arXiv research

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

In this article we extend cutting and blowing up to the nonrational symplectic toric setting. This entails the possibility of cutting and blowing up for symplectic toric manifolds and orbifolds in nonrational directions.

Let X be a compact toric surface. There exists a sequence of torus equivariant blow-ups of X such that the blown-up toric surface obtained admits a cscK metric.

Let $X$ be a compact toric extremal Kähler manifold. Using the work of Székelyhidi, we provide a combinatorial criterion on the fan describing $X$ to ensure the existence of complex deformations of $X$ that carry extremal metrics. As an example, we find new CSC metrics on 4-points blow-ups of $\C¶^1\times\C¶^1$.

We prove that $π_1(\text{Ham}(M))$ contains an infinite cyclic subgroup, where $\text{Ham}(M)$ is the Hamiltonian group of the one point blow up of ${\Bbb C}P^3$. We give a sufficient condition for the group $π_1(\text{Ham}(M))$ to contain an infinite cyclic subgroup, when $M$ is a general toric manifold.

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

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case o…

The paper extends toric variety correspondence to 4D almost complex torus manifolds.

Study real semi-stable degenerations and describe real loci via blow-ups.

We apply a local differential geometric framework from Kähler toric geometry to (re)construct Calabi's extremal Kähler metrics on $\bbC\bbP^n$ blown-up at a point from data on the moment polytope.

We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\C¶^n$, Hirzebruch surfaces, the blow up of $\CP^2$ at arbitrarily many points, and other toric varieties,…

Book teaches how Lagrangian torus fibration base geometry can be read off.

In this expository paper we review on the existence problem of Einstein-Maxwell Kähler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell Kähler metrics, and give extensions of volume minimization principle, the notion of toric K-stability and other related resu…

We describe the integral cohomology of $X/G$ where $X$ is a compact complex manifold and $G$ a cyclic group of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient conditions …

The modified J-flow with Calabi ansatz shows convergence or blow-up behavior based on topological constants.

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 construct a functor from the category of manifolds with generalized corners to the category of complexes of toric monoids, and for every `refinement' of the complex associated to a manifold, we show there is a unique `blow-up', i.e., a new manifold mapping to the original one, which satisfies a universal property an…

Given a compact Kahler manifold with an extremal metric (M,ω), we give sufficient conditions on finite sets points p_1,...,p_n and weights a_1,...a_n for which the blow up of M at p_1,...,p_n has an extremal metric in the Kahler class π^*[ω] - ε(a_1 PD[E_1] + .. + a_n PD[E_n]) for all εsufficiently small. In particular…

This note is a step towards demonstrating the benefits of a symplectic approach to studying equivariant Kähler geometry. We apply a local differential geometric framework from Kähler toric geometry due to Guillemin & Abreu to the case of the standard linear $\SU(n)$ action on $\bbC^n\setminus\{0\}$. Using this framewor…

Proves mass theorem up to dimension 19 using symmetrization and singularity techniques.

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga…

In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally étale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in th…

Study contact geometry of symplectic divisors, invariant under specific transformations.

We present a classification of compact Kaehler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest. The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a Kaehler manifo…

We pursue the symplectic description of toric Kahler manifolds. There exists a general local classification of metrics on toric Kahler manifolds equipped with Hamiltonian two-forms due to Apostolov, Calderbank and Gauduchon(ACG). We derive the symplectic potential for these metrics. Using a method due to Abreu, we rela…

The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.

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.

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…

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.

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

Toric quasifolds extend toric geometry to non-rational polytopes.

Cylindrical contact homology 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 $S^2 imes S^2$.

Paper generalizes toric concepts to nonrational settings.

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…

New SKT manifolds created using toric geometry.

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

Paper describes holomorphic polyvector fields on toric varieties.

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…

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…

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,…

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

Study shows certain toric arrangements have minimal topological complements.

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.