弦性不变量与环状Artin叠
Stringy invariants and toric Artin stacks
AI总结：
本文提出用光滑Artin叠弧上的动机积分计算Gorenstein测度与弦性Hodge数的猜想框架，并在fantastack（环状Artin叠）情形下证明，为环状簇奇点解消提供非分离方法。
摘要：
我们提出一个猜想性框架，用于通过光滑Artin叠的弧上的动机积分来计算Gorenstein测度和弦性Hodge数，并在fantastack的情形下验证该框架。fantastack是特定的环状Artin叠，为环状簇的奇点提供（非分离的）解消。具体地，设$\mathcal{X}$为具有良好模空间$π: \mathcal{X} \to X$的光滑Artin叠，并假设$X$是具有对数终端奇点的簇，$π$在$X$的某个非空开子集上诱导同构，且$π$的例外轨迹的余维数至少为2。我们猜想一个关于$\mathcal{X}$的动机测度的公式，该公式用$X$的Gorenstein测度以及一个衡量$π$非分离程度的函数来表示。我们还猜想，如果$\mathcal{X}$的稳定化子是Serre意义下的特殊群，那么$X$的几乎所有弧都提升为$\mathcal{X}$的弧，并解释在这种情况下，我们的猜想如何推导出$X$的弦性Hodge数的一个公式，该公式用$\mathcal{X}$的弧上的某个动机积分来表示。我们在$\mathcal{X}$为fantastack的情形下证明了这些猜想。
英文摘要：
We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin stacks, and we verify this framework in the case of fantastacks, which are certain toric Artin stacks that provide (non-separated) resolutions of singularities for toric varieties. Specifically, let $\mathcal{X}$ be a smooth Artin stack admitting a good moduli space $π: \mathcal{X} \to X$, and assume that $X$ is a variety with log-terminal singularities, $π$ induces an isomorphism over a nonempty open subset of $X$, and the exceptional locus of $π$ has codimension at least 2. We conjecture a formula for the motivic measure for $\mathcal{X}$ in terms of the Gorenstein measure for $X$ and a function measuring the degree to which $π$ is non-separated. We also conjecture that if the stabilizers of $\mathcal{X}$ are special groups in the sense of Serre, then almost all arcs of $X$ lift to arcs of $\mathcal{X}$, and we explain how in this case, our conjectures imply a formula for the stringy Hodge numbers of $X$ in terms of a certain motivic integral over the arcs of $\mathcal{X}$. We prove these conjectures in the case where $\mathcal{X}$ is a fantastack.
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