The theory of motives was created by Grothendieck in the 1960s as he searched for a universal cohomology theory for algebraic varieties. The theory of pure motives is well established as far as the construction is concerned. Pure motives are expected to have a number of additional properties predicted by Grothendieck's standard conjectures, but these conjectures remain wide open. The theory for mixed motives is still incomplete.
This book deals primarily with the theory of pure motives. The exposition begins with the fundamentals: Grothendieck's construction of the category of pure motives and examples. Next, the standard conjectures and the famous theorem of Jannsen on the category of the numerical motives are discussed. Following this, the important theory of finite dimensionality is covered. The concept of Chow-Kunneth decomposition is introduced, with discussion of the known results and the related conjectures, in particular the conjectures of Bloch-Beilinson type. We finish with a chapter on relative motives and a chapter giving a short introduction to Voevodsky's theory of mixed motives.
Jacob P. Murre, Universiteit Leiden, The Netherlands Jan Nagel, Universite de Bourgogne, Dijon Cedex, France Chris A. M. Peters, Universite Grenoble I, St. Martin d'Heres, France
Algebraic cycles and equivalence relations Survey of some of the main results on Chow groups Proof of the theorem of Voisin-Voevodsky Motives: Construction and first properties On Grothendieck's standard conjectures Finite dimensionality of motives Properties of finite dimensional motives Chow-Kunneth decomposition; The Picard and Albanese motive Chow-Kunneth decomposition in a special case On the conjectural Bloch-Beilinson filtration Relative Chow-Kunneth decomposition Surfaces fibered over a curve Beyond pure motives The category of motivic complexes Bibliography Index of notation Index