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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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Nigel Hitchin

Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of Oxford. Hitchin attended Ecclesbourne School, Duffield, and earned his BA in mathematics from Jesus College, Oxford, in 1968. After moving to Wolfson College, he received his D.Phil. in 1972. From 1971 to 1973 he visited the Institute for Advanced Study and 1973/74 the Courant Institute of Mathematical Sciences of New York University.

Hyperkähler manifold

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).

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