Toric流形的KO-理论

Classification of toric manifolds over an $n$-cube with one vertex cut

We say that a complete nonsingular toric variety (called a toric manifold in this paper) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds (or Bott towers) are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\mathrm{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Oda's $3$-fold, the simplest non-projective toric manifold, is over $\mathrm{vc}(I^n)$. In this paper, we classify toric manifolds over $\mathrm{vc}(I^n)$ $(n\ge 3)$ as vari…

Resolution of singularities of toric orbifolds and equivariant cobordism of contact toric manifolds

Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.

The rational homology of real toric manifolds

This is an extended abstract for a talk given at the mini-workshop "Cohomology rings and fundamental groups of hyperplane arrangements, wonderful compactifications, and real toric varieties", held in Oberwolfach, September 30-October 6, 2012. We describe a formula (obtained in joint work with Alvise Trevisan) for computing the Betti numbers of real toric manifolds, based on a study of cohomology with coefficients in rank one local systems for various polyhedral products. Applying this formula, we recover Henderson's computation of the Betti numbers of real Hessenberg varieties. We conclude wit…

On Transition Functions of Topological Toric Manifolds

We show that any topological toric manifold can be covered by finitely many open charts so that all the transition functions between these charts are Laurent monomials of $z_j$'s and $\bar{z}_j$'s. In addition, we will describe toric manifolds and some special class of topological toric manifolds in terms of transition functions of charts up to (weakly) equivariant diffeomorphism.

Toric manifolds over cyclohedra

We study the action of the dihedral group on the (equivariant) cohomology of the toric manifolds associated with cycle graphs.

Equivariant bordism of 2-torus manifolds and unitary toric manifolds--a survey

In this paper we survey results and recent progresses on the equivariant bordism classification of 2-torus manifolds and unitary toric manifolds.

Cohomological rigidity for toric Fano manifolds of small dimensions or large Picard numbers

The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with Picard number $\ge 2d-2$. In particular, we show that those manifolds except for two toric Fano $4$-folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID…

三维简单多面体上的紧致流形

Toric manifolds over 3-polytopes

本文研究了三维简单多面体上拟紧致流形的存在性，并指出某些多面体不对应于光滑射影紧致变种，揭示了富勒烯与紧致流形的关系。

Uniqueness of the direct decomposition of toric manifolds

In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as \emph{algebraic varieties} is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as \emph{smooth manifolds} is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension …

Wedge operations and torus symmetries

A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. More generally, it is known that some class of manifolds having well-behaved torus actions, called topological toric manifolds $M^{2n}$, can be classified in terms of combinatorial data containing simplicial complexes with $m$ vertices. We remark that topological toric manifolds are a generalization of smooth toric varieties. The number $m-n$ is known as the Picard number when $M^{2n}$ is a {compact smooth} toric variety. In this paper, we investigate the relationship between the topological t…

环面与齐性紧 Kähler 流形的强形式性

Strong formality of toric and homogeneous compact Kähler manifolds

本文研究紧 Kähler 流形的强形式性，证明完备光滑复环面簇与齐性紧 Kähler 流形能以相容方式同时满足理性形式性和强形式性。

On the KO-groups of toric manifolds

In this paper we consider the real topological K-groups of a toric manifold M, which turns out to be closely related to the topology of the small cover MR, the fixed points under the canonical conjugation on M . Following the work of Bahri and Bendersky [2], we give an explicit formula for the KO-groups of toric manifolds, and then we characterize the two extreme classes of toric manifolds according to the two A(1) modules shown in [2].

$K$-theory of $\mbox{hyperKähler}$ toric manifolds

Let $X$ be a toric $\mbox{hyperKähler}$ manifold. The purpose of this note is to describe the topological $K$-ring $K^*(X)$ of $X$. We give a presentation for the topological $K$-ring in terms of generators and relations similar to the known description of the cohomology ring of these manifolds.

环面流形上同调刚性的一则反例

A Counterexample to Cohomological Rigidity of Toric Manifolds

构造两个复维数四的环面流形，它们非同伦等价但整系数上同调环同构，从而给出上同调刚性的反例。

A generalization of the Davis-Januszkiewicz construction and applications to toric manifolds and iterated polyhedral products

The fundamental Davis-Januszkiewicz construction of toric manifolds is reinterpreted in order to allow for generalization. Applications involve the simplicial wedge $J$-construction and Ayzenberg's recent identities arising from composed simplicial complexes.

实 toric 流形的整数上同调

On the integral cohomology of real toric manifolds

本文研究实 toric 流形的整数上同调，通过组合数据推导出上同调群公式，并提出环结构的组合公式，进一步给出判断 spin^c 流形的准则。

The KO*-rings of BT^m, the Davis-Januszkiewicz Spaces and certain toric manifolds

This paper contains an explicit computation of the KO*-ring structure of an m-fold product of CP^{\infty}, the Davis-Januszkiewicz spaces and toric manifolds which have trivial Sq^2-homology.

Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold

To a direct sum of holomorphic line bundles, we can associate two fibrations, whose fibers are, respectively, the corresponding full flag manifold and the corresponding projective space. Iterating these procedures gives, respectively, a flag Bott tower and a generalized Bott tower. It is known that a generalized Bott tower is a toric manifold. However a flag Bott tower is not toric in general but we show that it is a GKM manifold, and we also show that for a given generalized Bott tower we can find the associated flag Bott tower so that the closure of a generic torus orbit in the latter is a b…

Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group $\mathbb{Z}_2^n$ and torus group $T^n$, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group $\mathcal{Z}_n(\mathbb{Z}_2^n)$ of 2-torus manifolds, and determine the dimention of $\mathcal{Z}_n(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups $\mathcal{Z}…

Toric origami manifolds and multi-fans

The notion of a toric origami manifold, which weakens the notion of a symplectic toric manifold, was introduced by Cannas da Silva-Guillemin-Pires \cite{ca-gu-pi11} and they show that toric origami manifolds bijectively correspond to origami templates via moment maps, where an origami template is a collection of Delzant polytopes with some folding data. Like a fan is associated to a Delzant polytope, a multi-fan introduced in \cite{ha-ma03} and \cite{masu99} can be associated to an oriented origami template. In this paper, we discuss their relationship and show that any simply connected compac…

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces are all non-free and noncompact, but still built via simple convex polytopes. We obtain an explicit formula of Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the $h$-vector of a simple convex polytope. As…

等变上同调区分环面流形

Equivariant cohomology distinguishes toric manifolds

具有群作用的空间的等变上同调不仅是作用群的分类空间的上同调环上的环，而且是其上的代数。证明了环面流形（即紧致光滑环面簇）同构为簇的充要条件是它们的等变上同调代数是弱同构的。我们还证明了拟流形，这可以被认为是一个拓扑对应环面流形，等变同胚当且仅当其等变上同调代数是同构的。

Cohomology of toric origami manifolds with acyclic proper faces

A toric origami manifold is a generalization of a symplectic toric manifold (or a toric symplectic manifold). The origami symplectic form is allowed to degenerate in a good controllable way in contrast to the usual symplectic form. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes, and the cohomology and equivariant cohomology rings of a symplectic toric manifold can be described in terms of the corresponding polytope. Recently, Holm and Pires described the cohomology of a toric origami manifold $M$ in terms of the orbit space $M/T$ when $M$ is orientable and …

A construction of SKT manifolds using toric geometry

We produce infinite families of SKT manifolds by using methods of toric geometry like the $J$-construction. These SKT manifolds are total spaces of certain principal $G$-bundles over smooth projective toric varieties, where $G$ is an even dimensional compact connected Lie group.

Young diagrams and intersection numbers for toric manifolds associated with Weyl chambers

We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of the cohomology of the toric manifold.

The topology of toric origami manifolds

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami symplectic manifolds, studied by Cannas da Silva, Guillemin and Pires, who classified toric origami manifolds by combinatorial origami templates. In this paper, we examine the topology of toric origami manifolds that have acyclic origami template and co-orientable folding hypersurface. We prove that the cohomology is concentrated in even degrees, and that the equivariant cohomology satisfies the GKM description. Fina…

The fundamental group and Betti numbers of toric origami manifolds

Toric origami manifolds are characterized by origami templates, which are combinatorial models built by gluing polytopes together along facets. In this paper, we examine the topology of orientable toric oigami manifolds with coorientable folding hypersurface. We determine the fundamental group. In our previous paper [HP], we studied the ordinary and equivariant cohomology rings of simply connected toric origami manifolds. We conclude this paper by computing some Betti numbers in the non-simply connected case.

Cohomology rings of good contact toric manifolds

A good contact toric manifold $M$ is determined by its moment cone $C$. We compute the equivariant cohomology ring with $\Z$ coefficient of $M$ in terms of the combinatorial data of $C$. Then under a smoothness criterion on the cone $C$, we compute the singular cohomology ring with $\Z$ coefficient of $M$ in terms of the combinatorial data of $C$.

Graph invariants and Betti numbers of real toric manifolds

For a graph $G$, a graph cubeahedron $\square_G$ and a graph associahedron $\triangle_G$ are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the $b$-number, and we show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to a graph cubeahedron. The $b$-number is a counterpart of the notion of $a$-number, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold $X^\mathbb{R}(\triangle_G)$ corresponding to a graph as…

典范胞腔流形的特征数及其应用

Characteristic numbers of canonical toric manifolds and their applications

本文计算了典范胞腔流形的陈数、米尔诺数和庞特里亚金数，及其实数对应物的斯蒂夫尔-惠特尼数，探讨了其在单位ary、定向和非定向 bordism 类的组合刻画、单位ary bordism 环生成元的几何代表、典范胞腔流形的边界判定准则以及其浸入欧几里得空间的维数估计。

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.

复曲面几何中的一些例子

Some examples in toric geometry

我们给出了复曲面几何中关于复曲面流形和准复曲面流形之间关系的两个例子，并提供了基本多胞形和特征映射上的充分条件，使得所得的准复曲面流形几乎是复杂的。

论环面辛流形中实拉格朗日子流形的拓扑

On the topology of real Lagrangians in toric symplectic manifolds

本文研究环面辛流形中由矩多面体对合对称性产生的实拉格朗日子流形，建立Delzant构造的实类比，证明其微分同胚类型由组合数据决定，并实现环面辛del Pezzo曲面上所有可能的连通实拉格朗日微分同胚类型。

关于来自简单复形的一类托里曼结构

On a class of toric manifolds arising from simplicial complexes

本文研究了一类来自简单复形的托里曼结构，提供了拓扑证明的Dehn-Sommerville关系，并分类了具有特定范畴的moment-angle manifold。

Toric spaces and face enumeration on simplicial manifolds

In this paper, we study the well-know $g$-conjecture for rational homology spheres in a topological way. To do this, we construct a class of topological spaces with torus actions, which can be viewed as topological generalizations of toric varieties. Along this way we prove that after doing stellar subdivision operations at some middle dimensional faces of an arbitrary rational homology sphere, the $g$-conjecture is valid. Furthermore, we give topological proofs of several fundamental algebraic results about Buchsbaum complexes and simplicial manifolds. In this process, we also get a few inter…