数理科学特論Ⅳ

The Bergman kernel plays a significant role in current research in several complex variables and complex geometry. In this course, I will introduce how to use fundamental complex analysis techniques to study the asymptotic behavior of the Bergman kernel in complex geometry. We will then utilize the asymptotic behavior of the Bergman kernel to prove (1) Demailly's Morse inequalities, (2) Tian's theorem about the convergence of the Fubini-Study metric, and (3) the Kodaira embedding theorem. If time permits, I will briefly introduce how to use methods from basic complex analysis and spectral theory to establish the full asymptotic expansion of the Bergman kernel (the Tian-Yau-Catlin-Zelditch asymptotic expansion).