Twisted toric structures

Twist tori and pseudo toric structures

Twist tori are examples of exotic monotone lagrangian tori, presented in [1]. This tree of examples grew up over the first one --- the torus $Θ\in \R^4$, constructured in [2] and [3]. On the other hand, in [4] and [5] we proposed a new structure which generalizes the notion of toric structure. One calls this generalization pseudo toric structure, and several examples were given which show that certain toric symplectic manifolds can carry the structre and that certain non toric symplectic manifolds do the same. Below we show that any twist torus $Θ^k \subset \R^{2k+2}$, defined in [1], can be c…

Construct $b$-symplectic toric manifolds from toric manifolds

In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies $b$-symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on $b$-symplectic manifolds. In this paper, we provide a new way to construct $b$-symplectic toric manifolds from usual toric manifolds. Conversely, through this way, we can also decompose a $b$-symplectic toric manifolds to usual toric manifolds. Finally, we will try to prove that this kind of decomposition is useful, although the symplectic structure for our decomposition o…

Toric Structures on Symplectic Bundles of Projective Spaces

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,ω)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $\bC P^r\times \bC P^s$ with any given symplectic form has a unique toric structure provided that $r,s\geq 2$. In contrast, the product $\bC P^r \times \bC P^1$ can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $ω$. In this paper we extend these results by considerin…

Unique toric structure on a Fano Bott manifold

We prove that if there exists a $c_1$-preserving graded ring isomorphism between integral cohomology rings of two Fano Bott manifolds, then they are isomorphic as toric varieties. As a consequence, we give an affirmative answer to McDuff's question on the uniqueness of a toric structure on a Fano Bott manifold.

On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$

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 find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we …

Toric generalized Kaehler structures. III

The paper clarifies some subtle points surrounding the definition of scalar curvature in generalized K$\ddot{a}$hler (GK) geometry. We have solved an open problem in GK geometry of symplectic type posed by R. Goto \cite{Go1} on relating the scalar curvature defined in terms of generalized pure spinors \emph{directly} to the underlying biHermitian structure. In particular, we apply this solution to toric GK geometry of symplectic type and prove that the scalar curvature suggested in this setting by L. Boulanger \cite{Bou} coincides with Goto's definition.

Poisson Cohomology of holomorphic toric Poisson manifolds. I

A holomorphic toric Poisson manifold is a nonsingular toric variety equipped with a holomorphic Poisson structure, which is invariant under the torus action. In this paper, we computed the Poisson cohomology groups for all holomorphic toric Poisson structures on $CP^n$, with the stand Poisson structure on $CP^n$ as a special case. We also computed the algebraic and the formal Poisson cohomology groups of holomorphic toric Poisson structures on $C^n$.

通过退化在K3曲面上的几乎环面纤维化

Almost toric fibrations on K3 surfaces via degenerations

本文通过Kulikov模型研究Kähler K3曲面，证明一般光滑纤维在交复形上具有几乎环面纤维化，并给出光滑Fano三维流形中反典范超曲面的辛Kulikov模型。

Simply Connected Manifolds with Infinitely Many Toric Contact Structures and Constant Scalar Curvature Sasaki Metrics

We study a class of simply connected manifolds in all odd dimensions greater than 3 that exhibit an infinite number of toric contact structures of Reeb type that are inequivalent as contact structures. We compute the cohomology ring of our manifolds by using the join construction for Sasaki manifolds and show that all such contact structures admit a ray of compatible Sasaki metrics of constant scalar curvature (CSC). Furthermore, infinitely many such structures admit at least 3 rays of constant scalar curvature Sasaki metrics.

Grossberg-Karshon twisted cubes and basepoint-free divisors

Let $G$ be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible $G$-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e. closed, convex) polytope; in this case, Grossberg and Karshon's ch…

A limit of toric symplectic forms that has no periodic Hamiltonians

We calculate the Riemann-Roch number of some of the pentagon spaces defined in [Klyachko,Kapovich-Millson,HK1]. Using this, we show that while the regular pentagon space is diffeomorphic to a toric variety, even symplectomorphic to one under arbitrarily small perturbations of its symplectic structure, it does not admit a symplectic circle action. In particular, within the cohomology classes of symplectic structures, the subset admitting a circle action is not closed.

The symplectic structure of a toric conic transform

Suppose that a compact $r$-dimensional torus $T^r$ acts in a holomorphic and Hamiltonian manner on polarized complex $d$-dimensional projective manifold $M$, with nowhere vanishing moment map $Φ$. Assuming that $Φ$ is transverse to the ray through a given weight $\boldsymbolν$, associated to these data there is a complex $(d-r+1)$-dimensional polarized projective orbifold $\hat{M}_{\boldsymbolν}$ (referred to as the $\boldsymbolν$-th \textit{conic transform} of $M$). Namely, $\hat{M}_{\boldsymbolν}$ is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polariz…

Toric origami structures on quasitoric manifolds

We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metric properties.

凹型辛流形的toric填充

Concave symplectic toric fillings

本文研究了接触toric3维流形的凹型辛填充结构，证明了其存在无限多个互不equivariant辛同构且不通过吹胀操作相关的填充。

Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$

I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold $S^2\times S^3$. In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, $Y^{p,q}$, discovered by physicists by showing that $Y^{p,q}$ and $Y^{p',q'}$ are inequivalent as contact structures if and only if $p\neq p'$.

Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

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. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials.

The cutting construction of toric symplectic and contact manifolds

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 a weakly convex good cone by the cutting construction. (Note that these toric contact manifolds can not be constructed by Delzant construction.) We further prove there are no toric Sasakian structures on these contact manifolds. From this, contact toric manifolds of toric K-cont…

The topology of toric symplectic manifolds

This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The …

Real and integral structures in quantum cohomology I: toric orbifolds

We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. Secondly, we use mirror symmetry to calculate the "most natural" integral structure in quantum cohomology of toric orbifolds. We show that the integral structure pulled back from the singularity B-model is described only in terms of topological data in the A-model; K-group and a characteristic class. Using integral structures, we g…

Spectral convergence in geometric quantization --- the case of toric symplectic manifolds

In this paper, we show the spectral convergence result of $\overline{\partial}$-Laplacians when $(X,ω)$ is a compact toric symplectic manifold equipped with the natural prequantum line bundle $L$. We consider a family $\{ J_s\}_s$ of $ω$-compatible complex structures tending to the large complex structure limit, and obtain the spectral convergence of $\overline{\partial}$-Laplacians acting on $L^k$.

An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the Gamma-class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau-Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan's crepant resolution conjecture.

单调吹胀在辛托罗形几何中的结构及麦杜夫的问题

The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff

研究单调多面体的吹胀结构，证明仅简单形和其二维吹胀体允许顶点吹胀，并回答麦杜夫2011年提出的问题。

Open WDVV方程与环面Calabi-Yau 3-fold的Frobenius结构

Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds

本文证明环面Calabi-Yau 3-fold上开Gromov-Witten理论的两种open WDVV方程，分别构造半单Frobenius流形与平坦$F$-流形。

Poisson Cohomology of holomorphic toric Poisson manifolds. II

In this paper, we give a description of holomorphic multi-vector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties. Based on the result, we compute the Poisson cohomology groups of holomorphic toric Poisson manifolds, i.e., toric varieties endowed with $T$-invariant holomorphic Poisson structures.

Uniqueness and examples of compact toric Sasaki-Einstein metrics

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 compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained fr…

Brane quantization of toric Poisson varieties

In this paper we propose a noncommutative generalization of the relationship between compact Kähler manifolds and complex projective algebraic varieties. Beginning with a prequantized Kähler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented f…