Introduction to Geometric Motivic Integration

The
concept of motivic integration was introduced by Kontsevich [Kon95] to (elegantly) show that birationally equivalent Calabi-Yau manifolds (i.e. smooth, compact, complex algebraic varieties with zero canonical divisor) have the same Hodge numbers. This generalized a Barytev result about the betti numbers, who used a more complicated machinery, e.g. the Weil conjectures [Bat99a].
Kontsevich constructed a certain motivic measure on the arc space of a complex variety (passing from “p-adic” to “t-adic”), taking values in the universal additive invariant of varieties: the Grothendieck ring of algebraic varieties. A whole theory of the subject was later developed by Denef and Loeser [DL98, DL99, DL02].

The
aim of this mini-course is to provide an introduction to the basic concepts and tools on motivic integration, and its related applications to zeta functions of singularities, providing concrete examples.

Program:
(1) Some arithmetic geometry problems. p-adic integration. The p-adic Igusa zeta function of a singularity.
(2) Arc and jet spaces. Constructible sets. The Grothendieck ring of varieties as universal additive invariant. Motivic measure.
(3) Motivic integration. Change of variables formula. Proof of Kontsevich’s Theorem on Hodge numbers of Calabi-Yau varieties.
(4) Application on singularities and motivic zeta functions.
(5) Some commentaries about the real case: Grothendieck ring of semi-algebraic arc-symmetric sets, virtual Poincar´e polynomial and zeta functions as in- variants for arc-analytic equivalent Nash function germs.

Referências
Victor V. Batyrev. Birational Calabi-Yau n-folds have equal Betti numbers. In New trends in algebraic geometry (Warwick, 1996), volume 264 of London Math. Soc. Lecture Note Ser., pages 1–11. Cambridge Univ. Press, Cambridge, 1999.
Alastair Craw. An introduction to motivic integration. In Strings and geometry, volume 3 of Clay Math. Proc., pages 203–225. Amer. Math. Soc., Providence, RI, 2004.
Jan Denef and Fran¸cois Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537.
Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232.
Motivic integration, quotient singularities and the McKay correspondence, Com- positio Math. 131 (2002), no. 3, 267–290.
Maxim Kontsevich. Lecture at Orsay. December 7, 1995.
Eduard Looijenga. Motivic measures. Ast´erisque, (276):267–297, 2002. S´eminaire Bour- baki, Vol. 1999/2000.
W. Veys. Arc spaces, motivic integration and stringy invariants. In Singularity theory and its applications, volume 43 of Adv. Stud. Pure Math., pages 529–572. Math. Soc. Japan, Tokyo, 2006.
Viu-Sos. An introduction to p-adic and motivic integration, zeta functions and new stringy invariants of singularities. Lecture Notes, available at http://jviusos.perso. univ-pau.fr/resources/motivic_integration.pdf, 2018.