Motives in Montpellier

In this course I will present the category of pure motives (i.e. for smooth projective varieties) as defined by Grothendieck. Some connections with the modern category of mixed motives will be made. I will spend a little time on generalities on the category and most of the time on applications to concrete questions, especially on algebraic cycles.
A preliminary list of results we will treat is the following:

• l-independency of l-adic cohomology (Katz-Messing 73),
• rational and numerical equivalence coincide for elliptic curves over finite fields (Kahn 2002 and Jannsen 2007),
• Bloch conjecture for some surfaces (Kimura 2005),
• signature of the intersection product on abelian fourfolds (Ancona 2021).

I will insist on the different phenomena between motives over the complex numbers and motives over finite fields.

This is an introductory course on the Grothendieck ring of varieties, which has been playing an increasingly important role in algebraic geometry in the last few decades, especially with the introduction of motivic integration by Kontsevich in the mid-90s. I will present the notion together with a few useful variants, and I will construct some of the usual motivic measures, showing how these allow to recover geometric information. Then I will explain the connections with birational geometry, in particular Bittner’s presentation by blow-up relations, and the theorem of Larsen and Lunts. Some time will be spent on Kapranov’s zeta function and questions around its (non) rationality. Time permitting, I will also discuss various motivic stabilisation results in this setting.

If motives are a (universal) cohomology theory for algebraic varieties and if, instead of individual cohomology groups H^n(X), one focuses on the total cohomology H^*(X), then motives naturally become derived or homotopical objects. This "derived" or "homotopical" turn can be dated back to the 1980s and 90s, following a suggestion by Beilinson and much effort by Voevodsky, Morel, and collaborators. The goals of this mini-course are to:

1. explain what such a turn entails and how one can still work effectively with these things;
2. introduce the stable A^1-homotopy theory and highlight some of its internal structure;
3. describe the relation with alternative motivic theories, including those from the other two mini-courses;
4. touch on a few of the many applications of this viewpoint.