Hodge Theory and Complex Algebraic Geometry II

The book is the second volume of a two-volume treatise on Hodge theory and its applications. The volume consists of three parts. The first part illustrates various aspects of the qualitative influence of Hodge theory on the topology of algebraic varieties. In particular, the Deligne's theorem on the degeneration of Leray spectral sequence of rational cohomology of a projective fibration at E2 and its consequences, e.g. surjectivity of the map from rational cohomology of the total space to (the base generated) monodromy invariant subspace of rational cohomology of the fiber, are discussed. The second part is devoted to the study of infinitesimal variations of Hodge structure for a family of smooth projective varieties and its applications, in particular those concerning the case of complete families of hypersurfaces of complete intersections of a given variety. The main explicit result in this part is Nori's connectivity theorem. The third (and final) part of the volume is devoted to relations between Hodge theory and algebraic cycles. Using infinitesimal techniques from the second part, certain cycle class maps and equivalence relations (rational, homological, algebraic and Abel-Jacobi equivalence) are established. In particular, variations on the theme of the relation of Chow groups and Hodge theory of smooth complex varieties are reviewed.

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