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to have a topology that is regionally akin to the subspace topology due to identification with open subsets of C n displaystyle mathbb C ^ n
. Yet another way by which Stein manifolds are similar to affine complex algebraic varieties is the fact that Cartan's theorems A and B hold for Stein manifolds.
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Complex geometry has substantial apps to theoretical physics, exactly where it is important in being familiar with conformal subject concept, string principle, and mirror symmetry. It is frequently a source of examples in other areas of mathematics, together with in representation idea exactly where generalized flag versions can be examined making use of complex geometry leading to the Borel–Weil–Bott theorem, or in symplectic geometry, where by Kähler manifolds are symplectic, in Riemannian geometry where complex manifolds supply samples of unique metric structures for example Calabi–Yau manifolds and hyperkähler manifolds, As well as in gauge idea, the place holomorphic vector bundles normally admit alternatives to big differential equations arising Discover outside of physics like the Yang–Mills equations. Complex geometry Furthermore is impactful in pure algebraic geometry, exactly where analytic brings about the complex environment like Hodge theory of Kähler manifolds encourage understanding of Hodge structures for varieties and techniques in addition to p-adic Hodge concept, deformation concept for complex manifolds evokes knowledge of the deformation idea of strategies, and effects concerning the cohomology of complex manifolds motivated the formulation of your Weil conjectures and Grothendieck's regular conjectures.
, and non-singular in any other case. A projective complex analytic range is usually a subset X ⊆ C P n displaystyle Xsubseteq mathbb CP ^ n
is Stein if it is holomorphically convex and holomorphically separable (begin to see the post on Stein manifolds for your complex definitions). It could be demonstrated on the other hand that this is akin to X displaystyle X
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to get compact. In such cases Yau's evidence in the Calabi conjecture implies that X displaystyle X