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.
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.
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.
Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
Formula found for minimum Ricci curvature in Fano bundles.
problem Finding the minimum Ricci curvature in Fano homogeneous toric bundles.
method Derived a formula for the greatest lower bound on Ricci curvature.
result Formula for the minimum Ricci curvature in Fano bundles.
Proves GAGA-style result for toric vector bundles.
problem None explicitly stated in the abstract.
method Algebraic construction of Frölicher approximating vector bundle.
result Proves GAGA-style result for toric vector bundles.
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.
A principal toric bundle M is a complex manifold equipped with a free holomorphic action of a compact complex torus T. Such a manifold is fibered over M/T, with fiber T. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety X⊂M o…
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…
The paper proves the existence of constant scalar metrics on stable toric bundles.
problem Existence of constant scalar metrics on stable toric bundles.
method Study of the generalized Abreu equation on Delzant polytopes.
result Existence of constant scalar metrics on stable homogeneous toric bundles.
The paper constructs toric vector bundles using spectral networks and non-abelianization.
problem Understanding how holomorphic vector bundles arise from spectral networks and non-abelianization.
method Constructing toric vector bundles on complete toric surfaces via spectral networks and non-abelianization.
result The moduli space of rank 2 toric vector bundles over toric surfaces admits an A-type X-cluster structure. 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.
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…
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…
It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.
Kähler-Ricci flow converges on specific toric bundles to solitons.
problem Analyzing Kähler-Ricci flow on homogeneous toric bundles.
method Normalized Kähler-Ricci flow with invariant initial form.
result Flow converges to Kähler-Ricci solitons modulo torus action.
Construct SU(3) structures on bundles over complex manifolds.
problem Creating globally-defined SU(3) structures on bundles. method Construction on smooth compact toric varieties and extensions to Kähler-Einstein manifolds.
result Extends SU(3) structures to non-toric Kähler-Einstein manifolds. A simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold GC/P is provided in terms of symplectic data.
This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric …
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.
New examples found of complex manifolds with special metrics.
problem Existence of Kähler-Einstein metrics on certain complex manifolds.
method Using Hultgren's polytope formulation, constructing explicit examples of toric Fano manifolds.
result Found examples of projective bundles that admit coupled Kähler-Einstein metrics but no ordinary Kähler-Einstein metrics.
We find sufficient conditions for a principal toric bundle over compact Kähler manifolds to admit Calabi-Yau connections with torsion. With the aids of a topological classification, we construct such geometry on $n(S^2\times S^4)#(n+1)(S^3\times S^3)$
In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on com…
In this note, stimulated by the existence result of Futaki-Ono-Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of P^1-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here the …
We study the contact equivalence problem for toric contact structures on S3-bundles over S2. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To f…
In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant f1 for the first…
Established a correspondence for toric fibrations using Delzant polytopes.
problem Existence of extremal Kähler metrics on toric fibrations.
method Using weighted constant scalar curvature Kähler metrics and uniform K-stability.
result Equivalence between extremal metrics and weighted uniform K-stability of Delzant polytopes.
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…
Study finds limits on Sasaki manifold curvature.
problem Determining bounds on Sasaki manifold curvature.
method Analogue of Li's work on toric Fano manifolds, using Wang-Zhu and Futaki-Ono-Wang's results.
result Greatest lower bounds on transverse Ricci curvature determined.
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.
The paper describes the Picard group and quantization in toric orbifolds.
problem Link between complex and symplectic aspects in orbifold setting.
method Combinatorial description of orbifold Picard group.
result Breakdown of identification of line bundles by Chern class.
This paper solves a question about curves on toric surfaces.
problem Which curves can be vanishing cycles for degenerations?
method Reformulated as a mapping class group problem and determined the monodromy group.
result A complete answer to the question of which curves can be vanishing cycles.
Formula for α-Futaki character on toric manifolds.
problem Obstruction to Kähler-Yang-Mills equations existence.
method Provided a formula and computed values for α-Futaki character.
result No solutions with α>0 on certain ample line bundles over toric manifolds.
We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…
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 studies reducibility properties in Sasakian geometry, classifying certain contact structures and extremal metrics.
problem Reducibility properties in Sasakian geometry, focusing on contact structures and extremal metrics.
method Developed the Sasaki version of the de Rham Decomposition Theorem, introduced cone reducible concept, and classified Sasakian structures.
result Classified all Sasakian structures up to contact isotopy on S3 bundles over a Riemann surface of genus greater than zero, and showed extremal Sasaki metrics split in the toric case. Study of hyperkähler reduction on abelian varieties and toric manifolds.
problem Understanding hyperkähler reduction on specific manifolds.
method Lifts canonical Kähler reduction to hyperkähler, studies on abelian varieties and toric manifolds.
result Obtains decoupling result, variational characterisation, relation to K-stability, and proves existence and uniqueness under suitable assumptions. Study shows how the energy of a metric on a toric variety relates to the volume of holomorphic sections.
problem Understanding the volume of holomorphic sections on projective toric varieties.
method Defined energy at equilibrium and showed its asymptotic behavior as a function of the volume of L2-norm unit balls. result The energy of a metric on a toric variety describes the asymptotic behavior of the volume of holomorphic sections.
In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Ou…
The paper studies non-Kähler LVMB manifolds and their metrics.
problem Existence and properties of special metrics on LVMB manifolds.
method Analyzes LVMB manifolds with holomorphic bundle structures over toric bases, using combinatorial data.
result Constructs a new example of a balanced metric on an LVMB manifold.
Study G2-manifolds from symplectic SU(3)-manifolds with T2-symmetry.
problem Understanding G2-manifolds from symplectic SU(3)-manifolds. method Cohomological lifting of multi-toric graphs.
result Compact part of G2-moment graph can be obtained cohomologically from the base. The mirror of a projective toric manifold XΣ is given by a Landau-Ginzburg model (Y,W). We introduce a class of Lagrangian submanifolds in (Y,W) and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over XΣ. Through this ge…
A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a …
This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifo…
Logarithmic connections on principal bundles over normal varieties are studied.
problem Existence and properties of logarithmic connections on principal bundles over normal varieties.
method Introducing logarithmic connections, showing equivalence to covariant derivatives, and proving existence conditions.
result Existence of logarithmic connections on principal bundles over normal varieties is equivalent to certain conditions on the associated vector bundles and adjoint bundles.
The study connects spectral theory of lens spaces with Ehrhart theory.
problem Determining isospectral lens spaces.
method Employing Ehrhart quasi-polynomials and lattice structures, the study introduces toric varieties and uses harmonic polynomial representations.
result Isospectral lens spaces have equal global sections of powers of a line bundle and the same general intersection number.
The paper proves spectral convergence for a specific type of geometric quantization.
problem Spectral convergence of ∂-Laplacians on toric symplectic manifolds. method Study of a family of compatible complex structures converging to the large complex structure limit.
result Spectral convergence of ∂-Laplacians acting on Lk. The paper extends metrics and solitons on toric Fano manifolds with irregular Sasaki-Einstein metrics.
problem Extension of metrics and solitons on toric Fano manifolds with irregular Sasaki-Einstein metrics.
method Verification of the extension of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space of positive rational powers of the canonical line bundle.
result The extended metric along the zero section has an expression that can be extended to the total space, and restricts to a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric.