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.
Let $X$ be a $4$-dimensional toric orbifold. If $H^3(X)$ has a non-trivial odd primary torsion, then we show that $X$ is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.
Integral bases for the second degree cohomology of 4-dimensional toric orbifolds
Institute of Mathematical Science, Pusan National University（釜山国立大学数学科学研究所）；Department of Mathematics Education, Pusan National University（釜山国立大学数学教育学系）
The integral cohomology rings of four-dimensional toric orbifolds
Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Institute of Mathematical Science, Pusan National University（釜山国立大学数学科学研究所）；Department of Mathematics Education, Pusan National University（釜山国立大学数学教育系）
We study the GKM theory for a equivariant stratified space having orbifold structures in tis successive quotients. Then, we introduce the notion of an \emph{almost simple polytope}, as well as a \emph{divisive toric variety} generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.
Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an invariant cell structure on it and call such a toric orbifold retractable. In this paper, our main goal is to study equivariant cohomology theories of retractable toric orbifolds. Our results extend the corresponding results on divisive weighted projective spaces.
We prove a version of homological mirror symmetry statement for toric Calabi-Yau $3$-orbifolds, thus extending arXiv:1604.06448 to the case of orbifolds under the mirror symmetry setting considered in arXiv:1604.07123. The B-model is the matrix factorization category for the toric Calabi-Yau $3$-orbifold with a superpotential; while the A-model is a topologically defined Fukaya-type category on its mirror curve.
Given a root system $Φ$ of type $A_n$, $B_n$, $C_n$, or $D_n$ in Euclidean space $E$, let $W$ be the associated Weyl group. For a point $p \in E$ not orthogonal to any of the roots in $Φ$, we consider the $W$-permutohedron $P_W$, which is the convex hull of the $W$-orbit of $p$. The representation of $W$ on the rational cohomology ring $H^\ast(X_Φ)$ of the toric variety $X_Φ$ associated to (the normal fan to) $P_W$ has been studied by various authors. Let $\{s_1,\ldots,s_n\}$ be a complete set of simple reflections in $W$. For $K \subseteq [n]$, let $W_K$ be the standard parabolic subgroup of …
We introduce a new definition of weighted Grassmann orbifolds. We study their several invariant $q$-cell structures and the orbifold singularities on these $q$-cells. We discuss when the integral cohomology of a weighted Grassmann orbifold has no $p$-torsion. We compute the equivariant $K$-theory ring of weighted Grassmann orbifolds with rational coefficients. We introduce divisive weighted Grassmann orbifolds and show that they have invariant cell structures. We calculate the equivariant cohomology ring, equivariant $K$-theory ring and equivariant cobordism ring of a divisive weighted Grassma…
