Arithmetic and geometric aspects of Voevodsky motives and Nori motives

• Applicant: Dr. Luca Terenzi
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 568083338

The aim of the theory of Mixed Motives is to define a universal cohomology theory for algebraic varieties. There exist two main unconditional approaches to this theory, due to Voevodsky and to Nori; the latter has been developed only over fields of characteristic zero so far. A deep conjecture asserts that Voevodsky's and Nori's theories of motives are essentially equivalent. This expectation has guided the development of both theories in the last 30 years. The overall goal of the proposed research programme is to improve the current understanding of Voevodsky's and Nori's theories and of their relation to classical cohomology theories. The focus is on two key aspects: the Tannakian formalism, related to the theory of motivic Galois groups, and the six functor formalism, related to the theory of mixed motivic sheaves. The research programme is divided into four independent projects addressing specific objectives. The goal of the first project is to provide a Tannakian description of the six functor formalism on mixed Hodge modules, using techniques inspired by motivic Galois theory; this will greatly simplify the construction of realizations of mixed motivic sheaves into mixed Hodge modules. The goal of the second project is to study motivic local systems over PEL Shimura varieties: the final result will be a motivic version of the classical formula computing the degeneration of local systems arising from group representations to the strata of the Baily--Borel compactification. The goal of the third project is to develop a well-behaved theory of Nori motives and motivic Galois groups over fields of positive characteristic, and to relate it to the theory of Voevodsky motives as in the characteristic zero setting; part of this project is inspired by the emerging motivic approach to independence-of-l questions. The goal of the fourth project is to study how general six functor formalisms of motivic origin can be rephrased in terms of quasi-coherent derived categories of suitable stacks; motivic Galois theory will play an important role in this study, with potential applications to the other projects. The proposed research programme will bring new evidence in support of the expected equivalence between Voevodsky's and Nori's approaches to Mixed Motives, even without attempting a direct attack on this deep conjecture.

• DFG Programme: WBP Position