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Complex geometry from Riemann to Kähler-Einstein and Calabi-Yau. (English) Zbl 1404.14002
Advanced Lectures in Mathematics (ALM) 38. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-352-4/pbk). iii, 664 p. (2018).

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Publisher’s description: Complex geometry has been extensively studied and developed since the 19th century. This volume examines the subject from a global, historical perspective.
It begins with an essay on the historical development of complex geometry, with extensive quotations of experts past and present, followed by a discussion of Calabi-Yau manifolds, and a look at Shing-Tung Yau as a writer and mathematician. Next, it presents a series of selected papers, beginning with Bernhard Riemann’s thesis on the foundation of complex analysis, and his masterpiece on the foundations of geometry, followed by numerous seminal papers of modern practitioners: Atiyah, Bott, Chern, Calabi, Chow, Donaldson, Hirzebruch, Kähler, Kodaira, Siu, Uhlenbeck, and Yau.
The volume concludes with a set of commentaries written by Yau, from his personal perspective, on the broad subject of complex geometry and its applications; and finally with an extensive list of papers on complex geometry from the 20th and 21st centuries, categorized by topic.
The articles of this volume will be reviewed individually.
Indexed articles:
Ji, Lizhen, A historical glimpse of some topics in complex geometry, 3-75 [Zbl 1415.32001]
Cheng, Shiu-Yuen; Ji, Lizhen; Wang, Liping; Xu, Hao, Calabi-Yau manifolds at the interface of mathematics and physics, 81-106 [Zbl 1415.32002]
Ji, Lizhen, Shing-Tung Yau, his mathematics and writings, 109-116 [Zbl 1409.01045]
Riemann, Bernhard, Foundations for a general theory of functions of a complex variable, 121-123 [Zbl 1415.30002]
Riemann, Bernhard, The hypotheses on which geometry is based, 125-128 [Zbl 1428.53006]
Riemann, Bernhard, A mathematical work that seeks to answer the question posed by the most distinguished Academy of Paris, 129 [Zbl 1464.53005]
Kähler, Erich, About a remarkable Hermitian metric, 133-145 [Zbl 1426.53004]
Chern, Shiing-Shen, Characteristic classes of Hermitian manifolds, 149-185 [Zbl 1415.32024]
Chow, Wei-Liang, On compact complex analytic varieties, 189-206 [Zbl 1415.32009]
Calabi, Eugenio, The space of Kähler metrics, 209-210 [Zbl 1426.53002]
Calabi, Eugenio, On Kähler manifolds with vanishing canonical class, 211-220 [Zbl 1426.53003]
Kodaira, Kunihiko, On a differential-geometric method in the theory of analytic stacks, 223-228 [Zbl 1415.32015]
Kodaira, Kunihiko, On Kähler varieties of restricted type, 231-250 [Zbl 1415.32016]
Hirzebruch, Friedrich; Kodaira, Kunihiko, On the complex projective spaces, 253-265 [Zbl 1415.32014]
Atiyah, Michael; Bott, Raoul, A Lefschetz fixed point formula for elliptic differential operators, 269-274 [Zbl 1451.58004]
Yau, Shing-Tung, Calabi’s conjecture and some new results in algebraic geometry, 277-281 [Zbl 1415.32020]
Yau, Shing-Tung, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, 285-345 [Zbl 1435.32028]
Siu, Yum-Tong; Yau, Shing-Tung, Compact Kähler manifolds of positive bisectional curvature, 349-364 [Zbl 1415.32018]
Siu, Yum-Tong, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, 367-408 [Zbl 1415.32017]
Donaldson, Simon, A new proof of a theorem of Narasimhan and Seshadri, 411-418 [Zbl 1410.14002]
Donaldson, Simon, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, 421-450 [Zbl 1427.14003]
Uhlenbeck, Karen; Yau, Shing-Tung, On the existence of Hermitian Yang-Mills connections in stable vector bundles, 453-486 [Zbl 1415.32019]
Yau, Shing-Tung, A brief history of Kähler geometry, 489-522 [Zbl 1415.32004]
Yau, Shing-Tung, Selected commentaries, 527-630 [Zbl 1417.00075]
Ji, Lizhen; Yau, Shing-Tung, Some classic papers in geometric analysis in the 20th century, 633-664 [Zbl 1415.32003]
MSC:
14-03 History of algebraic geometry
32-03 History of several complex variables and analytic spaces
53-03 History of differential geometry
01A75 Collected or selected works; reprintings or translations of classics
00B15 Collections of articles of miscellaneous specific interest
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