Abstract:

A quaternion-Kähler manifold is a 4n-dimensional Riemannian manifold whose holonomy group is contained in Sp(n)Sp(1), one of the 7 groups appearing in Berger’s list of Riemannian holonomy groups. They are automatically Einstein and can be regarded as a generalisation of hyperkähler manifolds. The Sp(1) factor in the holonomy is both a blessing and a curse. On the one hand, quaternion-Kähler manifolds are not in general Ricci-flat or even Kähler, which scraps a lot of complex-analytic and symplectic tools that otherwise could have been used to study their geometry. On the bright side, though, their tangent spaces are endowed with a quaternionic structure, which allows one to use methods from twistor theory and make a bridge to complex geometry and even complex-algebraic geometry. Amazingly, there are no known examples of compact nonsymmetric quaternion-Kähler manifolds that are not hyperkähler (Sp(n)Sp(1) is the only special holonomy group with no such examples!). At the same time, there are plenty of symmetric spaces – both compact and noncompact – which are quaternion-Kähler; in fact, there is precisely one such space for each compact simple Lie group. For instance, the compact symplectic group Sp(n+1) corresponds to the quaternionic projective space HP^n.

This talk is supposed to be a relatively gentle introduction to the theory of quaternion-Kähler manifolds. I will assume that the listeners are acquainted with the basics of differential geometry and don’t faint at the word ‘holonomy’ and will define everything else.