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.

中文翻译：

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从黎曼和小平到复杂流形的现代发展

我们研究了与黎曼曲面有关的复杂流形理论，霍奇理论，Chern类，小平嵌入和Hirzebruch–Riemann–Roch，以及均匀化定理，Kähler–Einstein度量和Donaldson–Uhlenbeck–Yau理论的现代发展在Hermitian Yang-Mills连接上。我们强调与物理学有关的数学思想。最后，我们确定了未来可能的研究方向，并提出了一些重要的开放性问题。