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**Differential geometry of complex vector bundles.**
*(English)*
Zbl 0708.53002

Publications of the Mathematical Society of Japan, 15; Kanô Memorial Lectures, 5. Princeton, NJ: Princeton University Press; Tokyo: Iwanami Shoten Publishers. xi, 304 p. $ 60.50 (1987).

In recent years, the theory of complex vector bundles has interested not only algebraic and differential geometers and complex analysts but also low-dimensional topologists and mathematical physicists working on gauge theory. This book is based on the author’s lectures given in Berkeley and Japan and it is an excellent textbook for researchers and graduate students in these fields of mathematics.

The first two chapters give a short introduction to connections and Chern classes. In chapter III, vanishing theorems of Bochner, Kodaira, Nakano, Vesenti, Gigante and Girbau, Bott,... are discussed. Among them, vanishing theorems of Bochner type are used extensively in subsequent chapters. The notion of Einstein-Hermitian vector bundles is introduced in Chapter IV, which is closely parallel to the stability. In Chapter V, it is proved that every irreducible Einstein-Hermitian vector bundle over a compact Kähler manifold is stable. It is proved in Chapter VI that if M is an algebraic manifold with an ample line bundle H, then every H- semistable vector bundle over M admits an approximate Einstein-Hermitian structure. Chapter VII is devoted to the study of the moduli spaces of holomorphic structures and Einstein-Hermitian connections in a complex vector bundle over a compact Kähler manifold.

The first two chapters give a short introduction to connections and Chern classes. In chapter III, vanishing theorems of Bochner, Kodaira, Nakano, Vesenti, Gigante and Girbau, Bott,... are discussed. Among them, vanishing theorems of Bochner type are used extensively in subsequent chapters. The notion of Einstein-Hermitian vector bundles is introduced in Chapter IV, which is closely parallel to the stability. In Chapter V, it is proved that every irreducible Einstein-Hermitian vector bundle over a compact Kähler manifold is stable. It is proved in Chapter VI that if M is an algebraic manifold with an ample line bundle H, then every H- semistable vector bundle over M admits an approximate Einstein-Hermitian structure. Chapter VII is devoted to the study of the moduli spaces of holomorphic structures and Einstein-Hermitian connections in a complex vector bundle over a compact Kähler manifold.

Reviewer: K.Ogiue

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

32Lxx | Holomorphic fiber spaces |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |