Chow rings, decomposition of the diagonal, and the topology of families

Cover......Page 1
Chow Rings, Decomposition of the Diagonal, and the Topology of Families......Page 4
ISBN-13: 9780691160504  ISBN-10: 0691160503  ISBN-13: 9780691160511  ISBN-10: 0691160511......Page 5
Contents......Page 6
Preface......Page 8
Chapter One Introduction......Page 10
1.1.1 Spread......Page 12
1.1.2 Spreading out rational equivalence......Page 13
1.1.3 Applications of Mumford-type theorems......Page 14
1.1.4 Another spreading principle......Page 15
1.2 THE GENERALIZED BLOCH CONJECTURE......Page 16
1.3 DECOMPOSITION OF THE SMALL DIAGONAL AND APPLICATION TO THE TOPOLOGY OF FAMILIES......Page 18
1.4 INTEGRAL COEFFICIENTS AND BIRATIONAL INVARIANTS......Page 20
1.5 ORGANIZATION OF THE TEXT......Page 22
2.1.1 Construction......Page 24
2.1.2 Localization exact sequence......Page 25
2.1.3 Functoriality and motives......Page 27
2.1.4 Cycle class......Page 31
2.2 HODGE STRUCTURES......Page 33
2.2.1 Polarization......Page 34
2.2.2 Hodge classes......Page 35
2.2.3 Standard conjectures......Page 36
2.2.4 Mixed Hodge structures......Page 39
2.2.5 Coniveau......Page 41
3.1 A GENERAL PRINCIPLE......Page 45
3.1.1 Mumford\'s theorem......Page 49
3.1.2 Further applications......Page 50
3.2.1 Generalized decomposition of the diagonal......Page 53
3.2.2 Generalized Bloch conjecture......Page 55
3.2.3 Nilpotence conjecture and Kimura\'s theorem......Page 57
4.1 HODGE CONIVEAU OF COMPLETE INTERSECTIONS......Page 64
4.1.1 Proof of Theorem 4.1 in the case of hypersurfaces......Page 68
4.1.2 Complete intersections......Page 70
4.2.1 A conjecture on effective cones......Page 73
4.2.2 On the generalized Hodge conjecture for coniveau 2 complete intersections......Page 74
4.3 EQUIVALENCE OF GENERALIZED BLOCH AND HODGE CONJECTURES FOR GENERAL COMPLETE INTERSECTIONS......Page 76
4.3.1 Varieties with trivial Chow groups......Page 77
4.3.2 A consequence of Conjecture 2.29......Page 79
4.3.3 A spreading result......Page 80
4.3.4 Proof of Theorem 4.16......Page 81
4.3.5 Further applications......Page 88
4.4 FURTHER APPLICATIONS TO THE BLOCH CONJECTURE ON 0-CYCLES ON SURFACES......Page 94
5.1 TAUTOLOGICAL RING OF A K3 SURFACE......Page 97
5.1.1 Other hyper-Kähler manifolds......Page 99
5.2 A DECOMPOSITION OF THE SMALL DIAGONAL......Page 105
5.2.1 Calabi-Yau hypersurfaces......Page 110
5.3.1 Deligne\'s decomposition theorem......Page 115
5.3.2 Multiplicative decomposition isomorphisms......Page 118
5.3.3 Families of abelian varieties......Page 122
5.3.4 A multiplicative decomposition theorem for families of K3 surfaces......Page 124
6.1.1 Atiyah-Hirzebruch-Totaro topological obstruction.......Page 132
6.1.2 Kollar\'s example......Page 134
6.2 RATIONALLY CONNECTED VARIETIES AND THE RATIONALITY PROBLEM......Page 136
6.2.1 The group Z 2n2(X)......Page 138
6.2.2 The group Z 4(X) and unramified cohomology......Page 141
6.3.1 Structure of the Abel-Jacobi map and birational invariants......Page 148
6.3.2 Decomposition of the diagonal and structure of the Abel-Jacobi map......Page 155
Bibliography......Page 164
Index......Page 172