-manifolds up to 12 vertices.
We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated…
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.
The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.
problem Compactifying character varieties of punctured surfaces.
method Projective compactifications using ideal triangulations and Komyo's method.
result Boundary divisors are toric varieties and the boundary complex is a sphere.
Study linear subvarieties of meromorphic differential strata, proving toric closures and new proofs of theorems.
problem Understanding linear subvarieties in strata of meromorphic differentials.
method Investigate closures in multi-scale compactification, prove restrictions on period coordinates.
result Prove closures are locally toric varieties, generalize cylinder deformation theorem.
New constraints found for Lagrangian embeddings in symplectic fillings.
problem Understanding Lagrangian embeddings in symplectic fillings with semisimple cohomology.
method Deriving constraints through Lagrangian embeddings and symplectic cohomology analysis.
result Existence of many non-toric monotone symplectic manifolds with proper wrapped Fukaya categories.
We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytop…
Solves a general class of free boundary Monge-Ampère equations.
problem Optimal transport with degenerate densities and geometric problems.
method Analyzes a specific class of Monge-Ampère equations and their applications.
result Solves the equations for a general class, including applications to optimal transport and geometric problems.
We study the half-form Kaehler quantization of a smooth symplectic toric manifold (X,ω), such that [ω/2π]−c1(X)/2∈H2(X,Z) and is nonnegative. We define the half-form corrected quantization of (X,ω) to be given by holomorphic sections of a certain hermitian line bundle L→X with Ch…
We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion c…
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.
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…
Various curvature conditions are studied on metrics admitting a symmetry group. We begin by examining a method of diagonalizing cohomogeneity-one Einstein manifolds and determine when this method can and cannot be used. Examples, including the well-known Stenzel metrics, are discussed. Next, we present a simplification…
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 contact geometry of symplectic divisors, invariant under specific transformations.
problem Understanding contact structures on symplectic divisors and their boundaries.
method Invariant analysis of contact structures under toric and interior blow-ups/blow-downs, open book decomposition construction.
result Contact structure on divisor boundaries is invariant under specified transformations.
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.
Homological mirror symmetry proved for symmetric squares of punctured spheres.
problem Proving homological mirror symmetry for symmetric squares of punctured spheres.
method Constructed quasi-equivalences between wrapped Fukaya categories and derived categories of coherent sheaves, using categorical resolutions and localisation.
result Wrapped Fukaya category of symmetric square quasi-equivalent to coherent sheaves on a singular surface.
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.
Let X be a compact toric Kähler manifold with −KX nef. Let L⊂X be a regular fiber of the moment map of the Hamiltonian torus action on X. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of X as virtual counts of holomorphic discs with Lagrangian boundary condition L. We prove a formula…
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.
Let M be a complete finite-volume hyperbolic 3-manifold with compact non-empty geodesic boundary and k toric cusps, and let T be a geometric partially truncated triangulation of M. We show that the variety of solutions of consistency equations for T is a smooth manifold or real dimension 2k near the point representing …
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…
We describe a natural strategy to enumerate compact hyperbolic 3-manifolds with geodesic boundary in increasing order of complexity. We show that the same strategy can be employed to analyze simultaneously compact manifolds and finite-volume manifolds having toric cusps. In opposition to this we show that, if one allow…
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.