達布定理在幾何上是很有趣的定理，實際上它是利用比較代數的手法－也就是層論－所證明出來的。在我的碩士論文中所回顧的幾個數學定理中，複向量叢版本的達布定理是其中最重要的定理之一。要證明複向量叢版本的達布定理，則必須要先證明達布定理，而這就是第一章所做的事情。

在第二章裡，我們會介紹何謂複向量叢和複向量叢上的聯絡。聯絡有很多有趣的性質。一個聯絡在局部上可以被寫成一個矩陣值的式，而這稱之為聯絡式。在進行計算的時候，聯絡式是一個很好用的工具。在一個複向量叢上的聯絡，可以很自然的衍生出一個在共軛複向量叢上的聯絡。最後，所有的計算都會被應用在一個特別的例子－複流型的切叢－上，而複流型的切叢本身就是一個複向量叢的例子。

在最後一章裡，我們會介紹主叢和主聯絡及其相關性質。而在最後一章中，最重要的結果是我們將會構造一個過程把複向量叢送到主叢，也會構造出一個過程把複向量叢上的聯絡送到主聯絡。換句話說，我們可以把複向量叢想成是一個主叢。

Dolbeault's theorem is a great result in geometry and is achieved by algebraic method. More precisely, Dolbeault's theorem is an application of sheaf theory. One of main theorems reviewed in my thesis is Dolbeault's theorem for complex vector bundles. To prove this, it is necessary to show Dolbeault's theorem first. This is what we review in the first section.

In the second section, we introduce complex vector bundles and connections on them. A connection can be locally written as a matrix-valued 1-form which is called a connection form. It is convenient for us to derive some formulas via connection forms. A connection in a complex vector bundle also may be naturally induced its corresponding dual bundle and conjugate bundle. Eventually, those computation can be applied to tangent bundles over complex manifolds.

We introduce the principal bundles and principal connections in the last section. The most important result in this section is that we can construct a process sending a complex vector bundle to a principal bundle and a process sending a connection in a complex vector bundle to a principal connection. In other words, complex vector bundles can be regarded as principal GL(r,C}-bundles.

1 Preliminary
1.1 Basic Definitions about Bundles
1.2 Sheaf Theory
2 Complex Vector Bundles
2.1 Connections on Complex Vector bundles over a Real Manifold
2.2 Connections on Complex Vector bundles over a Complex Manifold
2.3 Hermitian Structure
2.4 Connections in Associated Vector Bundles
2.5 Subbudles and Quotients Bundles
2.6 Hermitian Manifolds
2.7 Dolbeault's Theorem for Complex Vector Bundles
2.8 Classification of Smooth Complex Vector Bundles
3 Principal Bundles
3.1 Basic Definitions
3.2 Associated Bundles
3.3 Sections
3.4 Gauge Groups
3.5 Connections and Curvatures
3.6 Relation between Complex Vector Bundles and Principal Bundles

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S. Kobayashi, Differential geometry of complex vector bundles, Princeton Univ. Pr., 1932.
S. Kobayashi and K. Nomizu, Foundations of differential geometry, Wiley, 1932.
D. Husemoller, Fibre bundles, Springer-Verlag, 1994.
M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surface, Phil. Trans. Roy. Soc. London A 308 (1982), 528-559.