Introduction to motives

This article is based on the lectures of the same tittle given by the first author during the instructional workshop of the program "number theory and physics" at ESI Vienna during March 2009. An account of the topics treated during the lectures can be found in [50] where the categorical aspects of the theory are stressed. Although naturally overlapping, these two independent articles serve as complements to each other. In the present article we focus on the construction of the category of pure motives starting from the category of smooth projective varieties. The necessary preliminary material is discussed. Early accounts of the theory were given in Manin [42] and Kleiman [36], the material presented here reflects to some extent their treatment of the main aspects of the theory. We also survey the theory of endomotives developed in [11], this provides a link between the theory of motives and tools from quantum statistical mechanics which play an important role in results connecting number theory and noncommutative geometry. An extended appendix (by Matilde Marcolli) further elaborates these ideas and reviews the role of motives in noncommutative geometry.