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A Kähler manifold is really a complex manifold with a Riemannian metric and symplectic structure suitable Along with the complex composition. Each complex submanifold of a Kähler manifold is Kähler, and so specifically each individual non-singular affine or projective complex variety is Kähler, right after limiting the normal Hermitian metric on C n displaystyle mathbb C ^ n
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Complex geometry is anxious not only with complex spaces, but other holomorphic objects connected to them. The classification of holomorphic line bundles over a complex variety X displaystyle X
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of complex projective House that is definitely, in the identical way, domestically supplied with the zeroes of a finite collection of holomorphic capabilities on open subsets of C P n displaystyle mathbb CP ^ n
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Application of transcendental methods to algebraic geometry falls in this group, along with additional geometric facets of complex Evaluation.
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Complex geometry also will make use of approaches arising out of differential geometry and analysis. Such as, the Hirzebruch-Riemann-Roch theorem, a Specific situation on the Atiyah-Singer index theorem, computes the holomorphic Euler attribute of a holomorphic vector bundle with regards to characteristic lessons from the underlying clean complex vector bundle.