arXiv research

Classifies equivariant vector bundles over toric manifolds.

Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.

The paper constructs toric vector bundles using spectral networks and non-abelianization.

We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques.

Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.

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.

Defines and classifies toric co-Higgs bundles on projective toric varieties.

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 $X\subset M$ 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…

For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.

On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists f…

New SKT manifolds created using toric geometry.

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 $T^*Q\setminus{Q}$. If the geodesic flow is toric integrable, the cosphere bundle admit…

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ 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.

We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.

Extends geometric quantization to singular spaces.

Curvature on Kähler toric manifolds

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to Σ$ over a compact complex curve $Σ$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$ admits a Kähler metric of constant scalar curvature if and only if $E$ is polystable. We also…

Paper describes holomorphic polyvector fields on toric varieties.

We construct characteristic classes of smooth (Hamiltonian) fibrations as as fiber integrals of products of Pontriagin (or Chern) classes of vertical vector bundles over the total space of the universal fibration. We give explicit formulae of these fiber integrals for toric manifolds and get estimates of the dimension …

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.

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.

Researchers construct explicit bundles for ALF metrics, revealing rational patching matrices for gravitational instantons.

In this paper, we study the generalized Abreu equation on a Delzant ploytope $Δ\subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.

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 $n(S^2\times S^4)#(n+1)(S^3\times S^3)$

New examples found of complex manifolds with special metrics.

Proves weight polytope matches with energy vectors in toric varieties.

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 $S^3$-bundles over $S^2$. 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 $f_1$ for the first…

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…

This review discusses solutions to Einstein's equations using twistor theory.

Established a correspondence for toric fibrations using 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…

We construct globally-defined $SU(3)$ structures on smooth compact toric varieties (SCTV) in the class of $\mathbb{CP}^1$ bundles over $M$, where $M$ is an arbitrary SCTV of complex dimension two. The construction can be extended to the case where the base is Kähler-Einstein of positive curvature, but not necessarily t…

Study properties of contact structures on symplectic disk bundles with concave boundaries.

We determine the greatest lower bounds on the transverse Ricci curvature of compact toric Sasaki manifolds with positive basic first Chern class and with the first Chern class of the contact bundle being trivial. This is based on Wang-Zhu's and Futaki-Ono-Wang's works, and is an analogue of C. Li's work on toric Fano m…

Formula for α-Futaki character on 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.

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

Study of hyperkähler reduction on abelian varieties and toric manifolds.

Study BV operators on holomorphic polyvector fields on toric varieties.

Study the geometry and symmetries of moduli spaces of connections on KT manifolds.