Lectures on the Theory of Pure Motives (University Lecture)

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-Künneth 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 -- P. 4 of cover. 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-KГјnneth 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 Cover 1 Title page 2 Contents 4 Introduction 6 Algebraic cycles and equivalence relations 12 Survey of some of the main results on Chow groups 24 Proof of the theorem of Voisin-Voevodsky 30 Motives: Construction and first properties 34 On Grothendieck’s standard conjectures 46 Finite dimensionality of motives 54 Properties of finite dimensional motives 62 Chow-Künneth decomposition; The Picard and Albanese motive 78 Chow-Künneth decomposition in a special case 94 On the conjectural Bloch-Beilinson filtration 96 Relative Chow-Künneth decomposition 116 Surfaces fibered over a curve 130 Beyond pure motives 134 The category of motivic complexes 146 Bibliography 150 Index of notation 156 Index 158 Back Cover 162