class contains exactly one Ricci-flat metric. These are often called Calabi–Yau manifolds. However, the term is often used in slightly different ways...

manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979. Marcel Berger's 1955 paper on...

Kähler–Einstein metric would be Ricci flat, making the manifold a Calabi–Yau manifold. The Calabi conjecture was resolved in the case where c 1 ( X ) < 0 {\displaystyle...

geometry". Tosatti, Valentino; Weinkove, Ben; Yau, Shing-Tung (2008). "Taming symplectic forms and the Calabi-Yau equation". Proceedings of the London Mathematical...

complex submanifolds. This follows from the Wirtinger inequality. On a Calabi–Yau manifold, the real part of a holomorphic volume form (suitably normalized)...

Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat. The most important problem...

Donaldson with new algebro-geometric and analytic techniques for constructing Calabi–Yau manifolds with cylindrical ends, resulting in tens of thousands of diffeomorphism...

G2 manifold Calabi–Yau manifold Bonan, Edmond (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", Comptes Rendus de l'Académie...

Kähler–Einstein metrics, a result closely related to the Calabi conjecture. The latter result, established by Yau, provides the largest class of known examples of...

SU(2n) ⊂ U(2n) ⊂ SO(4n), so every hyperkähler manifold is a Calabi–Yau manifold, every Calabi–Yau manifold is a Kähler manifold, and every Kähler manifold...

Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such...

MR 1265307. Batyrev, Victor V. (1994). "Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties". Journal of Algebraic Geometry: 493–535...

form) has been generalized by Yujiro Kawamata and others to families of Calabi–Yau varieties of any dimension. A logarithmic transformation (of order m with...

holomorphically symplectic, Kähler manifold is hyperkähler, as follows from the Calabi–Yau theorem. Hilbert schemes of points on the K3 surface and on a 4-dimensional...