Chern classes of Deligne-Mumford stacks and their coarse moduli spaces

Let $X$ be a complex projective algebraic variety with Gorenstein quotient singularities and \X a smooth Deligne-Mumford stack having $X$ as its coarse moduli space. We show that the CSM class $c^{SM}(X)$ coincides with the pushforward to $X$ of the total Chern class c(T_{I\X}) of the inertia stack I\X. We also show that the stringy Chern class $c_{str}(X)$ of $X$, whenever is defined, coincides with the pushforward to $X$ of the total Chern class c(T_{II\X}) of the double inertia stack II\X. Some consequences concerning stringy/orbifold Hodge numbers are deduced.Comment: v1:Preliminary version, comments are very welcome! v2: Revised and slightly expanded versio

On degree zero elliptic orbifold Gromov-Witten invariants

We compute, by two methods, the genus one degree zero orbifold Gromov-Witten invariants with non-stacky insertions which are exceptional cases of the dilaton and divisor equations. One method involves a detailed analysis of the relevant moduli spaces. The other method, valid in the presence of torus actions with isolated fixed points, is virtual localization. These computations verify the conjectural evaluations of these invariants. Some genus one twisted orbifold Gromov-Witten invariants are also computed.Comment: v1: 17 pages; v2: revision, 18 pages; v3: minor changes, 19 page

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.Comment: 26 page

Note on orbifold Chow ring of semi-projective toric Deligne-Mumford stacks

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}.Comment: The proof of a proposition is revise

On computations of genus zero two-point descendant Gromov-Witten invariants

We present a method of computing genus zero two-point descendant Gromov-Witten invariants via one-point invariants. We apply our method to recover some of calculations of Zinger and Popa-Zinger, as well as to obtain new calculations of two-point descendant invariants.Comment: 17 page

Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves

We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required R-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the previously determined analytic continuation of the fundamental solution of the QDE of the Hilbert scheme. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov-Witten theory of the symmetric product of the plane is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture.Comment: 52 pages, 2 diagrams, Forum of Math. Pi (to appear

K-theoretic quasimap invariants and their wall-crossing

For each positive rational number $\epsilon$, we define $K$-theoretic $\epsilon$-stable quasimaps to certain GIT quotients W\sslash G. For $\epsilon>1$, this recovers the $K$-theoretic Gromov-Witten theory of W\sslash G introduced in more general context by Givental and Y.-P. Lee. For arbitrary $\epsilon_1$ and $\epsilon_2$ in different stability chambers, these $K$-theoretic quasimap invariants are expected to be related by wall-crossing formulas. We prove wall-crossing formulas for genus zero $K$-theoretic quasimap theory when the target W\sslash G admits a torus action with isolated fixed points and isolated one-dimensional orbits.Comment: 20 page

Integrating Lie algebroids via stacks

Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into a Weinstein groupoid.Comment: a proof improve

The Crepant Transformation Conjecture implies the Monodromy Conjecture

In this note we prove that the crepant transformation conjecture for a crepant birational transformation of Lawrence toric DM stacks studied in \cite{CIJ} implies the monodromy conjecture for the associated wall crossing of the symplectic resolutions of hypertoric stacks, due to Braverman, Maulik and Okounkov.Comment: 18 pages, Referee's corrections and improvement

Integrating Poisson manifolds via stacks

A symplectic groupoid $G.:=(G_1 \rightrightarrows G_0)$ determines a Poisson structure on $G_0$. In this case, we call $G.$ a symplectic groupoid of the Poisson manifold $G_0$. However, not every Poisson manifold $M$ has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita equivalence in the category of all Poisson manifolds. In this paper, we construct symplectic Weinstein groupoids which provide a solution to the above problem (Theorem \ref{main}). More precisely, we show that a symplectic Weinstein groupoid induces a Poisson structure on its base manifold, and that to every Poisson manifold there is an associated symplectic Weinstein groupoid