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Vanishing theorems on complex manifolds. (English) Zbl 0578.32055
Progress in Mathematics, Vol. 56. Boston - Basel - Stuttgart: Birkhäuser. XIII, 170 p. DM 55.00 (1985).
The subject of cohomology vanishing theorems for holomorphic vector bundles on complex manifolds plays an important role in several complex variables and algebraic geometry. The theory began in 1953 with the Kodaira vanishing theorem, but its roots can be traced back to Riemann and Roch for the case of curves and to Picard for surfaces. Although the theory straddles complex analysis and algebraic geometry, this book takes primarily an analytic approach.
The authors restrict their attention to compact Kaehler manifolds.
This book provides a good survey on this subject. The specific contents of the book are as follows: The first chapter treats complex differential geometry leading up to the Kodaira-Nakano identity.
Chapter II deals with the vanishing theorems of Kodaira and Nakano for positive and negative line bundles, the imbedding theorem of Kodaira and the Hodge theory of harmonic forms. In contrast with the Kodaira vanishing theorem, the Nakano vanishing theorem does not hold for semi- positive or semi-negative line bundles. Chapter III presents some generalizations of the Nakano vanishing theorem. Introducing the notion of k-positive and k-negative line bundles, the authors give some generalizations of the Nakano vanishing theorem to these line bundles. They also give another generalization by introducing a powerful inductive technique of slicing by hyperplane sections. Chapter III also explains the relationship between the first Lefschetz theorem and the Nakano vanishing theorem.
Chapter IV discusses vanishing theorems for line bundles on a complex projective space and on smooth complete intersections, and a vanishing theorem of Le Potier on Grassmann manifolds.
There are two methods of generalizing the concept of positivity and vanishing theorems for line bundles to vector bundles. In chapter V, the concept of ampleness for vector bundles is defined and various vanishing theorems for vector bundles are proved.
Chapter VI gives more vanishing theorems for vector bundles by using the differential-geometric approach of Nakano and Griffiths.
Chapter VII treats some generalizations of the Kodaira vanishing theorem due to Ramanujam, Kawamata and Viehweg and some vanishing theorems for singular spaces due to Mumford and Grauert-Riemenschneider.
Reviewer: K.Ogiue

MSC:
32L20 Vanishing theorems
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32Q99 Complex manifolds
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
53C55 Global differential geometry of Hermitian and Kählerian manifolds