Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences, and mixed complexes of sheaves: a survey

This is a survey of author’s results on weight structures and Voevodsky’s motives. Weight structures are natural counterparts of t-structures (for triangulated categories) introduced by the author. They allow to construct weight complexes, weight filtrations, and weight spectral sequences for various cohomology theories. Partial cases of the latter are: ‘classical’ weight spectral sequences (for singular and etale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences. All of those are mentioned in the current paper. The details, proofs, and several more results could be found in other papers of the author (cited here). We also mention a certain triangulated category of comotives that contains reasonable (co)motives for all function fields (and also of other projective limits of smooth varieties).