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## Book Description

The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard-Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry.

1. Cover
2. Half Title
3. Title Page
5. Contents
6. Introduction
7. I: The Topology of Algebraic Varieties
1. 1. The Lefschetz Theorem on Hyperplane Sections
1. 1.1 Morse theory
2. 1.2 Application to affine varieties
3. 1.3 Vanishing theorems and Lefschetz’ theorem
4. Exercises
2. 2. Lefschetz Pencils
1. 2.1 Lefschetz pencils
2. 2.2 Lefschetz degeneration
3. 2.3 Application to Lefschetz pencils
4. Exercises
3. 3. Monodromy
1. 3.1 The monodromy action
2. 3.2 The case of Lefschetz pencils
3. 3.3 Application: the Noether-Lefschetz theorem
4. Exercises
4. 4. The Leray Spectral Sequence
1. 4.1 Definition of the spectral sequence
2. 4.2 Deligne’s theorem
3. 4.3 The invariant cycles theorem
4. Exercises
8. II: Variations of Hodge Structure
1. 5. Transversality and Applications
1. 5.1 Complexes associated to IVHS
2. 5.2 The holomorphic Leray spectral sequence
3. 5.3 Local study of Hodge loci
4. Exercises
2. 6. Hodge Filtration of Hypersurfaces
1. 6.1 Filtration by the order of the pole
2. 6.2 IVHS of hypersurfaces
3. 6.3 First applications
4. Exercises
3. 7. Normal Functions and Infinitesimal Invariants
1. 7.1 The Jacobian fibration
2. 7.2 The Abel-Jacobi map
3. 7.3 The case of hypersurfaces of high degree in Pn
4. Exercises
4. 8. Nori’s Work
1. 8.1 The connectivity theorem
2. 8.2 Algebraic equivalence
3. 8.3 Application of the connectivity theorem
4. Exercises
9. III: Algebraic Cycles
1. 9. Chow Groups
1. 9.1 Construction
2. 9.2 Intersection and cycle classes
3. 9.3 Examples
4. Exercises
2. 10. Mumford’s Theorem and its Generalisations
1. 10.1 Varieties with representable CH0
2. 10.2 The Bloch-Srinivas construction
3. 10.3 Generalisation
4. Exercises
3. 11. The Bloch Conjecture and its Generalisations
1. 11.1 Surfaces with pg = 0
2. 11.2 Filtrations on Chow groups
3. 11.3 The case of abelian varieties
4. Exercises
10. References
11. Index