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Concept
Hyperkähler manifold
Summary
In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.
Hyperkähler manifolds were defined by Eugenio Calabi in 1979.
Equivalently, a hyperkähler manifold is a Riemannian manifold of dimension whose holonomy group is contained in the compact symplectic group Sp(n).
Indeed, if is a hyperkähler manifold, then the tangent space TxM is a quaternionic vector space for each point x of M, i.e. it is isomorphic to for some integer , where is the algebra of quaternions. The compact symplectic group Sp(n) can be considered as the group of orthogonal transformations of which are linear with respect to I, J and K. From this, it follows that the holonomy group of the Riemannian manifold is contained in Sp(n). Conversely, if the holonomy group of a Riemannian manifold of dimension is contained in Sp(n), choose complex structures Ix, Jx and Kx on TxM which make TxM into a quaternionic vector space. Parallel transport of these complex structures gives the required complex structures on M making into a hyperkähler manifold.
Every hyperkähler manifold has a 2-sphere of complex structures with respect to which the metric is Kähler. Indeed, for any real numbers such that
the linear combination
is a complex structures that is Kähler with respect to . If denotes the Kähler forms of , respectively, then the Kähler form of is
A hyperkähler manifold , considered as a complex manifold , is holomorphically symplectic (equipped with a holomorphic, non-degenerate, closed 2-form). More precisely, if denotes the Kähler forms of , respectively, then
is holomorphic symplectic with respect to .
Conversely, Shing-Tung Yau's proof of the Calabi conjecture implies that a compact, Kähler, holomorphically symplectic manifold is always equipped with a compatible hyperkähler metric.
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