论文标题
弦乐不变和感谢
Stringy invariants and toric Artin stacks
论文作者
论文摘要
我们提出了一个猜想的框架，用于计算Gorenstein测量和尖锐的Hodge数字，以动机整合光滑的Artin堆栈的弧度，并且在幻想中，我们验证了该框架，这些框架是某些曲折的Artin堆栈，这些堆栈提供了（不可分离的）奇异品种的（不分开）奇异品种的分辨率。具体而言，让$ \ natercal {x} $成为一个平稳的artin堆栈，承认一个好的模量空间$π：\ nathcal {x} \ to x $，并假设$ x $是一种具有对数末端的奇异性的多样性，$π$ to $ x $诱导了与$ x $ $ x $ $ x $ use的iSomorphist a in Ismorphist the $ x $ $ x $ $ x $ usim y和x $ usim usim y和x $ usim $ x $ usim y和usim usim odim odim odim odim $ x $。 $ \ Mathcal {X} $的动机度量的公式，根据Gorenstein量度的$ x $，以及一个函数，可衡量$π$未分离的程度。我们还推测，如果$ \ nathcal {x} $的稳定器是特殊的组，那么几乎所有$ x $ lift to to a arcs $ \ nathcal {x} $的弧线的所有弧线，我们在这种情况下，我们的猜想意味着在这种情况下，我们的猜想是如何在某些$ x $中的nodge nectal of x $ x $ x $ x $ x $ x集成。 $ \ MATHCAL {X} $。在$ \ Mathcal {x} $是幻想中，我们证明了这些猜想。
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.
