arXiv research

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.

Computes Weyl group of Kähler toric manifold isometries.

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

New toric Fano manifolds found without extremal Kähler metrics.

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,…

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…

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

Strong formal properties for toric and homogeneous Kähler manifolds.

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…

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…

Proof confirms condition for Kähler-Einstein metrics on toric Fano manifolds.

The first eigenvalue of invariant functions on toric Kähler manifolds has no critical points.

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

Geometrically decomposes Kähler functions on toric manifolds.

In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial $(S^1)^n$-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manif…

Kahler toric manifolds linked to dually flat spaces via affine isometry.

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

Study on Kähler-Ricci solitons on toric manifolds, proving rigidity.

Extends Kähler metrics theory to symplectic manifolds with toric actions.

The study finds conditions for the existence of extremal toric almost Kähler metrics.

Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.

We extend Calabi ansatz over Kähler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kähler metric on Kähler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat Kähler metric on the toric Kähler cone manifold…

New examples found of complex manifolds with special metrics.

This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold

Sharp inequalities and extremizers for J functional on Kähler manifolds.

Study minimal Lagrangian tori on Kähler manifolds, answering questions about their existence and stability.

Researchers found unique scalar-flat Kähler metrics on toric surfaces.

This paper classifies instantons with closed reductions and provides examples of non-closed reductions.

Paper proves Kähler-Ricci solitons on a new type of variety.

6D nearly Kähler manifolds with $\mathbb{T}^3$ actions are uniquely determined by a function on $\mathbb{R}^3$.

Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In…

Solves complex Monge-Ampère equation for Kähler-Ricci solitons.

We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.

Paper proves Kähler-Ricci solitons on specific compactifications.

Study of Hermitian structures on toric suspensions of balanced manifolds.

Classifies scalar-flat toric Kähler instantons in 4D.

This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.

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…

Positive holomorphic sectional curvature on rational surfaces is characterized by Kähler metrics.

Formula found for Kähler-Einstein metric existence obstruction.

Study of finite energy quasiplurisubharmonic functions on toric Kähler manifolds.

Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different …

A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2π\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $φ:M\to\R^n$, a …

Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how …

We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture of Lejmi and the second author in this case. We also strengthen existing …