Motivic measures.

*(English)*Zbl 0996.14011
Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque 276, 267-297, Exp. No. 874 (2002).

This report concerns very recent developments in the theory of arc spaces, motivic integration, and the McKay correspondence. To a large extent, these developments have been propelled by the work of J. Denef and F. Loeser, during the last decade, and it is the author’s main goal to provide a profound survey on their results published in numerous papers and preprints. However, this report is by far more than just a survey, since the author presents some of their results somewhat differently, very much so for the benefit of a better understanding of this comparatively new, rapidly progressing and highly advanced field of research in algebraic geometry. Therefore this report contains much more detailed proofs than usual for a seminar talk, and that is why it may be regarded as a research paper, too.

An \(n\)-jet of an arc in an algebraic variety \(X\) defined over an algebraically closed field \(k\) is a \(k[[t]]/(t^{n+1})\)-valued point of \(X\). The set of such \(n\)-jets are the closed points of a variety \(L_n(X)\) over \(k\) and the arc space \(L(X)\) of the variety \(X\) is by definition the projective limit \(\displaystyle \varprojlim_n L_n(X)\).

The first systematic study of arc spaces goes back to the work of J. Nash, written in 1968 and finally published in 1995 [Duke Math. J. 81, 31–38 (1995; Zbl 0880.14010)]. The renewed interest in arc spaces arose in the 1990’s, namely in the context of mirror symmetry and V. Batyrev’s study of Calabi-Yau manifolds [in: New trends in algebraic geometry. Set. Pap. Euro Conf., Warwick 1996, Lond. Math. Soc. Lect. Note Ser. 264, 1–11 (1999; Zbl 0955.14028)], on the one hand, and in the study of \(p\)-adic integration techniques in complex algebraic geometry by J. Denef and F. Loeser [Invent. Math. 135, 201–232 (1999; Zbl 0928.14004)], on the other hand.

The underlying techniques developed here, now going under the name “motivic integration”, have led to an avalanche of far-reaching application in the meantime, including the so-called “stringy invariants” of singularities, a complex analogue of Igusa’s local zeta function, a motivic version of the classical Thom-Sebastiani property, and the motivic McKay correspondence.

In the present article, the author masterly succeeds in covering, explaining, and partially revising these recent achievements, guided by the various original research papers that appeared during the past ten years. The paper is divided into eight main sections treating the following topics:

1. The arc space and its measure;

2. The transformation rule (of Denef-Loeser-Kontsevich);

3. The basic formula (in the Grothendieck ring);

4. The motivic nearby fiber;

5. The motivic zeta function (of Denef-Loeser);

6. The motivic convolution (including the abstract Thom-Sebastiani property);

7. The McKay correspondence (after Batyrev, Denef-Loeser, Reid);

8. A (new) proof of the Denef-Loeser-Kontsevich transformation rule.

All in all, this extended report provides a highly valuable introduction to and a systematic overview of the recent and extremely powerful method of motivic integration. Also, this article is a very useful guide through the vast original literature on the subject.

For the entire collection see [Zbl 0981.00011].

An \(n\)-jet of an arc in an algebraic variety \(X\) defined over an algebraically closed field \(k\) is a \(k[[t]]/(t^{n+1})\)-valued point of \(X\). The set of such \(n\)-jets are the closed points of a variety \(L_n(X)\) over \(k\) and the arc space \(L(X)\) of the variety \(X\) is by definition the projective limit \(\displaystyle \varprojlim_n L_n(X)\).

The first systematic study of arc spaces goes back to the work of J. Nash, written in 1968 and finally published in 1995 [Duke Math. J. 81, 31–38 (1995; Zbl 0880.14010)]. The renewed interest in arc spaces arose in the 1990’s, namely in the context of mirror symmetry and V. Batyrev’s study of Calabi-Yau manifolds [in: New trends in algebraic geometry. Set. Pap. Euro Conf., Warwick 1996, Lond. Math. Soc. Lect. Note Ser. 264, 1–11 (1999; Zbl 0955.14028)], on the one hand, and in the study of \(p\)-adic integration techniques in complex algebraic geometry by J. Denef and F. Loeser [Invent. Math. 135, 201–232 (1999; Zbl 0928.14004)], on the other hand.

The underlying techniques developed here, now going under the name “motivic integration”, have led to an avalanche of far-reaching application in the meantime, including the so-called “stringy invariants” of singularities, a complex analogue of Igusa’s local zeta function, a motivic version of the classical Thom-Sebastiani property, and the motivic McKay correspondence.

In the present article, the author masterly succeeds in covering, explaining, and partially revising these recent achievements, guided by the various original research papers that appeared during the past ten years. The paper is divided into eight main sections treating the following topics:

1. The arc space and its measure;

2. The transformation rule (of Denef-Loeser-Kontsevich);

3. The basic formula (in the Grothendieck ring);

4. The motivic nearby fiber;

5. The motivic zeta function (of Denef-Loeser);

6. The motivic convolution (including the abstract Thom-Sebastiani property);

7. The McKay correspondence (after Batyrev, Denef-Loeser, Reid);

8. A (new) proof of the Denef-Loeser-Kontsevich transformation rule.

All in all, this extended report provides a highly valuable introduction to and a systematic overview of the recent and extremely powerful method of motivic integration. Also, this article is a very useful guide through the vast original literature on the subject.

For the entire collection see [Zbl 0981.00011].

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14F42 | Motivic cohomology; motivic homotopy theory |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14G05 | Rational points |

14J10 | Families, moduli, classification: algebraic theory |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |