Hyperkähler manifold

A hyperkähler manifold is a Riemannian manifold $(M^{4n},g)$ equipped with complex structures $I,J,K$ satisfying $IJ=K=-JI$ , each orthogonal for $g$ , 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 ordered triple is fixed by the full holonomy group, so $\operatorname{Hol}_p(g)$ is contained in the compact symplectic group $\operatorname{Sp}(n)$ . Conversely, containment of the full holonomy in a chosen copy of $\operatorname{Sp}(n)$ produces a global parallel triple. Containment only of the restricted holonomy $\operatorname{Hol}_p^0(g)$ gives such triples locally (and on the universal cover); monodromy from the other components of the full holonomy can obstruct a global ordered triple. The complex two-form

\Omega_I=\omega_J+i\omega_K

is a parallel holomorphic symplectic form on the complex manifold $(M,I)$ .

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.

Morphisms

A triholomorphic map preserves the specified ordered triple but need not preserve the metric. A hyperkähler isometric immersion preserves both the metric and the ordered triple. If it is a diffeomorphism, it is a hyperkähler isometry. A rotating hyperkähler isometry instead permits one constant $SO(3)$ -rotation of the triple and is not strict unless that rotation is the identity.

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 specifies a complex manifold, not a particular metric or quaternionic triple. Yau's theorem supplies a compatible hyperkähler metric after a Kähler class is chosen. The IHS convention excludes flat complex tori and nontrivial products. The core uses the broader Riemannian convention.

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.