## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularity Theory and Its Applications, S. Izumiya, G. Ishikawa, H. Tokunaga, I. Shimada and T. Sano, eds. (Tokyo: Mathematical Society of Japan, 2006), 529 - 572

### Arc spaces, motivic integration and stringy invariants

#### Abstract

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 $\mathbb{R}$, 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.

#### Article information

**Dates**

Received: 26 March 2004

Revised: 21 September 2004

First available in Project Euclid:
3 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546543251

**Digital Object Identifier**

doi:10.2969/aspm/04310529

**Mathematical Reviews number (MathSciNet)**

MR2325153

**Zentralblatt MATH identifier**

1127.14004

#### Citation

Veys, Willem. Arc spaces, motivic integration and stringy invariants. Singularity Theory and Its Applications, 529--572, Mathematical Society of Japan, Tokyo, Japan, 2006. doi:10.2969/aspm/04310529. https://projecteuclid.org/euclid.aspm/1546543251