it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were first given this...

uniqueness theorem for hyperkähler metrics on compact Kähler manifolds admitting holomorphically symplectic structures. Examples of hyperkähler metrics on noncompact...

super-space extensions, was developed by J. Harnad and S. Shnider. Hyperkähler manifolds of dimension 4 k {\displaystyle 4k} also admit a twistor correspondence...

K3 surface is Kähler (by Siu). Almost complex manifold Hyperkähler manifold Quaternion-Kähler manifold K-energy functional Cannas da Silva (2001), Definition...

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

connected self-dual Einstein 4-manifolds are coincide with the 4-dimensional, reverse-oriented hyperkähler manifolds in the Ricci-flat case, but are...

A hyperkähler manifold is a quaternionic manifold with a torsion-free Sp ⁡ ( n ) {\displaystyle \operatorname {Sp} (n)} -structure. A hyperkähler manifold...

complex Lagrangian submanifolds of hyperkähler manifolds, fixed points of a real structure of Calabi–Yau manifolds. The SYZ conjecture deals with the...

Calabi–Yau manifold into simpler pieces and considering the effects of T-duality on these pieces. The simplest example of a Calabi–Yau manifold is a torus...

differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as...

In differential geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G...

compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of...

not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex. Every hyperkähler manifold is also hypercomplex. The converse...

describes a field Σ that takes on values in a nonlinear manifold called the target manifold  T. The non-linear σ-model was introduced by Gell-Mann &...

}_{2}} . Although the above loose version of the definition includes hyperkähler manifolds, the standard convention of excluding these will be followed by...

examples of exotic metric structures such as Calabi–Yau manifolds and hyperkähler manifolds, and in gauge theory, where holomorphic vector bundles often admit...

mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group G is the quotient of a fiber of a hyperkähler moment map M → g...

sheaves on one Calabi–Yau manifold is equivalent in a certain sense to the Fukaya category of a completely different Calabi–Yau manifold. This equivalence provides...

M^{[n]}} by Hartogs' principle. A holomorphically symplectic, Kähler manifold is hyperkähler, as follows from the Calabi–Yau theorem. Hilbert schemes of points...

SO(4n), so every hyperkähler manifold is a Calabi–Yau manifold, every Calabi–Yau manifold is a Kähler manifold, and every Kähler manifold is orientable....

Sasaki–Einstein; if it is hyperkähler, M {\displaystyle M} is called 3-Sasakian. Any 3-Sasakian manifold is both an Einstein manifold and a spin manifold. If M is...

algebraic surfaces, and via the ADHM construction, or hyperkähler reduction (see hyperkähler manifold), a geometric invariant theory procedure. The groundbreaking...

Strominger in his 1990 paper Special Geometry. The Higgs branch is a hyperkähler manifold as was shown by Luis Alvarez-Gaume and Daniel Freedman in their 1981...

constructing new invariants of smooth manifolds, the construction of exotic geometric structures such as hyperkähler manifolds, as well as giving alternative...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

of hyperkähler manifolds and locally conformally Kähler manifolds. He proved an analogue of the global Torelli theorem for hyperkähler manifolds and...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

compact extra dimensions must be shaped like a Calabi–Yau manifold. A Calabi–Yau manifold is a special space which is typically taken to be six-dimensional...

between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are...

Generalized complex manifold Calabi–Yau manifold Hyperkähler manifold K3 surface hypercomplex manifold Quaternion-Kähler manifold Symplectic topology...