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

conjecture that any compact Kähler manifold with vanishing first Chern class also admits a (typically different) Kähler metric with vanishing Ricci tensor...

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...

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...

{\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...

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...

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

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...

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

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

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

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

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

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...

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

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...

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

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...

Kodaira's result, a compact Kähler manifold M embeds as an algebraic variety. This result gives an example of a complex manifold with enough meromorphic functions...

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...

so every hyperkähler manifold is a Calabi–Yau manifold, every Calabi–Yau manifold is a Kähler manifold, and every Kähler manifold is orientable. The strange...

g)} is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex...

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

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

a Kähler manifold. This implies that, for any projective algebraic variety X, X ( C ) {\displaystyle X(\mathbb {C} )} is a compact Kähler manifold. The...

doctoral dissertation, titled "Isometric complex analytic imbedding of Kähler manifolds", was done under the supervision of Salomon Bochner. From 1951 to 1955...