Topic
Calabi conjecture
About: Calabi conjecture is a research topic. Over the lifetime, 187 publications have been published within this topic receiving 7630 citations.
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01 Jan 2000
TL;DR: The first known examples of these manifolds were discovered by the author in 1993-5 as mentioned in this paper, and much previously unpublished material which significantly improves the original constructions was presented in this book.
Abstract: The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kahler geometry. Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperkahler manifolds). These are constructed and studied using complex algebraic geometry. The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated. The first known examples of these manifolds were discovered by the author in 1993-5. This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.
1,181 citations
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TL;DR: A proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold is announced and some new results in algebraic geometry and differential geometry are proved, including that the only Köhler structure on a complex projective space is the standard one.
Abstract: We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kahler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For example, we prove that the only Kahler structure on a complex projective space is the standard one.
864 citations
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01 Sep 1987
462 citations
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372 citations