Hyperkähler manifold - Imported Knowlpedia Content

Definition

A hyperkähler manifold is a Riemannian manifold $(M^{4n},g)$ equipped with complex structures $I,J,K$ satisfying $IJ=K=-JI$ , such that

\nabla I=\nabla J=\nabla K=0

for the Levi-Civita connection $\nabla$ of $g$ . Thus $(M,I,J,K,g)$ is hyper-Hermitian, and each pair $(g,I)$ , $(g,J)$ , and $(g,K)$ is Kähler. The ordered triple and metric are part of the structure. Equivalently, one may require the three fundamental forms $\omega_I,\omega_J,\omega_K$ to be closed. No compactness, simple connectedness, or irreducibility hypothesis is included.

Holonomy and complex-symplectic structure

The parallel triple reduces the holonomy group to the compact symplectic group $\operatorname{Sp}(n)$ . Conversely, a Riemannian $4n$ -manifold with holonomy contained in $\operatorname{Sp}(n)$ locally carries such a parallel triple; a global triple requires the corresponding global reduction. The complex two-form

\Omega_I=\omega_J+i\omega_K

is a parallel holomorphic symplectic form on the complex manifold $(M,I)$ . These equivalences are treated in Joyce, Chapter 7.

Structure and examples

Hyperkähler metrics are Ricci-flat because $\operatorname{Sp}(n)\subseteq\operatorname{SU}(2n)$ . Every unit vector $(a,b,c)\in S^2$ gives another parallel complex structure $aI+bJ+cK$ and another Kähler form $a\omega_I+b\omega_J+c\omega_K$ .

Flat quaternionic space $\mathbb H^n$ is the basic noncompact example. A Riemannian manifold that is Kähler only for $I$ is a near miss: the additional parallel anticommuting structures $J,K$ are essential.

Terminology

In algebraic geometry, “compact hyperkähler manifold” is sometimes used for an irreducible holomorphic symplectic manifold: a compact simply connected Kähler manifold whose holomorphic two-forms are generated by a nondegenerate one. That convention excludes flat complex tori and nontrivial products. The core uses the broader Riemannian convention; the relationship is discussed in Huybrechts, §1.

References

Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. Oxford DOI record. Relevant: Chapter 7, hyperkähler manifolds.
Daniel Huybrechts, “Compact Hyperkähler Manifolds: Basic Results,” Inventiones Mathematicae 135 (1999), 63–113. DOI record. Relevant: §1, the irreducible holomorphic-symplectic convention.