The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933.
Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.
on a complex manifold is called strictly plurisubharmonic if the real closed (1,1)-form is positive, that is, a Kähler form.
There is no comparable way of describing a general Riemannian metric in terms of a single function.
For a compact Kähler manifold X, the volume of a closed complex subspace of X is determined by its homology class.
In a sense, this means that the geometry of a complex subspace is bounded in terms of its topology.
Explicitly, Wirtinger's formula says that where Y is an r-dimensional closed complex subspace and ω is the Kähler form.
In particular, ωn is not zero in H2n(X, R), for a compact Kähler manifold X of complex dimension n. A related fact is that every closed complex subspace Y of a compact Kähler manifold X is a minimal submanifold (outside its singular set).
Even more: by the theory of calibrated geometry, Y minimizes volume among all (real) cycles in the same homology class.
[6] The identities form the basis of the analytical toolkit on Kähler manifolds, and combined with Hodge theory are fundamental in proving many important properties of Kähler manifolds and their cohomology.
, Hodge theory gives an interpretation of the splitting above which does not depend on the choice of Kähler metric.
So this Hodge decomposition theorem connects topology and complex geometry for compact Kähler manifolds.
This is not true for compact complex manifolds in general, as shown by the example of the Hopf surface, which is diffeomorphic to S1 × S3 and hence has b1 = 1.
[9] A related result is that every compact Kähler manifold is formal in the sense of rational homotopy theory.
The Kodaira embedding theorem characterizes smooth complex projective varieties among all compact Kähler manifolds.
Namely, a compact complex manifold X is projective if and only if there is a Kähler form ω on X whose class in H2(X, R) is in the image of the integral cohomology group H2(X, Z).
Equivalently, X is projective if and only if there is a holomorphic line bundle L on X with a hermitian metric whose curvature form ω is positive (since ω is then a Kähler form that represents the first Chern class of L in H2(X, Z)).
One may ask whether every compact Kähler manifold can at least be deformed (by continuously varying the complex structure) to a smooth projective variety.
Kunihiko Kodaira's work on the classification of surfaces implies that every compact Kähler manifold of complex dimension 2 can indeed be deformed to a smooth projective variety.
[17] An alternative proof of this result which does not require the hard case-by-case study using the classification of compact complex surfaces was provided independently by Buchdahl and Lamari.
The reference to Einstein comes from general relativity, which asserts in the absence of mass that spacetime is a 4-dimensional Lorentzian manifold with zero Ricci curvature.
Although Ricci curvature is defined for any Riemannian manifold, it plays a special role in Kähler geometry: the Ricci curvature of a Kähler manifold X can be viewed as a real closed (1,1)-form that represents c1(X) (the first Chern class of the tangent bundle) in H2(X, R).
It follows that a compact Kähler–Einstein manifold X must have canonical bundle KX either anti-ample, homologically trivial, or ample, depending on whether the Einstein constant λ is positive, zero, or negative.
Shing-Tung Yau proved the Calabi conjecture: every smooth projective variety with ample canonical bundle has a Kähler–Einstein metric (with constant negative Ricci curvature), and every Calabi–Yau manifold has a Kähler–Einstein metric (with zero Ricci curvature).
[20] By contrast, not every smooth Fano variety has a Kähler–Einstein metric (which would have constant positive Ricci curvature).
However, Xiuxiong Chen, Simon Donaldson, and Song Sun proved the Yau–Tian–Donaldson conjecture: a smooth Fano variety has a Kähler–Einstein metric if and only if it is K-stable, a purely algebro-geometric condition.
At the other extreme, the open unit ball in Cn has a complete Kähler metric with holomorphic sectional curvature equal to −1.
For holomorphic maps between Hermitian manifolds, the holomorphic sectional curvature is not strong enough to control the target curvature term appearing in the Schwarz lemma second-order estimate.
This motivated the consideration of the real bisectional curvature, introduced by Xiaokui Yang and Fangyang Zheng.
[23] This also appears in the work of Man-Chun Lee and Jeffrey Streets under the name complex curvature operator.