CohFT代数的态射与Kirwan映射的量子化

Quantum Kirwan morphism and Gromov-Witten invariants of quotients II

This is the second in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. In this part we construct virtual fundamental classes on the moduli spaces used in the construction of the quantum Kirwan map and the gauged Gromov-W…

Quantum Kirwan morphism and Gromov-Witten invariants of quotients I

This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive group $G$ to the orbifold quantum cohomology $QH(X//G)$ of its geometric invariant theory quotient $X//G$, and prove that it intertwines the genus zero gauged Gromov-Witten potential of X with the genus zero Gromov-Witten graph potential of $X//G$.

Quantum Kirwan morphism and Gromov-Witten invariants of quotients III

This is the third in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. We also give a formula for a solution to the quantum differential equation in terms of a localized gauged potential. These results overlap with those …

上同调场论的形变理论

Deformation theory of Cohomological Field Theories

本文发展了上同调场论的形变理论，引入同伦与量子扩展，构建了新的图复形以研究作用于其模空间的泛形变群，该群包含前幂单Grothendieck--Teichmüller群和Givental群。

Hecke圈的模空间的上同调

Cohomology of the moduli space of Hecke cycles

设$X$是亏格为$g \ge 3$的光滑射影曲线，$M_0$是秩为2的$X$上半稳定丛的平凡行列式的模空间. Seshadri \cite{Se1}、Narasimhan-Ramanan \cite{NR}和Kirwan \cite{k5}构建了三种不同的$M_0$去奇异化。本文构造了Kirwan去奇异化到Narasimhan-Ramanan去奇异化的一个双有理态射，并证明了Narasimhan-Ramanan去奇异化（称为Hecke圈的模空间）是最近在\cite{KL}中证明的Kirwan去奇异化与Seshadri去奇异化之间的中间簇.作为副产品，我们计算的模空间的Hecke圈的上同调。

Canonical reduction of stabilizers for Artin stacks with good moduli spaces

We present a complete generalization of Kirwan's partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if $\mathcal{X}$ is an irreducible Artin stack with stable good moduli space $\mathcal{X} \to X$, then there is a canonical sequence of birational morphisms of Artin stacks $\mathcal{X}_n \to \mathcal{X}_{n-1} \to \ldots \to \mathcal{X}_0 = \mathcal{X}$ with the following properties: (1) the maximum dimension of a stabilizer of a point of $\mathcal{X}_{k+1}$ is strictly smaller than the maximum dimension of a stabilizer of $\mathcal{X}_k$ and the final s…