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From Riemann and Kodaira to modern developments on complex manifolds

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Japan. J. Math. 11, 265–303 (2016) DOI: 10.1007/s11537-016-1565-6

From Riemann and Kodaira to modern developments on complex manifolds

?

Shing-Tung Yau

Received: 9 March 2016 / Revised: 19 May 2016 / Accepted: 9 June 2016 Published online: 9 August 2016

© The Mathematical Society of Japan and Springer Japan 2016 Communicated by: Hiraku Nakajima

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 mathemati- cal ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.

Keywords and phrases: Kähler–Einstein metric, Donaldson–Uhlenbeck–Yau correspondence, mirror symmetry, Calabi–Yau manifold

Mathematics Subject Classification (2010): 53C55, 32Q25

? This article is based on the 16th Takagi Lectures that the author delivered at the University of Tokyo on November 28 and 29, 2015.

S.-T. YAU

Department of Mathematics, Harvard University, One Oxford Street, Cambridge MA 02138, USA

(e-mail: )

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