Singular symplectic flops and Ruan cohomology

Q-Gorenstein环面接触流形的接触不变量、Ehrhart多项式与Chen-Ruan上同调

Contact invariants of Q-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

本文研究Q-Gorenstein环面接触流形的柱形接触同调不变量，建立其与环面图的Ehrhart（拟）多项式、对应环面孤立奇点的crepant环面轨形消解的Chen-Ruan上同调，以及生成该流形的单调环面辛轨形底空间的Chen-Ruan上同调的关联。

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S of T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen-Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen-Ruan cohomology in terms of orbifold fixed point data.

Orbifold cohomology of abelian symplectic reductions and the case of weighted projective spaces

These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand their topology. We then show how these results can be used to compute the Chen-Ruan orbifold cohomology ring of abelian symplectic reductions. We conclude by comparing the several rings associated to a w…

Transformation of algebraic Gromov-Witten invariants of three-folds under flops and small extremal transitions, with an appendix from the stringy and the symplectic viewpoint

We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its applications. This gives an algebro-geometric account of Witten's wall-crossing formula for correlation functions of the descendant nonlinear sigma model in adjacent geometric phases of a gauge linear sigma model and of the symplectic approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem. A terse account from the stringy and the symplecti…

Double bubble plumbings and two-curve flops

We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $(-1,-1)$-curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.

Virtual moduli cycles and Gromov-Witten invariants of noncompact symplectic manifolds

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtua…

The Full Orbifold $K$-theory of Abelian Symplectic Quotients

In their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifold $K$-theory of an orbifold ${\mathfrak X}$, analogous to the Chen-Ruan orbifold cohomology of ${\mathfrak X}$ in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when ${\mathfrak X}$ arises as an abelian symplectic quotient. Our methods are integral $K$-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the $K$-theoretic Kirwa…

Orbifold cohomology of torus quotients

We introduce the_inertial cohomology ring_ NH^*_T(Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring H_{CR}^*(Y/T) of the quotient orbifold Y/T. For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that NH^*_T(Y) has a natural ring surjection onto H_{CR}^*(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment ma…

Duality theorems for étale gerbes on orbifolds

Let $G$ be a finite group and $\Y$ a $G$-gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe $c$ on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the $G$-gerbe $\Y$ and the geometry of $\hat{\Y}$ {\em twisted by} $c$. We prove several results verifying this duality in the contexts of noncommutative geometry and symplectic topology. In particular, we prove that the category of sheaves on $\Y$ is equivalent to the category of $c$-twisted sheaves on $\hat{\Y}$. When $\Y…

Orbifold cohomology of hypertoric varieties

Hypertoric varieties are hyperkähler analogues of toric varieties, and are constructed as abelian hyperkähler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrangements. By developing hyperkähler analogues of symplectic techniques developed by Goldin, Holm, and Knutson, we give an explicit combinatorial description of the Chen-Ruan orbifold cohomology of an orbifold hypertoric variety in terms of the combinatorial data of a rational coorie…