Japanese Journal of Mathematics

, Volume 11, Issue 2, pp 265–303

From Riemann and Kodaira to Modern Developments on Complex Manifolds

Takagi Lectures

DOI: 10.1007/s11537-016-1565-6

Cite this article as:
Yau, ST. Jpn. J. Math. (2016) 11: 265. doi:10.1007/s11537-016-1565-6


We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and Hirzebruch–Riemann–Roch, and some modern development of uniformization theorems, Kähler–Einstein metric and the theory of Donaldson–Uhlenbeck–Yau on Hermitian Yang–Mills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.

Mathematics Subject Classification (2010)

53C55 32Q25 

Keywords and phrases

Kähler–Einstein metric Donaldson–Uhlenbeck–Yau correspondence mirror symmetry Calabi–Yau manifold 

Copyright information

© The Mathematical Society of Japan and Springer Japan 2016

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA