Stacky GKM图与Orbifold Gromov-Witten理论
Stacky GKM Graphs and Orbifold Gromov-Witten Theory
摘要：
光滑GKM栈是一个光滑的Deligne-Mumford栈，它具有代数环面$T$的作用，只有100多个零维和一维轨道。(i)在任意一维$T$ -轨道闭包至少包含一个$T$不动点的温和假设下，定义了光滑GKM栈的栈GKM图. Stacky GKM图是一个装饰图，它包含足够的信息来重构1-骨架（零维和一维$T$ -轨道的并集）的$T$ -等变形式邻域，作为一个形式光滑的DM栈，并配备一个$T$ -作用. (ii)我们公理化了堆叠GKM图的定义，并引入了抽象堆叠GKM图，它比诚实光滑GKM堆叠的堆叠GKM图更一般。从一个抽象的GKM图，我们构造了一个正式的光滑GKM堆栈。(iii)我们定义了光滑GKM栈的等变orbifold Gromov-Witten不变量，以及正式光滑GKM栈的正式等变orbifold Gromov-Witten不变量。这些不变量可以通过虚拟定位来计算，并且只依赖于堆栈GKM图或抽象堆栈GKM图。光滑GKM栈$\mathcal{X}$的栈GKM图的形式等变轨道Gromov-Witten不变量是$\mathcal{X}$的等变轨道Gromov-Witten不变量的改进.
英文摘要：
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 $T$-action. (ii) We axiomize the definition of a stacky GKM graph and introduce abstract stacky GKM graphs which are more general than stacky GKM graphs of honest smooth GKM stacks. From an abstract GKM graph we construct a formal smooth GKM stack. (iii) We define equivariant orbifold Gromov-Witten invariants of smooth GKM stacks, as well as formal equivariant orbifold Gromov-Witten invariants of formal smooth GKM stacks. These invariants can be computed by virtual localization and depend only the stacky GKM graph or the abstract stacky GKM graph. Formal equivariant orbifold Gromov-Witten invariants of the stacky GKM graph of a smooth GKM stack $\mathcal{X}$ are refinements of equivariant orbifold Gromov-Witten invariants of $\mathcal{X}$.
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