Construction of Hyperkähler Metrics for Complex Adjoint Orbits

2.3 Hyperkähler Manifolds 16
2.3 Hyperkähler Manifolds
We briefly recall some basic facts about hyperkähler manifolds, and estab-
lish the notation we shall use throughout this work. The basic reference is
[HKLR87].
A manifold M is hyperkähler if it is equipped with a metric g and complex
structures I, J, K satifying the quaternionic relations
I 2 = J 2 = K 2 = IJK = −1,
and such that the metric is Kähler with respect to each of these complex
structures. We thus obtain three Kähler forms,
ω1(X, Y ) = g(IX, Y ); ω2(X, Y ) = g(JX, Y ); ω3(X, Y ) = g(KX, Y ).
Hyperkähler manifolds posess some remarkable properties, which in a
sense reflect their “rigidity”. We list some of these below.
(i) If we just assume that I, J, K define almost complex structures, their
integrability follows from the closedness of the forms ω1, ω2, ω3. This is es-
pecially useful when constructing hyperkähler manifolds as moduli spaces of
instantons. We shall face one such example in Chapter 4.
(ii) The form ωc = ω2 + iω3 is a holomorphic symplectic form with re-
spect to the complex structure I. Thus, if we are looking for hyperkähler
manifolds, complex manifolds with holomorphic symplectic forms are natural
candidates.
(iii) If G is a Lie group acting on a hyperkähler manifold so as to preserve
its hyperkähler structure, there is a hyperkähler quotient construction. If
µ1, µ2, µ3 are the three moment maps correponding to the symplectic forms
ω1, ω2, ω3 respectively, then the quotient is obtained by taking (µ −1
1 (0) ∩
µ −1
2 (0) ∩ µ −1
3 (0))/G, with its quotient metric.