Wim Veys (Leuven, Belgium)

An Introduction to Motivic Integration and its Applications

Lecture 1: Spaces of jets and arcs
number-theoretic pre-history: numbers of solutions of polynomial congruences;
spaces of jets and arcs, examples and properties;
Grothendieck ring of algebraic varieties.
Lecture 2: Motivic integration
motivic measure on non-singular and on singular varieties;
Kontsevich's completed Grothendieck ring;
motivic integrals;
Kontsevich's application: birationally equivalent Calabi-Yau manifolds have the same Hodge numbers;
motivic volume.
Lecture 3: Motivic and related zeta functions
modification formula for the topological Euler characteristic;
motivic zeta functions;
topological zeta functions;
monodromy conjecture.
Lecture 4: Batyrev's stringy invariants
special singularities: terminal, canonical, log terminal, log canonical;
Batyrev's stringy Euler number and stringy E-function for log terminal singularities: well defined using motivic integration;
stringy invariants for general singularities.


1. W. Veys: Arc spaces, motivic integration and stringy invariants, math.AG/0401374.
2. M. Blickle: A short course on geometric motivic integration, math.AG/0507404.