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On the flip side, final results and procedures from several of those fields often feed back into complex geometry, and one example is developments within the mathematics of string theory and mirror symmetry have unveiled much about the nature of Calabi–Yau manifolds, which string theorists predict must have the framework of Lagrangian fibrations with the SYZ conjecture, and the development of Gromov–Witten principle of symplectic manifolds has triggered advances in enumerative geometry of complex types.
A Kähler manifold is really a complex manifold having a Riemannian metric and symplectic construction suitable With all the complex structure. Every complex submanifold of a Kähler manifold is Kähler, and so in particular every non-singular affine or projective complex wide range is Kähler, just after limiting the typical Hermitian metric on C n displaystyle mathbb C ^ n
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Recognize that because just about every biholomorphism is a diffeomorphism, and C n displaystyle mathbb C ^ n
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