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.