Ph.D Thesis | |

Ph.D Student | Hase Antonius Carl |
---|---|

Subject | Quaternion-Kahler Symmetric Spaces |

Department | Department of Mathematics |

Supervisors | PROF. Michah Sageev |

DR. Tobias Hartnick |

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 d^{c}: Ω^{k}(*D*,
ℝ) → Ω^{k}(*D*,ℝ) as ω=dd^{c}
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 D^{c}: Ω^{k}(*D*,ℝ)
→ Ω^{k}(*D*,ℝ) on quaternionic domains, which
was defined by Oskar Hamlet and Tobias Hartnick. The operator D^{c} is
called the quaternionic conjugate differential. On quaternionic hyperbolic
space we show that Ω is given as Ω=dD^{c} 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 H^{2}_{c}(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 H^{4}_{c}(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.