Quantum cohomology and toric minimal model programs
Quantum cohomology and toric minimal model programs
复制标题
DOI：
10.1016/j.aim.2019.07.004
复制
发表时间：
2012-07
影响因子：
1.7
通讯作者：
Eduardo Gonzalez;C. Woodward
中科院分区：
文献类型：
--
作者：
Eduardo Gonzalez;C. Woodward
我们给出了一个量子版本的Danilov-Jurkiewicz介绍的上同调的紧复曲面orbifold与投射粗模空间。更确切地说，我们从[4]中的Batyrev环的形式版本到正则体变形下的量子orbifold上同调构造了一个正则同构。这个同构推广了Givental [23]，Iritani [34]和Shuaya-Oh-Ohta-Ono [21]关于环面流形的结果和Coates-Lee-Corti-Tseng [11]关于加权射影空间的结果。该证明使用量子版本的Kirwan满射性（定理2.6）和使用环面最小模型程序（tmmp）推导出的维数相等（定理4.19）。我们发现，有一个自然的分解的量子上同调的和对应的奇异性的TMMP，其中每一个都产生了一个收集的哈密顿不可置换拉格朗日环面。
We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring from [4] to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results of Givental [23], Iritani [34] and Fukaya-Oh-Ohta-Ono [21] for toric manifolds and Coates-Lee-Corti-Tseng [11] for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity (Theorem 2.6 below) and an equality of dimensions (Theorem 4.19 below) deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp, each of which gives rise to a collection of Hamiltonian non-displaceable Lagrangian tori.