From Riemann and Kodaira to modern developments on complex manifolds

S.-T. Yau

Abstract: 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.