Niveau: Supérieur, Doctorat, Bac+8
Seattle lectures on motivic integration Franc¸ois Loeser Preliminary notes (January 17, 2006) These notes are the written-up version of a series of 4 lectures given at the Summer Institute. Though I tried as much as possible to keep the basic structure of the lectures as well as their rather informal style, some flesh has also been added to the bones. Motivic integration being born exactly ten years ago, nothing could be more timely than the proposition by the organizers of the Institute to review the achievements of the past decade in a series of lectures. I would like to thank them for providing me such a unique opportunity. Lecture 1: Before Motivic Integration 1.1. Modifications. One may start the whole story of motivic integration with a somewhat intriguing result obtained by Jan Denef and myself in 1987 and only published in 1992 [23]. At the time we certainly would never have guessed the fantastic developments that would arise later. Let us consider a smooth complex algebraic variety X and a closed nowhere dense subscheme F . By a log-resolution h : Y ? X of (X,F ) we mean a proper morphism h : Y ? X with Y smooth such that the restriction of h: Y \ h?1(Fred)? X \ Fred is an isomorphism, and h?1(Fred) is a divisor with simple normal crossings.
- almost all
- prime ideals
- theorems between -adic
- complex algebraic
- rational function
- local ring
- varieties