HQDIAG

From Riemann to Poincaré, topological methods pervaded the development of geometry, algebra and arithmetic. Homology and homotopy, sheaf theory and spectral sequences, derived and model categories have shaped a wealth of invariants applied to a wide variety of problems, crowned by Grothendieck’s refoundation of algebraic geometry. In this line, Voevodsky’s motivic theory has burst out in the nineties led by the proof of the Bloch-Kato conjecture. After Morel and Voevodsky’s foundation of motivic homotopy, the theory has strongly evolved around its close connection with algebraic topology, abutting to important foundational results such as cohomological orientation theory (motivic cohomology, algebraic cobordism), and the discovery of the quadratic nature of the A1-homotopical invariants: Morel’s computations of stable homotopy sheaves of spheres, Panin and Walter’s generalized orientations leading to Levine’s quadratic enumerative geometry. The aim of the project is to extend and apply A1-homotopical methods in three main complementary directions, a unifying motto being the role of characteristic classes, especially that of the fundamental class of the diagonal: - The study of non necessarily proper algebraic varieties through A1-homotopical methods, with the aim of establishing a theory of "A1-homotopy at infinity". A cornerstone would be the understanding of the fundamental class of the diagonal of open algebraic varieties. Our first target is to develop methods of computations from different perspectives, which could also give a new approach for computing these fundamental classes. We also hope to develop this line of thoughts in order to initiate an A1-homotopical study of singularities, knots and links following Mumford, Milnor, and others. A long-term motivation is to build unstable A1-homotopical invariants at infinity, and formulate an A1-homotopical analogue of the Poincaré conjecture. - Secondly, we want to apply the quadratic invariants of A1-homotopy theory to arithmetic problems. A first direction, of an arithmetic nature, is the development of a quadratic Riemann-Roch formula, based on the quadratic invariants of A1-homotopy such as Hermitian K-theory, and Chow-Witt groups. Further, we intend to develop the recent notion of formal ternary laws, an A1-homotopical analogue of formal group laws. Given the central role of the latter, we think this study is promising for the future of stable A1-homotopy. We also propose a more exploratory route, to uncover the possible role of quadratic invariants of A1-homotopy in Beilinson's conjectures on special values of L-functions. - Our last theme propose to develop new decomposition theorems for relative motives. Indeed, it can be formulated in terms of decomposition of the fundamental class of the diagonal, this time in the proper case. Some of our recent results give new cases where the relative Chow-Künneth conjecture can be proved. We hope to be able to extend this result to more arithmetical cases, by working over number fields or even number rings. We also plan to develop the theory of relative Nori motives and its links with Voevodsky's theory. It is finally possible to transport the methods and definitions of A1-homotopy at infinity to the category of Nori motives and to take advantage of this abelian theory.