Arc spaces, motivic integration and stringy invariants
Arc spaces, motivic integration and stringy invariants
复制标题
弧空间、动机积分和弦不变量
DOI：
10.2969/aspm/04310529
复制
发表时间：
2004
影响因子：
0.9
通讯作者：
W. Veys
中科院分区：
文献类型：
--
作者：
W. Veys
动机积分的概念是由Kontsevich发明的，以证明双有理等价的卡-丘流形具有相同的霍奇数。他建造了一定的措施弧空间的代数品种，motivic措施，微妙和关键的财产，它采取的价值不是在R，但在Grothendieck环的代数品种。一个完整的理论在这个问题上，然后开发的Denef和Loeser在各种文件，与几个应用程序。Batyrev介绍了与motivic集成技术新的奇异不变量，弦不变量，代数品种温和的奇异性，更准确地说，日志终端奇异性。他用他们例如制定一个拓扑镜像对称测试对奇异卡-丘品种。我们将这些不变量推广到几乎任意的奇异变种，假设Mori的最小模型程序。这些笔记的目的是对这些概念进行一个温和的介绍。[2018 - 08 - 18][2018 - 08][2018 - 08 - 18][2018 - 08][2018 - 08 - 18][2018 - 08][2018 - 08 - 08][2018 - 08][2018 - 08 - 08][2018 - 08][2018 - 08 - 08][2018 - 08][2018 - 08 - 18][2018 - 08][2018 - 08 - 08][2018 - 08][2018 - 08][2018 - 08][2018 - 08][20 - 09][2018 - 08][20 - 08][2018 - 08][20 - 09]][2018 - 09]][2018在这里，我们只想解释的基本概念和第一个结果，包括p-adic数理论的理论前的历史，并提供具体的例子。本文是作者在札幌第12届MSJ-IRI "奇点理论及其应用"（2003年）上演讲的"扩展摘要"的稍微改编版本。最后，我们列出了最近的各种结果。
The concept of motivic integration was invented by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructed a certain measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in R, but in the Grothendieck ring of algebraic varieties. A whole theory on this subject was then developed by Denef and Loeser in various papers, with several applications. Batyrev introduced with motivic integration techniques new singularity invariants, the stringy invariants, for algebraic varieties with mild singularities, more precisely log terminal singularities. He used them for instance to formulate a topological Mirror Symmetry test for pairs of singular Calabi-Yau varieties. We generalized these invariants to almost arbitrary singular varieties, assuming Mori’s Minimal Model Program. The aim of these notes is to provide a gentle introduction to these concepts. There exist already good surveys by Denef-Loeser [DL8] and Looijenga [Loo], and a nice elementary introduction by Craw [Cr]. Here we merely want to explain the basic concepts and first results, including the p-adic number theoretic pre-history of the theory, and to provide concrete examples. The text is a slightly adapted version of the ‘extended abstract’ of the author’s talks at the 12th MSJ-IRI ”Singularity Theory and Its Applications” (2003) in Sapporo. At the end we included a list of various recent results.