凸函數與可積調和形式

Abstract

本篇論文我們討論關於在複數空間上的可微形式的一些內積結構。給定一個L2 結構，我們考慮在可微形式上的Dolbeault算子的伴隨形式。而所謂的調和形式是指落在Dolbeault算子的核與其伴隨形式的核中的可微形式。 我們知道所有在L2 結構下收斂的調和形式會形成一個有限維度的向量空間。而這篇論文的主要結果就是得出這個向量空間的維度。

This thesis studies some inner product structures related to differential forms on the complex space. From a class of L2 structure, we consider the formal adjoint of the Dolbeault operator on the differential forms. The differential forms in the kernels of the Dolbeault operator and its adjoint are known as the harmonic forms. We shall see that the harmonic forms which converge under the L2 structure become a finite dimensional vector space. As the main result of this thesis, we obtain the dimension of this vector space.