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It is correct that each complex manifold is particularly a real clean manifold. This is because the complex plane C displaystyle mathbb C
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, and non-singular if not. A projective complex analytic variety is often a subset X ⊆ C P n displaystyle Xsubseteq mathbb CP ^ n
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In addition, the additional framework of complex geometry will allow, specifically in the compact setting, for international analytic final results to become tested with wonderful achievement, which includes Shing-Tung Yau's evidence of your Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge correspondence, and existence success for Kähler–Einstein metrics and frequent scalar curvature Kähler metrics. These benefits often feed back into complex algebraic geometry, and one example is lately the classification of Fano manifolds employing K-steadiness has benefited immensely equally from approaches in Assessment As well as in pure birational geometry.
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Serre's GAGA theorem asserts that projective complex analytic versions are actually algebraic. While it's not strictly true for affine varieties, There's a class of complex manifolds that act a great deal like affine complex algebraic versions, called Stein manifolds. A manifold X displaystyle X Find out more
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