From Riemann and Kodaira to modern developments on complex manifolds
The article presents a rich exposition of the modern theory of complex manifolds, done through a presentation of the main classical techniques and a storytelling based on the author's long and outstanding experience. Every paragraph presents an idea or a problem, along with detailed references to the original works; nevertheless there are no proofs, computations or new results. At the end many illuminating open questions are posed and possible paths for further developments of the theory are proposed. The paper starts with an exposure of the foundational ideas due to Riemann, Hodge, Hirzebruch and Kodaira. Then the stress is put on the problem of existence of Kähler-Einstein metrics on various classes of manifolds and algebraic varieties: The leitmotiv is the extensive work that Yau and other mathematicians did on Calabi's conjecture. The article continues with the generalization of the ideas of the previous part to the setting of holomorphic bundles over complex manifolds. This leads to Hermitian Yang-Mills connections: The history of the study of this subject is presented, along with the techniques used to investigate their existence and the structure of some moduli space of theirs. At the end of the exposition, the modern ideas of mirror symmetry are presented. The article is intended to be a survey of the historical development of the subject. The references are precise and very numerous, making it a valuable support for the study and an inspiration for the future research.
scientific article; zbMATH DE number 6647351
scientific article