Six Lectures on Motives

These lecture notes are taken from my lecture series in the Asian-French summer school on motives and related topics. My goal in my lectures was two-fold: to give first of all a sketch of Voevodsky’s foundational construction of the triangulated category of motives and its basic properties, and then to give an idea of some of the applications and wider vistas this construction has made possible. In doing this, wanted also to point out some of the origins of this theory, coming from both the categorical side involving aspects of sheaf theory and triangulated categories, as well as the input from algebraic geometry, mainly through algebraic cycles. This latter aspect lead me to devote an entire lecture to so-called moving lemmas, as I felt this subject captured much of the geometric side of the theory. I also reviewed much of the necessary material about triangulated categories and sheaves on a Grothendieck site, with the intention of making the discussion as accessible as possible. I chose mixed Tate motives for illustrating applications,. This subject touches on a broad range of subjects, including the theory of t-structures, Tannakian categories, rational homotopy theory, Grothendieck-Teichmüller theory, moduli of curves, polylogarithms and multiple zeta values. For this reason, I felt that an overview would be of interest to a fairly wide audience. I have also included two lectures on the extension of motives given by the motivic stable homotopy category of Morel-Voevodsky, giving a sketch of the construction as well as a discussion of the motivic Postnikov tower. I have made only minor changes and additions to my original lectures in these notes; I hope this will transmit the informal nature of the lectures to the reader. The rewriting of these notes let me recall how much I enjoyed the summer school at the I.H.E.S and gives me the opportunity of thanking most heartily the organizers of summer school, Jean-Marc Fontaine and Jean-Benoit Bost, for putting together a truly worthwhile conference.