Noncommutative motives and their applications

Summary

This survey is based on lectures given by the authors during the program “Non-commutative algebraic geometry and representation theory” at MSRI in the Spring 2013. It covers recent work by the authors on noncommutative motives and their applications, and is intended for a broad mathematical audience. In Section 1 we recall the main features of Grothendieck’s theory of motives. In Sections 2 and 3 we introduce several categories of noncommutative motives and describe their relation with the classical commutative counterparts. In Section 4 we formulate the noncommutative analogues of Grothendieck’s standard conjectures of type C and D, of Voevodsky’s smash-nilpotence conjecture, and of Kimura–O’Sullivan finite-dimensionality conjecture. Section 5 is devoted to recollections of the (super-)Tannakian formalism. In Section 6 we introduce the noncommutative motivic Galois (super-)groups and their unconditional versions. In Section 7 we explain how the classical theory of (intermediate) Jacobians can be extended to the noncommutative world. Finally, in Section 8 we present some applications to motivic decompositions and to Dubrovin’s conjecture.

We recall here the main features of Grothendieck’s classical theory of pure motives, which will be useful when passing to the noncommutative world. These facts are well-Let 𝓥 (k) be the category of smooth projective k-schemes. The category of pure motives is obtained from 𝓥 (k) known and we refer the reader to [André 2004; Jannsen et al. 1994; Manin 1968] for more detailed treatments. Let k be a base field and F a field of coefficients.by linearization, idempotent completion, and inversion of the Lefschetz motive.