Hamiltonian torus actions on symplectic orbifolds and toric varieties

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.

     

The coherent-constructible correspondence and Fourier-Mukai transforms

As evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In a previous paper, the authors showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on ℝ ⁿ . In this paper we investigate and reprove some of Kawamata’s results from this perspective.     

Convergence of Kähler-Einstein orbifolds

We proved the convergence of a sequence of two-dimensional compact Kähler-Einstein orbifolds with rational quotient singularities and with some uniform bounds on the volumes and on the Euler characteristics of our orbifolds, to a Kähler-Einstein two-dimensional orbifold. Our limit orbifold can have worse singularities than the orbifolds in our sequence. We will also derive some estimates on the norms of the sections of plurianticanonical bundles of our orbifolds in the sequence that we are considering and our limit orbifold.     

Torsion in the Cohomology of Torus Orbifolds

The authors study torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X with actions of the n-dimensional compact torus.Compact simplicial toric varieties are in our family.For a prime number p,the authors find a necessary condition for the integral cohomology of X to have no p-torsion.Then it is proved that the necessary condition is sufficient in some cases.The authors also give an example of X which shows that the necessary condition is not sufficient in general.     

Representations of orbifold groupoids

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an almost free action of a proper bundle of topological groups.     

Calabi–Yau algebras and weighted quiver polyhedra

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi–Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3-dimensional Calabi–Yau algebras can be constructed from dimer models on a torus. Toric orders are examples of a much broader class of algebras: positively graded cancellation algebras. For these algebras the CY-3 condition implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold.     

Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Comparing to the construction of stringy cohomology ring of equivariant stable almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds, the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold. The authors show that for a finite group G and a G-equivariant stable almost complex manifold X, the G-invariant part of the stringy cohomology ring of (X, G) is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold [X/G]. Similar result holds when G is a torus and the action is locally free. Moreover, for a compact presentable stable almost complex orbifold, they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.     

Compact anti-self-dual orbifolds with torus actions

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat K?hler orbifold. We use this observation to classify ALE scalar-flat K?hler 4-orbifolds whose isometry group contain a 2-torus.     

Orbifold homeomorphism finiteness based on geometric constraints

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman’s stability theorem must preserve orbifold structure.     

The Chen-Ruan cohomology of almost contact orbifolds

Comparing to the Ch-~Ruan cohomology theory for the almost complex orbifolds, we study the orbifold cohomology theory for almost contact orbifolds. We define the Chen-Ruan cohomology group of any almost contact orbifold. Using the methods for almost complex orbifolds, we define the obstruction bundle for any 3-multisector of the almost contact orbifolds and the Chen~Ruan cup product for the Che-Ruan cohomology. We also prove that under this cup product the direct sum of all dimensional orbifold cohomology groups constitutes a cohomological ring. Finally we calculate two examples.     

Local torus actions modeled on the standard representation

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which includes the topological classifications of quasi-toric manifolds by Davis and Januszkiewicz and of effective T²-actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We discuss locally toric Lagrangian fibrations from the viewpoint of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes.     

Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.     

F-structure on collapsed orbifolds

In this paper, we prove the existence of nilpotent Killing structures and F-structures on collapsed Riemannian orbifolds. Therefore a sufficiently collapsed orbifold X is the union of orbits, each orbit is an infranil orbifold of positive dimension; in particular, the F-structure provides a decomposition of X into compact flat orbifolds.     

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 -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 [Yongbin Ruan, The cohomology ring of crepant resolutions of orbifolds, in: Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 117-126].     

A Brody theorem for orbifolds

We study the Kobayashi pseudodistance for orbifolds, proving an orbifold version of Brody’s theorem and classifying which one-dimensional orbifolds are hyperbolic.     

The orbifold Chow ring of toric Deligne-Mumford stacks

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

Torus graphs and simplicial posets

For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated simplicial poset. This extends a series of previous results on the equivariant cohomology of torus manifolds. As a primary combinatorial application, we show that a simplicial poset is Cohen-Macaulay if its face ring is Cohen-Macaulay. This completes the algebraic characterisation of Cohen-Macaulay posets initiated by Stanley. We also study blow-ups of torus graphs and manifolds from both the algebraic and the topological points of view.     

Singular set of some Kähler orbifolds

We consider some examples of orbifolds with positive first Chern class. Applying a result of Ding and Tian, we show that the singularities must be very mild if the orbifold admits a Kähler-Einstein metric.