Complex geometry concerns the holomorphic properties of complex manifolds. In this course we shall focus on the analytic and geometric techniques in this study. The celebrated Kodaira's vanishing and embedding theorems give the criterions of when a Kähler manifold can be realized as a projective variety, so that algebraic tools can be brought into the study of such manifolds. The tools of Kodaira mainly are linear analysis such as L2-estimate of dbar operator and the regularity of solutions of linear elliptic systems.
The results discussed in this course will be mainly on the following questions (mainly within the Kähler category) with mostly nonlinear tools: (1) Given two complex (Kähler) manifolds which are diffeomorphic to each other, when they are biholomorphic? (2) What are the `best' (canonical) metrics on complex manifolds or holomorphic vector bundles? Under what conditions they do exist? What are the consequences? (3) Does the Kählerity condition survives small deformations of complex manifolds? How about the projectivity? The first problem can be viewed a holomorphic version of Poincaré conjecture, which is still wildly open outside the Kähler category.
No knowledge of PDE is assumed. Some background of theory of holomorphic functions will be helpful, but not necessary. All results involving nonlinear analysis (such as the existences of harmonic maps and solutions of Monge-Amperè type equations) will be introduced with care. We shall review the Kähler identities and the proof of Kodaira's theorems at the beginning.
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Text: Complex Manifolds, by J. Morrow and K. Kodaira, AMS.
Principles of of Algebraic Geometry, by Griffiths-Harris, John Wiley and Sons.
Differential Analysis on Complex Manifolds, by Wells. Springer.
Complex differential manifolds, by Zheng, AMS/IP.
Complex geometry, by Huybrechts, Springer.
A quick introduction on SCV and baby complex geometric notations
A proof of the De Rham-Dolbeault theorem
A quick introduction to tensor algebra, smooth manifolds and vector bundles