Spring 2018 Complex Manifolds and Kaehler Geometry

Reference：

Books
【1】《Lectures on Kaehler geometry》 by  Andrei Moroianu
【2】《Complex Geometry, An Introduction》by Daniel Huybrechts
【3】《Canonical Metrics in Kähler Geometry》 by Tian Gang
【4】《Complex Manifold without Potential Theory》by Chern Shiing-Shen
【5】《Differential Analysis on Complex Manifolds》by  R.O.Wells
Chapters
【1】《黎曼几何引论》第八章 by 陈维桓 李兴校
【2】《Foundations of Differential Geometry》Chapter IX by Shoshichi Kobayashi and Katsumi Nomizu
【3】《Chern-Weil Theory and Witten Deformation》Chapter 1 by Zhang Weiping
【4】《Holomorphic Morse Inequalities and Bergman Kernels》Chapter 1,5 by Ma Xiaonan and George Marinescu
【5】《Complex Analytic and Differential Geometry》Chapter V, VI, VII by Jean-Pierre Demailly

Preliminary: Geometry and Topology I ，familiar with the definition and the basic properties of differential manifolds, vector bundles, connections and curvatures. No sheaf theory is used in this lecture.

Plan：
Chapter I  Foundations of Kähler Geometry,  including the definition of Kähler manifold，Chern connection and holomorphic sectional curvature;
Chapter II Topology of Kähler Manifolds,  including Chern-Weil theory，Chern class，Dolbeault cohomology，Kähler identity， Hodge decomposition，Hard lefschetz theorem, etc;
Chapter III Positive vector bundles and Vanishing Theorem;
Chapter IV Calabi-Yau Manifolds and Kähler-Einstein Metric.  

Note (PDF version, not finished)

• Lesson 1: Kaehler Manifolds
• Lesson 2: Chern connection
• Lesson 3: Holomorphic sectional curvature
• Lesson 4: Chern class
• Lesson 5: Hodge theory
• Lesson 6: Relations of Hodge numbers
• Lesson 7: Bochner methods and vanishing
• Lesson 8: Proof of Hodge Theorem
• Lesson 9: Kodaira embedding theorem