Lev A. Borisov, R. Paul Horja
We explicitly calculate the Grothendieck $K$-theory ring of a smooth toric Deligne-Mumford stack and define an analog of the Chern character. In addition, we calculate $K$-theory pushforwards and pullbacks for weighted blowups of reduced smooth toric DM stacks.
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The purpose of this paper and its prequel (Toric Stacks I) is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well as classical toric varieties. While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that any normal variety with an action of a dense open torus arises from a fan. In [FMN09, Theorem 7.24], it is shown that a smooth separated DM stack with an action o…
The purpose of this paper and its sequel (Toric Stacks II) is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well as classical toric varieties. In this paper, we define a \emph{toric stack} as a quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a \emph{stacky fan}. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stress…
Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck $K$-theory of simplicial toric stack bundles and study the Chern character homomorphism.
Any toric Deligne-Mumford stack is a $μ$-gerbe over the underlying toric orbifold for a finite abelian group $μ$. 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.
We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.
We prove that the automorphism group of a toric Deligne-Mumford stack is isomorphic to the $2$-group associated to the stacky fan.
We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.
We define Reichstein transforms to be certain birational transformations of Artin stacks with good moduli spaces. Our main technical result is that the Reichstein transform of an Artin toric stack is again an Artin toric stack. This leads to our main theorem which states that for Artin toric stacks there is a canonical sequence of Reichstein transforms which produces a Deligne-Mumford toric stack. When the good moduli space of an Artin toric stack is projective our procedure can be interpreted in terms of Kirwan's partial desingularization of geometric invariant theory quotients.
The purpose of this paper is to prove that integral Chow rings of toric stacks are naturally isomorphic to Stanley-Reisner rings.
There is a well developed intersection theory on smooth Artin stacks with quasi-affine diagonal. However, for Artin stacks whose diagonal is not quasi-finite the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space - extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stack…
We give a precise relation between the mirror transformation and the Seidel elements for weak Fano toric Deligne-Mumford stacks. Our result generalizes the corresponding result for toric varieties proved by González and Iritani in \cite{GI}.
The Integral Chow Ring of Smooth Non-strict Toric Stacks
研究将Stanley-Reisner环扩展至非严格堆叠风扇，证明其与光滑非严格堆叠的积分Chow环同构，推广了更一般堆叠的Chow环结果。
Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. 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.
For a toric Deligne-Mumford (DM) stack, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism on a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the toric DM stack. We also choose a full strong exceptional collection from the set of direct summands of the push-forward in several examples of two dimensional toric DM orbifolds.
In this paper, we develop several techniques for computing the higher G-theory and K-theory of quotient stacks. Our main results for computing these groups are in terms of spectral sequences. We show that these spectral sequences degenerate in the case of many toric stacks, thereby providing an efficient computation of their higher K-theory. We apply our main results to give explicit description for the higher K-theory for many smooth toric stacks. As another application, we describe the higher K-theory of toric stack bundles over smooth base schemes.
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.
Let X be an irreducible affine T-variety. We consider families of affine stable toric T-varieties over X and give a description of the corresponding moduli space as the quotient stack of an open subscheme in a certain toric Hilbert scheme under the action of a torus.
A Functorial Classification of Toric Stacks
本文通过引入栈扇2-范畴$\textsf{StkFan}$，构造其与环面栈2-范畴$\textsf{TStk}$的等价，解决了环面栈范畴化构造的错误或局限问题，还发展了环面范畴理论并为Gale对偶提供公理框架。
We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.
We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles and prove that the I-function gives points in the Lagrangian cone, namely we construct an explicit slice of the Lagrangian cone defined by the genus $0$ Gromov-Witten theory of a toric stack bundle.
We study the bounded derived categories of torus-equivariant coherent sheaves on smooth toric varieties and Deligne-Mumford stacks. We construct and describe full exceptional collections in these categories. We also observe that these categories depend only on the PL homeomorphism type of the corresponding simplicial complex.
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.
We study line bundles on toric DM stacks $\mathbb{P}_{\mathbfΣ}$ of dimension two. We give a combinatorial criterion of when infinitely many line bundles on $\mathbb{P}_{\mathbfΣ}$ have trivial cohomology. We further discuss the structure of the set of such line bundles.
In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.
We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.
We compute the Grothendieck and Picard groups of a complete smooth toric Deligne-Mumford stack by using a suitable category of graded modules over a polynomial ring.
We prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.
We calculate explicit formulas for the general equivariant Bondal-Orlov functors on the localized K-theory groups for a crepant birational transformation of toric DM stacks. We recall some facts that the Bondal-Orlov functors give equivalences on the bounded derived categories. Applying twice of these functors we get the Seidel-Thomas spherical twists for the derived category.
On the derived category of a toric stack bundle
本文研究了分裂环带叠片丛上环面等变凝聚层导出范畴的性质，并给出了其半正交分解。
We prove a Givental-style mirror theorem for toric Deligne--Mumford stacks X. This determines the genus-zero Gromov--Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen--Ruan orbifold cohomology of X.
For any vector bundle $V$ on a toric Deligne-Mumford stack $\ix$ the formalism of \cite{EJK:16} defines two intertial products $\star_{V^{+}}$ and $\star_{V^{-}}$ on the Chow group of the inertia stack. We give an explicit presentation for the integral $\star_{V^+}$ and $\star_{V^-}$ Chow rings, extending earlier work of Boris-Chen-Smith \cite{BCS:05} and Jiang-Tsen \cite{JiTs:10} in the orbifold Chow ring case, which corresponds to $V = 0$. We also describe an {\em asymptotic} product on the rational Chow group of the inertia stack obtained by letting the rank of the bundle $V$ go to infinity…
We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $\mathbb{P}_{\mathbfΣ}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_{\mathbfΣ}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $\mathbb{P}_{\mathbfΣ}$.
We investigate moduli stacks of pointed chains of projective lines related to the Losev-Manin moduli spaces and show that these moduli stacks coincide with certain toric stacks which can be described in terms of the Cartan matrices of root systems of type A. We also consider variants of these stacks related to root systems of type B and C.
Asymptotics of immaculate line bundles on smooth toric Deligne-Mumford stacks
本文研究了光滑环面 Deligne-Mumford 栈上的无瑕线丛，给出了存在无穷多无瑕线丛的判据，肯定回答了 Borisov 和 Wang 的问题，并描述了其渐近行为。
Gluing moduli spaces of quantum toric stacks via secondary fan
本文通过在量子框架内调整 secondary fan 的构造，构建了一个涵盖给定维度环境空间下所有量子环面叠模空间的统一模空间，并描述了不同模空间间的穿墙现象。
We construct new "virtually smooth" modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov-Witten theory is also included.
On $\mathrm{H}-$trivial line bundles on toric DM stacks of dim $\geq3$
本研究针对任意维数光滑环面DM栈上的线丛展开，给出无穷多线丛具平凡上同调的充分条件，并证明三维下共线射线对不超过一对时该条件为充要条件。
We outline a proof that the stack of symplectic toric G-manifolds over a fixed orbit space W is a torsor for the stack of symplectic toric G-principal bundles over W.
We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d $\mathcal{N} = 2$ abelian mirror symmetry construction in physics. Given some toric data, we introduce the $K$-theoretic $I$-function with effective level structure for the associated toric stack. When a particular stability condition is chosen, it restricts to the $I$-function for the particular toric GIT quotient. The mirror of a toric stack is defined by the Gale dual of the original toric data. We then proved the mirror conjecture that the $I$-functions of a mirror pair coincide, under the mirror map, which…
Perverse sheaves on smooth toric varieties and stacks
本文将光滑环面簇及其轨道分层上的反常层范畴显式刻画为某代数上的有限维模范畴，并推广到等变情形以描述环面轨形上的反常层，同时比较了相关导出范畴。
We present a decomposition of the space of twisted arcs of a toric stack. As a consequence, we give a combinatorial description of the motivic integral associated to a torus-invariant divisor of a toric stack.
In this note, we prove that the Grothendieck group of a smooth complete toric Deligne-Mumford stack is torsion free. This statement holds when the generic point is stacky. We also construct an example of open toric stack with torsion in K-theory. This is a part of the author's Ph.D thesis.
Orbifold pseudo-effective cones of toric stacks
本文研究了分裂折线堆叠的折线伪有效锥的结构与性质。
In this paper, we prove that there exists an equivalence between 2-category of smooth Deligne-Mumford stacks with torus-embeddings and actions, and the 1-category of stacky fans. For this purpose, we obtain two main results. The first is to investigate a combinatorial aspect of the 2-category of toric algebraic stacks defined in \cite{I2}. We establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second is to give a geometric characterization theorem for toric algebraic stacks.
We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.
Equivariant principal bundles over toric Deligne--Mumford stacks
本文对环面Deligne--Mumford栈上的带框环面主丛进行分类，利用分片线性映射描述等变自同构群及结构群约化条件，并证明加权射影栈上主丛的等变分裂结果。
In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for algebraic curves motivated by Berkovich's construction of skeletons of analytic curves. Under certain assumptions, we construct a one-to-one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects, called "stacky tropical reductions", that can be enumerated in terms of tropical curves satisfying linear c…
Starting from a collection of line bundles on a projective toric orbifold X, we introduce a stacky analogue of the classical linear series. Our first main result extends work of King by building moduli stacks of refined representations of labelled quivers. We associate one such stack to any collection of line bundles on X to obtain our notion of a stacky linear series; as in the classical case, X maps to the ambient stack by evaluating sections of line bundles in the collection. As a further application, we describe a finite sequence of GIT wall crossings between [A^n/G] and G-Hilb(A^n) for a …
The nonequivariant coherent-costructible correspondence is a microlocal-geometric interpretation of homological mirror symmetry for toric varieties conjectured by Fang-Liu-Treumann-Zaslow. We prove a generalization of this conjecture for a class of toric stacks which includes any toric varieties and toric orbifolds. Our proof is based on gluing descriptions of $\infty$-categories of both sides.
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves and representations of G. This expression will play a crucial role in the proof of the remodeling conjecture (arXiv:0709.1453, arXiv:0807.0597) for affine toric Calabi-Yau 3-orbifolds …
The Chow quotient of a toric variety by a subtorus, as defined by Kapranov-Sturmfels-Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a fixed projective toric variety, as constructed by Alexeev and Brion. We show that, after endowing both spaces with the structure of a logarithmic stack, the resulting spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties that it satisfies.
We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as symplectic quotients of C^NxBG, where G is a finite non-abelian group.
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
Nonsimplicial toric Nullstellensatz and stacky GKZ theory
引入基于Q-Cartier除子的Cox环变体，解决非单纯环空簇的缺陷，实现理想-簇对应、子概形与层的显式分类、相关环空Deligne-Mumford堆栈及GKZ/Mori理论的堆栈扩展。
A Resolution of the Diagonal for Toric Deligne-Mumford Stacks
本文通过形变Bayer-Popescu-Sturmfels胞腔复形，给出光滑环面簇对应环面Deligne-Mumford栈的族形式对角线消解，还构造了有限阿贝尔群商对应环面Deligne-Mumford栈的对角线消解。
In this paper, we classify smooth toric Fano 5-folds of index 2. There exist exactly 10 smooth toric Fano 5-folds of index 2 up to isomorphisms.
We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules (and of the all-genus Gromov-Witten potentials) under a discrepant toric wall-crossing. In the case of weighted blowups of weak-Fano compact toric stacks along toric centres, we show that an analytic lift of the formal decomposition corresponds, via the $\widehatΓ$-integral structure, to an Orlov-type semiorthogonal decomposition of topologic…
Moduli spaces of quantum toric stacks and their compactification
针对环面簇因格有理性条件而具有等变刚性的问题，本文研究以 $\mathbb{R}^d$ 有限生成子群替代格的量子环面叠，并阐述其模空间与紧化。
We completely classify toric weakened Fano 3-folds, that is, smooth toric weak Fano 3-folds which are not Fano but are deformed to smooth Fano 3-folds. There exist exactly 15 toric weakened Fano 3-folds up to isomorphisms.
We extend our previous work arXiv:1007.0053 on coherent-constructible correspondence for toric varieties to include toric Deligne-Mumford (DM) stacks. Following Borisov-Chen-Smith, a toric DM stack $\cX_\bSi$ is described by a "stacky fan" $\bSi=(N,\Si,β)$, where $N$ is a finitely generated abelian group and $\Si$ is a simplicial fan in $N_\bR=N\otimes_{\bZ}\bR$. From $\bSi$ we define a conical Lagrangian $Λ_\bSi$ inside the cotangent $T^*M_\bR$ of the dual vector space $M_\bR$ of $N_\bR$, such that torus-equivariant, coherent sheaves on $\cX_\bSi$ are equivalent to constructible sheaves on $M…
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