### MA8108 Advanced complex analysis

Here is the lecture notes (will be updated after each lecture)

In the proof of the Riemann mapping theorem and the uniformization theorem, the key is the theory of maximal (Perron type) envelopes. In 1987, Ohsawa and Takegoshi proved the famous L2 extension theorem for holomorphic functions, which opens the door of studying maximal envelopes using L2 methods (e.g. the proof of the Suita conjecture and the Bergman kernel approximation of the maximal envelopes). The main tool in their proof is a highly non-trivial twisted version of the H\"ormander L2-theory (around 1960). In recent years (after 2014), a new way of looking at the Ohsawa-Takegoshi theorem was found by Berndtsson-Lempert based on a complex version of the classical Brunn-Minkowski theory (established by Berndtsson around 2005). In this course, we will introduce the complex Brunn-Minkowski theory, see how it naturally leads us to the H\"ormander L2-estimate in several complex variables.

Below is an ambitious version of the course plan. We plan to cover Part 1-5 in the class

Part 1: Convex analysis background - The classical Brunn-Minkowski inequality and the Alexandrov-Fenchel inequality (use Legendre transform to explain the theorem, but will not give the proof)

Part 2: An invitation to toric varieties - Delzant polytope and Bernstein-Kushnirenko theorem (use the dictionary between convex polytopes and algebraic varieties to explain the Bernstein-Kushnirenko theorem, but will not give the proof)

Part 3: Brascamp-Lieb proof of the Prekopa theorem - We will prove the Prekopa theorem which implies the Brunn-Minkowski inequality

Part 4: A short several complex variables course - We will mainly follow the H\"ormander book, the Blocki notes http://gamma.im.uj.edu.pl/~blocki/publ/ln/scv-poznan.pdf and chapter 2 of the Berndtsson notes http://www.math.chalmers.se/~bob/not3.pdf

Part 5: Subharmonicity of Bergman kernel (Berndtsson 2005) - This is a highly non-trivial complex generalization of the Prekopa theorem. Here we will introduce the H\"ormander L2-theory and prove results in section 4 of the H\"ormander book

Part 6: Some application of the H\"ormander L2-theory in algebraic geometry - Vanishing theorems, Bergman kernel approximation and the proof of the Bernstein-Kushnirenko theorem

Part 7: Berndtsson-Lempert approach to Ohsawa-Takegoshi and its applications - Demailly's Bergman kernel approximation and Siu's invariance of plurigenera

Part 8: From Ohsawa-Takegoshi to potential theory and positivity theory in algebraic geometry - Theory of maximal envelopes

Part 9: Ross-Witt Nystr\"om correspondence - The dictionary between maximal envelopes and geodesic rays

We shall mainly follow Berndtsson's notes http://www.math.chalmers.se/~bob/7nynot.pdf and my toric notes in this course.

Other books that will be used in this course:

1. (The origin of H\"ormander L2-theory, the textbook for self-learning) An Introduction to Complex Analysis in Several Variables, L.H\"ormander
2. (This openbook provides background for further studies on H\"ormander L2-theory) Complex Analytic and Differential Geometry, J. P. Demailly
3. (The origin of Deformation theory) Complex manifolds and deformation of complex structures, K.Kodaira

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