# Differential Geometry of Complex Vector Bundles

Shoshichi Kobayashi
Pages: 316
https://www.jstor.org/stable/j.ctt7zv1h0

1. Front Matter
(pp. I-VI)
2. Preface
(pp. VII-VIII)
S. Kobayashi
(pp. IX-XI)
4. Chapter I Connections in vector bundles
(pp. 1-29)

Although our primary interest lies in holomorphic vector bundles, we begin this chapter with the study of connections in differentiable complex vector bundles. In order to discuss moduli of holomorphic vector bundles, it is essential to start with differentiable complex vector bundles. In discussing Chern classes it is also necessary to consider the category of differentiable complex vector bundles rather than the category of holomorphic vector bundles which is too small and too rigid.

Most of the results in this chapter are fairly standard and should be well known to geometers. They form a basis for the subsequent chapters. As...

5. Chapter II Chern classes
(pp. 30-48)

In order to minimize topological prerequisites, we take the axiomatic approach to Chern classes. This enables us to separate differential geometric aspects of Chern classes from their topological aspects; for the latter the reader is referred to Milnor-Stasheff [1], Hirzebruch [1] and Husemoller [1]. Section 2 is taken from Kobayashi-Nomizu [1; Chapter XII]. For the purpose of reading this book, the reader may take as definition of Chern classes their expressions in terms of curvature. The original approach using Grassmannian manifolds can be found in Chern’s book [1].

The Riemann-Roch formula of Hirzebruch, recalled in Section 4, is used only...

6. Chapter III Vanishing theorems
(pp. 49-97)

In Section 1 we prove Bochner’s vanishing theorems and their variants. For Bochner’s original theorems, see Yano-Bochner [1]. For a modern exposition, see Wu [1]. These theorems are on vanishing of holomorphic sections or 0-th cohomology groups of holomorphic bundles under some “negativity” conditions on bundles. Vanishing theorems are proved also for Einstein-Hermitian vector bundles which will play a central role in subsequent chapters.

In Section 2 we collect definitions of and formulas relating various operators on Kähler manifolds. The reader who needs more details should consult, for example, Weil [1]. These operators and formulas are used in Section 3...

7. Chapter IV Einstein-Hermitian vector bundles
(pp. 98-132)

Given an Hermitian vector bundle (E, h) over a compact Kähler manifold (M, g), we have a field of endomorphismsKofEwhose components are given by$K_j^i = \sum {{g^{\alpha \bar \beta }}R_{j\alpha \bar \beta }^l}$. This field, which we call the mean curvature, played an important role in vanishing theorems for holomorphic sections, (see Section 1 of Chapter III). In this chapter we consider the Einstein condition, i.e., the condition thatKbe a scalar multiple of the identity endomorphism ofE. When the Einstein condition is satisfied,Eis called an Einstein-Hermitian vector bundle. In Section 1 we prove some basic properties of Einstein-Hermitian vector...

8. Chapter V Stable vector bundles
(pp. 133-192)

In this chapter we shall prove the theorem that every irreducible Einstein-Hermitian vector bundle over a compact Kähler manifold is stable. In Sections 1 and 2 we consider the special case where the base space is a compact Riemann surface. For in this case, the definition of stability (due to Mumford, see Mumford and Fogarty [1]) can be given without involving coherent sheaves and the theorem can be proven as a simple application of Gauss’ equation for subbundles.

However, the definition of stable vector bundle over a higher dimensional base space necessiates the introduction of coherent sheaves. Sections 3 through...

9. Chapter VI Existence of approximate Einstein-Hermitian structures
(pp. 193-236)

In this chapter we explain results of Donaldson [2] and prove the theorem to the effect that ifMis an algebraic manifold with an ample line bundleH, then everyH-semistable vector bundleEoverMadmits an approximate Einstein-Hermitian structure, (see (10.13)).

The partial differential equation expressing the Einstein condition is similar to that of harmonic maps. The best reference for analytic tools (in Sections 4 through 7) is therefore the lecture notes by Hamilton [1], The reader who reads Japanese may find Nishikawa’s notes (Nishikawa-Ochiai [1]) also useful.

Let Herm(r) denote the space ofrXr...

10. Chapter VII Moduli spaces of vector bundles
(pp. 237-290)

Atiyah-Hitchin-Singer [1] constructed moduli spaces of self-dual Yang-Mills connections in principal bundles with compact Lie groups over 4-dimensional compact Riemannian manifolds and computed their dimensions. Itoh [1], [2] introduced Kähler structures in moduli spaces of anti-self-dual connections inSU(n)-bundles over compact Kähler surfaces. Kim [1] introduced complex structures in moduli spaces of Einstein-Hermitian vector bundles over compact Kähler manifolds.

LetEbe a fixed${C^\infty }$complex vector bundle over a compact Kähler manifoldM, andha fixed Hermitian structure inE. In Section 1 we relate the mod space of holomorphic structures inEwith the moduli space of...

11. Bibliography
(pp. 291-298)
12. Index
(pp. 299-302)
13. Notations
(pp. 303-304)
14. Back Matter
(pp. 305-305)