variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics. Since Kähler manifolds are equipped with...

that is Kähler with respect to g {\displaystyle g} . If ω I , ω J , ω K {\displaystyle \omega _{I},\omega _{J},\omega _{K}} denotes the Kähler forms of...

Kähler manifold with a vanishing first real Chern class has a Kähler metric in the same class with vanishing Ricci curvature. (The class of a Kähler metric...

almost Kähler manifold. Any symplectic manifold admits a compatible almost complex structure making it into an almost Kähler manifold. A Kähler manifold is...

simply-connected Kähler manifold, a Kähler metric is Ricci-flat if and only if the holonomy group is contained in the special unitary group. On a general Kähler manifold...

differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a...

sign of the first Chern class of the Kähler manifold: When the first Chern class is negative, there is always a Kähler–Einstein metric, as Thierry Aubin...

complex manifold Complex Poisson manifold Hyper-Kähler manifold Kähler quotient Hyperkähler quotient Kähler–Einstein metric Nearly Kähler manifold Quaternion-Kähler...

curvature Kähler metric (cscK metric) is a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is a Kähler–Einstein...

Examples of Kähler manifolds include smooth projective varieties and more generally any complex submanifold of a Kähler manifold. The Hopf manifolds are examples...

but are sometimes called quaternion Kähler manifolds otherwise. Higher-dimensional Lorentzian Einstein manifolds are used in modern theories of gravity...

{\displaystyle M} . In particular, a Kähler manifold is nearly Kähler. The converse is not true. For example, the nearly Kähler six-sphere S 6 {\displaystyle...

Frölicher–Nijenhuis bracket Kähler manifold – Manifold with Riemannian, complex and symplectic structure Poisson manifold – Mathematical structure in...

submanifold of a Kähler manifold is Kähler, and so in particular every non-singular affine or projective complex variety is Kähler, after restricting...

Kähler (1864–1941), German politician Kähler Keramik, a Danish ceramics manufacturer Kähler manifold, an important geometric complex manifold Kahler (disambiguation)...

Kähler manifold is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of complex manifolds....

functions. On a Kähler manifold, the (p, q) components of a harmonic form are again harmonic. Therefore, for any compact Kähler manifold X, the Hodge theorem...

following questions: Hodge conjecture for Kähler varieties, vector bundle version. Let X be a complex Kähler manifold. Then every Hodge class on X is a linear...

1954.[Y78a] As a special case, this showed that Kähler-Einstein metrics exist on any closed Kähler manifold whose first Chern class is nonpositive. Yau's...

manifold G2 manifold Kähler manifold Calabi–Yau manifold Hyperkähler manifold Quaternionic Kähler manifold Riemannian symmetric space Spin(7) manifold The Wikibook...

{\partial }}} -lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold. Sometimes it is also known as the d d c {\displaystyle...

compact Kähler manifold, applying hp,q = hq,p recovers the earlier definition for projective varieties. By using hp,q = hq,p for compact Kähler manifolds this...

However, Kähler manifolds already possess holonomy in U ( n ) {\displaystyle U(n)} , and so the (restricted) holonomy of a Ricci-flat Kähler manifold is contained...

structure). A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is a Kähler manifold with Kähler form t 2 d θ + 2 t d t ∧ θ...

In complex geometry, the Kähler identities are a collection of identities between operators on a Kähler manifold relating the Dolbeault operators and...

prove existence of Kähler-Einstein metrics on closed Kähler manifolds with positive first Chern class, also known as "Fano manifolds." Tian and Yau extended...

space must not only be Kähler, but also the Kähler form must lift to integral cohomology. Such manifolds are called Hodge manifolds. The first example appeared...

over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials....

single tangent space to the entire manifold, and many special metrics such as constant scalar curvature metrics and Kähler–Einstein metrics are constructed...

Calabi–Eckmann manifold M is non-Kähler, because H 2 ( M ) = 0 {\displaystyle H^{2}(M)=0} . It is the simplest example of a non-Kähler manifold which is simply...