This work has been of great interest both to topologists and to number theorists. The first part of this book describes some of the work of Kuo-Tsai Chen on iterated integrals and the fundamental group of a manifold. The author attempts to make his exposition accessible to beginning graduate students. He then proceeds to apply Chen's constructions to algebraic geometry, showing how this leads to some results on algebraic cycles and the Abel-Jacobi homomorphism. Finally, he presents a more general point of view relating Chen's integrals to a generalization of the concept of linking numbers, and ends up with a new invariant of homology classes in a projective algebraic manifold.
Author(s): Bruno Harris, K. t Chen,
Year: 2004
Language: English
Pages: 120
Interated Integrals and Cycles on Algebraic Manifolds......Page 4
Contents......Page 12
Preface......Page 8
1.1 Introduction......Page 14
1.2 Differential equations......Page 15
1.3 Program......Page 20
1.4 Lie algebras......Page 21
1.5 Chen's Lie algebra and connection......Page 27
1.6 Some work of Quillen......Page 32
1.7 Group homology......Page 34
1.8 The basic isomorphisms......Page 38
1.9 Lattices in nilpotent Lie groups......Page 39
1.10 Some Hodge theory......Page 41
2.2 Generalities on Riemann surfaces and iterated integrals......Page 48
2.3 Harmonic volumes and iterated integrals......Page 55
2.4 Use of the Jacobian......Page 58
2.5 Variational formula for harmonic volume......Page 60
2.6 Algebraic equivalence and homological equivalence of algebraic cycles......Page 65
Homological versus algebraic equivalence in a Jacobian......Page 68
2.8 Currents and Hodge theory......Page 80
De Rham’s results for currents:......Page 83
3.1 Introduction......Page 86
3.2 The main theorem......Page 97
Appendix: Orientations, Fiber Integration......Page 108
THE REFERENCES......Page 114
List of Notations......Page 116
Index......Page 120
