The integral (orbifold) Chow ring of toric Deligne–Mumford stacks

In this paper we study the integral Chow ring of toric Deligne–Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne–Mumford stack is isomorphic to the Stanley–Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples.     

A note on finite abelian gerbes over toric Deligne-Mumford stacks

Any toric Deligne-Mumford stack is a -gerbe over the underlying toric orbifold for a finite abelian group . In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks.

     

Mellin-Barnes integrals as Fourier-Mukai transforms

We study the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky and its relationship with the toric Deligne-Mumford (DM) stacks recently studied by Borisov, Chen and Smith. We construct series solutions with values in a combinatorial version of the Chen-Ruan (orbifold) cohomology and in the K-theory of the associated DM stacks. In the spirit of the homological mirror symmetry conjecture of Kontsevich, we show that the K-theory action of the Fourier-Mukai functors associated to basic toric birational maps of DM stacks are mirrored by analytic continuation transformations of Mellin-Barnes type.     

Mirror fibrations and root stacks of weighted projective spaces

We show that the orbifold Chow ring of a root stack over a well-formed weighted projective space can be naturally seen as the Jacobian algebra of a function on a singular variety.     

The Chow rings for 3-dimensional toric varieties with one bad isolated singularity

The properties of a toric variety have strong connection with the combinatorial structure of the corresponding fan and the relations among the generators. Using this fact, we have described explicitly the Chow ring for aQ-factorial toric variety as the Stanley-Reisner ring for the corresponding fan modulo the linear equivalence relation. In this paper, we calculate the Chow ring for 3-dimensionalQ-factorial toric varieties having one bad isolated singularity.     

The intrinsic normal cone

Let be an algebraic stack in the sense of Deligne-Mumford. We construct a purely -dimensional algebraic stack over (in the sense of Artin), the intrinsic normal cone . The notion of (perfect) obstruction theory for is introduced, and it is shown how to construct, given a perfect obstruction theory for , a pure-dimensional virtual fundamental class in the Chow group of . We then prove some properties of such classes, both in the absolute and in the relative context. Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one. Oblatum 26-II-1996 & 27-VI-1996     

On the conjecture of King for smooth toric Deligne-Mumford stacks

We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks.     

A Model for the Orbifold Chow Ring of Weighted Projective Spaces

We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.     

The Chow ring of punctual Hilbert schemes on toric surfaces

Let X be a smooth projective toric surface, and the Hilbert scheme parametrizing the length d zero-dimensional subschemes of X. We compute the rational Chow ring . More precisely, if is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.     

Weighted Ehrhart theory and orbifold cohomology

We introduce the notion of a weighted δ-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted δ-vectors from a combinatorial perspective. We present a version of Ehrhart Reciprocity and prove a change of variables formula. We deduce a new geometric interpretation of the coefficients of the Ehrhart δ-vector. More specifically, they are sums of dimensions of orbifold cohomology groups of a toric stack.     

The Chow group of the moduli space of marked cubic surfaces

Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow-ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space. As an application we relate a recent result of Freitag on the Hilbert polynomial of a certain ring of modular forms to the Riemann–Roch theorem for the moduli space. Dedicated to the memory of our friend Fabio BardelliMathematics Subject Classification (2000) 14J15     

Fiber Fans and Toric Quotients

The GIT chamber decomposition arising from a subtorus action on a polarized quasiprojective toric variety is a polyhedral complex. Denote by Σ the fan that is the cone over the polyhedral complex. In this paper we show that the toric variety defined by the fan Σ is the normalization of the toric Chow quotient of a closely related affine toric variety by a complementary torus.     

Compactifying the space of stable maps

In this paper we study a notion of twisted stable map, from a curve to a tame Deligne-Mumford stack, which generalizes the well-known notion of stable map to a projective variety.

     

The Poincaré problem for foliations on compact toric orbifolds

We give an optimal upper bound of the degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional holomorphic foliation on a compact toric orbifold, i.e. on a complete simplicial toric variety. This bound depends only on the degree of the foliation and of the degrees of the toric homogeneous coordinates.

     

Tautological Classes of the Stack of Rational Nodal Curves

This is the second in a series of three papers in which we investigate the rational Chow ring of the stack 𝔐₀ consisting of nodal curves of genus 0. Here we define the basic classes: the classes of strata and the Mumford classes.     

The integral cohomology of toric manifolds

We prove that the integral cohomology of a smooth, not necessarily compact, toric variety X _Σ is determined by the Stanley-Reisner ring of Σ. This follows from a formality result for singular cochains on the Borel construction of X _Σ. As a consequence, we show that the cycle map from Chow groups to Borel-Moore homology is split injective.     

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.     

Equivariant Cobordism of Torus Orbifolds

Torus orbifolds are topological generalizations of symplectic toric orbifolds.The authours give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using a toric topological method. As a result, they show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. They also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.     

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.     

The loop orbifold of the symmetric product

Using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas-Sullivan orbifold product structure in the homology of the free loop space of the Borel construction of the symmetric product induces a ring structure in the homology of the inertia orbifold of the symmetric product. For a general almost complex orbifold, we define a new ring structure on the cohomology of its inertia orbifold which we call the virtual intersection ring. Finally we show that under Poincaré duality in the case of the symmetric product orbifold, both ring structures are isomorphic.