Note for Complex Geometry(NOT COMPLETED)
I made this note when learning the video course of complex geometry taught by Prof. Zhang Xi in 2020SP semester. The video can be found here. The note is divided into two parts.
Part I: Introduction to multivariable complex analysis, basic definition of complex manifold and Kahler manifold, connection and curvature, differential operator on complex manifold.
Part II: Hodge theory, Chern-Weil theory, sheaf theory and stability of vector bundle, line bundle and vanishing theorem, Hermitian-Einstein bundle, Donaldson-Uhlenbeck-Yau theorem and Calabi-Yau theorem.
One may refer to Differential Geometry of Complex Vector Bundles by Kobayashi for Chapter 1 to 7 of this course.