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Seattle lectures on motivic integration. (English) Zbl 1181.14017
Abramovich, D. (ed.) et al., Algebraic geometry, Seattle 2005. Proceedings of the 2005 Summer Research Institute, Seattle, WA, USA, July 25–August 12, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4703-9/hbk; 978-0-8218-4057-3/set). Proceedings of Symposia in Pure Mathematics. 80, Pt. 2, 745-784 (2009).
This article is an expository one based on lectures at the Summer Institute in Algebraic Geometry, Seattle, 2005. It gives a review of the main achievements in the subject.
The first lecture includes a brief review of \(p\)-adic integration, a result of Denef-Loeser giving a formula for the Euler characteristic of a smooth complex variety \(X\) in terms of its log resolution and the independence of the zeta function of a pair from the log resolution. There is also a result of Batyrev that the Betti numbers of two birationally equivalent Calabi-Yau varieties are equal, which led to his conjecture about the equality of their Hodge numbers, and so gave the main motivation for the invention of the motivic integration by Kontsevich.
The second lecture gives the basics of motivic integration. It discusses briefly arc spaces, additive invariants and Grothendieck rings. Then the change of variables formula and some of its major applications are given. Among them is a formula for the class of \(X\) in terms of its log resolution, the proof of the Batyrev conjecture by Kontsevich, the independence of the stringy invariant \(E_{st}(X)\) of a normal terminal \(\mathbb{Q}\)-Gorenstein variety on the log resolution, and the theorem of Mustată giving a formula for the log canonical threshold in terms of codimensions of some jet spaces.
In lecture 3, after introducing briefly the Milnor fiber and the nearby cycle functor, a relation between Euler characteristic and Lefschetz numbers is given. Then the motivic analogue of Igusa’s local zeta function is defined, and the monodromy conjecture is formulated. After introducing briefly the Hodge spectrum and convolution product, two versions of Thom-Sebastani theorem are formulated, one for the Hodge spectrum, and another as the motivic version of the theorem. Also, the motivic version of a result of Saito proving a Steenbrink conjecture is given.
In the last lecture a general setting for motivic integration is developed, based on a series of works of the author with R. Cluckers. After an introduction to semialgebraic geometry and some notions from model theory, the Denef-Pas cell decomposition theorem is stated. Using the language of constructible motivic functions, the general motivic measure satisfying some axioms is defined, the general change of variables formula is stated, and the motivic analogues of exponential functions are introduced. In the last part is obtained a general transfer principle, allowing to transfer relations between integrals from \(\mathbb{Q}_p\) to \(\mathbb{F}_p((t))\), and vice versa. As a special case one has the Ax-Kochen-Eršov theorem.
For the entire collection see [Zbl 1158.14004].

14E18 Arcs and motivic integration