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基本説明
Continues Voisin's study of topology of families of algebraic varieties and the relationships between Hodge theory and algebraic cycles.
Full Description
The 2003 second volume of this 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.
Contents
Introduction. Part I. The Topology of Algebraic Varieties: 1. The Lefschetz theorem on hyperplane sections; 2. Lefschetz pencils; 3. Monodromy; 4. The Leray spectral sequence; Part II. Variations of Hodge Structure: 5. Transversality and applications; 6. Hodge filtration of hypersurfaces; 7. Normal functions and infinitesimal invariants; 8. Nori's work; Part III. Algebraic Cycles: 9. Chow groups; 10. Mumford' theorem and its generalisations; 11. The Bloch conjecture and its generalisations; References; Index.
