The toric manifolds in question were invented by Bott and studied by Grossberg and Karshon under the name "Bott towers". Interest in them comes from their relation to characters of semisimple Lie groups and geometric quantization. We offer another construction of them, introduce a family symplectic structures as curvatures of line bundles, work out canonical coordinates (action-angle variables), derive positivity criteria for the curvature, and determine the range of the moment map. A character of the torus group is introduced as an integral in the spirit of Kirillov's character formula and shown to equal the sum over the integral points in the moment polytope by methods from convex analysis and equivariant cohomology. The result is compared with an explicit calculation of a path integral, based on a definition devised for the occasion.
Let $(M, α)$ be a $2n+1$-dimensional connected compact contact toric manifold of Reeb type. Suppose the contact form $α$ is regular, we find conditions under which $M$ is homeomorphic to $S^{2n+1}$.
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…
We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of such a prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern number of the symplectic toric manifold. The proof relies on the theory of generating functions coupled with equivariant cohomology, whereby we adapt Givental's approach to the Arnold conjecture f…
环面折纸流形的概念，削弱了辛环面流形的概念，由Cannas da Silva-Guillemin-Pires \cite{ca-gu-pi11}引入，他们通过矩映射证明了环面折纸流形与折纸模板的双射对应，其中折纸模板是具有一些折叠数据的Delzant多面体的集合。类似于扇形与德尔赞多面体相关联，在\cite{ha-ma03}和\cite{masu99}中引入的多扇形可以与定向折纸模板相关联。本文讨论了它们之间的关系，证明了任何单连通紧致光滑4-流形具有光滑作用$T^2$都可以是环面origami流形。我们还刻画了产品的偶数维领域，可以复曲面折纸流形。
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.
In this paper we consider the oscillatory integrals on Lefschetz thimbles in the Landau-Ginzburg model as the mirror of a toric Fano manifold. We show these thimbles represent the same relative homology classes as the characteristic cycles of the corresponding constructible sheaves under the equivalence of \cite{GPS18-2}. Then the oscillatory integrals on such thimbles are the same as the integrals on the characteristic cycles and relate to genus $0$ Gromov-Witten descendant potential for $X$, and this leads to a proof of Gamma II conjecture for toric Fano manifolds.
We give an explicit formula for the Gromov width for a class of symplectic toric manifolds constructed from simple graphs. As a corollary, we show a version of non-squeezing theorem with respect to the inclusion of connected graphs.
We study the equivariant disc potentials for immersed SYZ fibers in toric Calabi-Yau manifolds. The immersed Lagrangians play a crucial role in the partial compactification of the SYZ mirrors. Morever, their equivariant disc potentials have a close relation with that of Aganagic-Vafa branes. We show that the potentials can be computed by using an equivariant version of isomorphisms in the Fukaya category.
We show that existence and uniqueness of solutions to transported Monge-Ampere problem on complex compact toric manifold follows easily from the real theory of optimal transportation.
流形上的折叠辛形式是沿着超曲面具有最温和可能退化的闭2-形式。折纸辛流形是一类特殊的折叠辛流形，由Cannas da Silva，Guillemin和Pires研究，他们通过组合折纸模板对环面折纸流形进行了分类。本文研究了具有非循环折纸模板和可共定向折叠超曲面的环面折纸流形的拓扑。我们证明了上同调集中在偶数次，等变上同调满足GKM描述。最后，我们证明了具有共同定向折叠超曲面的环面折纸流形提供了一类Masuda和Panov环面流形的例子。
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 Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, i.e. any toric symplectic filling with zero first Chern class.
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.
The goal of this paper is to classify symplectic toric stratified spaces with isolated singularities. This extends a result of Burns, Guillemin, and Lerman which carries out this classification in the compact connected case. In making this classification, it is necessary to classify symplectic toric cones. Via a well-known equivalence between symplectic toric cones and contact toric manifolds, this allows for the classification of contact toric manifolds as well, extending Lerman's classification of compact connected contact toric manifolds.
Due to a previous result which states that contact varieties are isomorphic to certain varieties, the momentum polytopes of contact manifolds are convex.
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.
