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Concept# Kähler manifold

Summary

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.
Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.
Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view:
A Kähler manifold is a symplectic manifold (X, ω) equipped with an integrable almost-complex structure J which is compatible with the symplectic form ω, meaning that the bilinear form
on the tangent space of X at each point is symmetric and positive definite (and hence a Riemannian metric on X).
A Kähler manifold is a complex manifold X with a Hermitian metric h whose associated 2-form ω is closed. In more detail, h gives a positive definite Hermitian form on the tangent space TX at each point of X, and the 2-form ω is defined by
for tangent vectors u and v (where i is the complex number ). For a Kähler manifold X, the Kähler form ω is a real closed (1,1)-form. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric g defined by
Equivalently, a Kähler manifold X is a Hermitian manifold of complex dimension n such that for every point p of X, there is a holomorphic coordinate chart around p in which the metric agrees with the standard metric on Cn to order 2 near p.

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A compact Kahler manifold X is shown to be simply connected if its 'symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective connected holomorphic map f : X -> S between connected manifolds induces an isomorphism of fundamental groups if its smooth fibres are as above, and if X is Kahler.

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