Ph.D Thesis


Ph.D StudentHase Antonius Carl
SubjectQuaternion-Kahler Symmetric Spaces
DepartmentDepartment of Mathematics
Supervisors PROF. Michah Sageev
DR. Tobias Hartnick


Abstract

Hermitian symmetric spaces are well-studied objects at the crossroads of Lie theory, differential geometry and complex analysis. If we replace the complex numbers by the division algebra of quaternions in the definition of Hermitian symmetric spaces,

we obtain the class of quaternion-Kähler symmetric spaces. A lot less is known about these spaces. It is the intent of this thesis to adapt a few selected facts about Hermitian symmetric spaces to quaternion-Kähler symmetric spaces as far as possible.


The isometry invariant almost complex structure on Hermitian symmetric spaces is integrable. Hermitian symmetric spaces of non-compact type can even be realized as bounded symmetric domains in complex vector spaces as shown by Borel and Harish-Chandra. Contrarily, we will show that the only quaternion-Kähler symmetric spaces with integrable almost quaternionic structure are quaternionic vector space, quaternionic hyperbolic space and quaternionic projective space.

Hermitian symmetric spaces carry an invariant parallel 2-form, namely the Kähler form ω. On a bounded symmetric domain D the Kähler form has a Kähler potential φ. That is, ω is given in terms of the invariant conjugate differential dc: Ωk(D, ℝ) → Ωk(D,ℝ) as ω=ddc φ. The Kähler potential φ on D is given by φ(p)=log κ(p,p), where κ is the Bergman kernel. Instead of an invariant parallel 2-form, quaternion-Kähler symmetric spaces carry an invariant parallel 4-form Ω, which is called the fundamental 4-form. We study a new operator Dc: Ωk(D,ℝ) → Ωk(D,ℝ) on quaternionic domains, which was defined by Oskar Hamlet and Tobias Hartnick. The operator Dc is called the quaternionic conjugate differential. On quaternionic hyperbolic space we show that Ω is given as Ω=dDc φ, where φ is a real multiple of log K(q,q) for a quaternionic analogue of the Bergman kernel K.

The Kähler form ω on a Hermitian symmetric space defines a class in the second continuous cohomology H2c(G,ℝ). This class turns out to be bounded and its Gromov seminorm has been computed, which is of vital importance in questions of Kähler rigidity or higher Teichmüller theory. Analogously the fundamental 4-form Ω on a quaternion-Kähler symmetric space defines a class in the fourth continuous cohomology H4c(G,ℝ). It is known that the Gromov seminorm of this class is bounded, but unfortunately there is no known bound on the norm. We will explain why a recent attempt to compute this seminorm on quaternionic hyperbolic spaces is flawed.