arXiv research

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.

This paper provides a new method to construct $b$-symplectic toric manifolds from toric manifolds.

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…

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…

New SKT manifolds created using toric geometry.

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.

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…

Cylindrical contact homology linked to Ehrhart polynomials and Chen-Ruan cohomology.

Study of special Kato manifolds derived from toric geometry.

We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show,…

New toric Fano manifolds found without extremal Kähler metrics.

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and…

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisma…

Study global geometry of toric nearly Kähler manifolds using multi-moment maps.

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…

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…

Quantizes $b$-symplectic toric manifolds using $T$-modules.

Computes Weyl group of Kähler toric manifold isometries.

Moment polytope of toric exponential families is a projection of a simplex.

New submanifolds found in toric manifolds with specific actions.

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

We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.

Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational balls useful for symplectic surgeries. The main result of the paper is a complete c…

Classifies equivariant vector bundles over toric manifolds.

In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative $K$-stability and the properness of modified $K$-energy. In …

Uniform K-stability ensures existence of special metrics on toric manifolds.

The paper derives a formula for Chow weights of toric blow-ups.

We show that any $(\C ^*)^n$-invariant stably complex structure on a topological toric manifold of dimension $2n$ is integrable. We also show that such a manifold is weakly $(\C ^*)^n$-equivariantly isomorphic to a toric manifold.

The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.

Real Lagrangians in toric manifolds are classified by combinatorial data.

Strong formal properties for toric and homogeneous Kähler manifolds.

The paper constructs submanifolds with corners in Delzant polytopes from affine subspaces.

The paper studies a generalized Pythagorean theorem on dually flat spaces via toric geometry.

Classifies toric fibers in $S^2 imes S^2$.

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

Classifies toric dually flat manifolds into complex space forms.

Paper generalizes toric concepts to nonrational settings.

In this paper we study the relative Chow and $K$-stability of toric manifolds in the toric sense. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford cr…

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper f…

The paper examines the topology of quaternionic toric actions on manifolds.

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…

Study shows certain toric arrangements have minimal topological complements.

Proves Stolz conjecture for specific types of manifolds.

Researchers find a list of non-isometric toric para-Kaehler-Einstein manifolds.

We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki…

We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the n-torus and the 2-sphere.