On Mumford Orbifolds

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants.

Construction of p-adic Hurwitz spaces

Moduli spaces for Galois covers of p-adic Mumford curves by Mumford curves are constructed using Herrlich's Teichmuller spaces, Andre's orbifold fundamental groups, and Kato's graphs of groups encoding ramification data of charts for Mumford orbifolds.

The Orbifold Cohomology Ring of Simplicial Toric Stack Bundles

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.

Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds

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.

扭曲射流、母题测度与轨形上同调

Twisted jets, motivic measure and orbifold cohomology

本文通过引入Deligne-Mumford栈上的扭曲射流并研究其母题测度，证明了具有Gorenstein商奇点的双有理极小模型具有相同的带霍奇结构的轨形上同调。

Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi

We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived al…

Equivariant Gromov-Witten Theory of Affine Smooth Toric Deligne-Mumford Stacks

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 …

Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks

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.

Mumford curves and Mumford groups in positive characteristic

A Mumford group is a discontinuous subgroup $Γ$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $Γ$ is a curve of genus 0. As abstract group $Γ$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphi…

导出的德林费尔德-穆尔德堆栈的霍奇-康斯坦-罗斯伯格同构

Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks

本文研究了导出德林费尔德-穆尔德堆栈的霍奇-康斯坦-罗斯伯格同构，建立了其与微分形式上同调之间的自然同构，并扩展了相关同调理论的结果。

Toric Deligne-Mumford stacks and the better behaved version of the GKZ hypergeometric system

We generalize the combinatorial description of the orbifold (Chen--Ruan) cohomology and of the Grothendieck ring of a Deligne--Mumford toric stack and its associated stacky fan in a lattice $N$ in the presence of a deformation parameter $β\in N \otimes {\mathbb C}.$ As an application, we construct a topological mirror symmetry map that produces a complete system of $Γ$--series solutions to the better behaved version of the GKZ hypergeometric system for $β\in N \otimes {\mathbb C}.$

Stacky GKM Graphs and Orbifold Gromov-Witten Theory

A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. (i) We define the stacky GKM graph of a smooth GKM stack, under the mild assumption that any one-dimensional $T$-orbit closure contains at least one $T$ fixed point. The stacky GKM graph is a decorated graph which contains enough information to reconstruct the $T$-equivariant formal neighborhood of the 1-skeleton (union of zero-dimensional and one-dimensional $T$-orbits) as a formal smooth DM stack equipped with a $T$-actio…

On generating series of classes of equivariant Hilbert schemes of fat points

In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring of complex quasi-projective varieties) of Hilbert schemes of zero-dimensional subschemes on smooth varieties and on orbifolds in terms of certain local data and the, so called, power structure over the corresponding ring. Here we give an analogue of these formulae for equivariant (with respect to an action of a finite group on a smooth variety) Hilbert schemes of zero-dimensional subschemes and compute some local generating series for an action of the cyclic group on a smooth surface.

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild…

A bicategory of reduced orbifolds from the point of view of differential geometry - I

We describe a bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we first define a $2$-category $(\mathcal{R}ed\,\mathcal{A}tl)$ whose objects are reduced orbifold atlases (on any paracompact, second countable, Hausdorff topological space). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl,…

Open-closed Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford (DM) stacks (with possibly non-trivial generic stabilizers and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack with boundaries mapped into a Aganagic-Vafa brane. All genus open-closed Gromov-Witten invariants are defined by torus localization and depend on the choice …

Stringy K-theory and the Chern character

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natura…