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.
Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
problem Characterizing and applying toroidal and semi-toric compactifications to weak K-moduli.
method Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
result Different proof of a theorem of Alexeev-Engel on weak K-moduli compactifications.
Paper proves Kähler-Ricci solitons on specific compactifications.
problem Existence of Kähler-Ricci solitons on compact manifolds.
method Continuity method applied to wonderful group compactifications.
result Existence of Kähler-Ricci solitons on certain compactifications.
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.
Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off thei…
The paper determines the bifurcation set of a real polynomial function of two variables using Newton polygons.
problem Determining the bifurcation set of a real polynomial function of two variables.
method Using toric compactification and toric modifications to count singular phenomena at infinity.
result An upper bound of the number of elements in the bifurcation set is given in terms of its Newton polygon.
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.
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 proves a compactification for polyhedral norms.
problem Horofunction compactification for polyhedral norms.
method Establishing a criterion for converging sequences and generalizing the moment map.
result The horofunction compactification of a polyhedral norm is homeomorphic to the dual unit ball.
We develop methods to study the singularities of certain G2 cones related to toric hyperkahler spaces and Einstein selfdual orbifolds. This allows us to determine the low energy gauge groups of chiral N=1 compactifications of M-theory on a large family of such backgrounds, which includes the models recently studied …
The paper solves a geometric P=W conjecture for SL(2,C) using Thurston's compactification.
problem Addressing the geometric P=W conjecture for SL(2,C) in projective compactifications of character varieties of closed surfaces.
method Using Thurston's compactification of Teichmüller space and new results, the paper constructs a projective compactification of the SL(2,C)-character variety of any closed surface of genus g>1.
result The boundary divisors are toric varieties and the dual intersection complex is a sphere.
Introduces new limit spaces for degenerating Calabi-Yau families.
problem Understanding degenerating Calabi-Yau families and their limit structures.
method Introduces galaxy spaces as dense subspace of infinite open Calabi-Yau varieties.
result Galaxy spaces are projective limits of toroidal compactifications.
Study connects mirror symmetry invariants to K-stability for toric manifolds.
problem Relating invariants from mirror symmetry to K-stability for toric polarized manifolds.
method Analyzes expansions involving base loci of linear systems from Landau-Ginzburg potentials.
result Shows Z-stability naturally arises from mirror symmetry considerations.
We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of o…
New methods compute geometry of hyperKähler metrics at infinity.
problem Understanding the geometry of hyperKähler metrics at infinity.
method Quasi-asymptotically conical metrics, Taub-NUT deformations, compactification by manifolds with corners.
result Identifies unique tangent cones and cohomology groups.
The paper constructs new G2 manifolds and studies their mirror symmetry.
problem Constructing and understanding G2 manifolds for string theory. method Using twisted connected sums and dual tops, the authors construct and study mirror symmetry for these manifolds.
result Novel dual superstring backgrounds and exact dualities among 2d N=1 sigma models are conjectured.
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.
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…
Study on likelihood functions, associative equations, and Frobenius manifolds.
problem Maximum likelihood estimation and associativity equations in statistical models.
method Analyzes the cone of concentration matrices, log-likelihood function, and Frobenius manifolds.
result Maximum likelihood degree is indexed by components of Frobenius residuals.
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.
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.
Study shows Satake compactifications as horofunctions with polyhedral metrics.
problem Understanding horofunction compactifications of symmetric spaces.
method Investigated invariant Finsler metrics and polyhedral metrics for compactification.
result Generalized Satake compactifications realized as horofunction compactifications.
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.
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…
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.
We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…
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…
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 constructs examples of compactifications for Einstein metrics.
problem Compactifying Einstein metrics with specific properties.
method Constructing projective and c--projective compactifications. result Neutral signature Einstein metrics can be compactified canonically.