Thesis Defense
Friday August 10th, 2018
Time: 2:30 PM
Title: Quaternionic Geometry and Special Holonomy
Speaker: Joseph Thurman, Stony Brook University
Location: Math Tower 5-127
This thesis studies connections with special holonomy group arising from quaternionic manifolds. The focus is on two previously described constructions that produce such connections from positive quaternion-Kähler manifolds with an isometric and quaternionic circle action. The first, due to Hitchin [17], yields quaternionic connections with a preferred complex structure, while the second, due to Haydys [14], yields Kähler metrics. In particular, both constructions produce Kähler metrics in real dimension four, and therefore generalize a theorem of Pontecorvo [38] that produces scalar-flat Kähler metrics from anti-self-dual Hermitian surfaces in real dimension 4.

The goal of this work is to explore the relationship between these two constructions. Although they are superficially similar, the main result of this dissertation shows that they are in fact distinct. This result is obtained by describing a simplification of Haydys’s construction that allows for explicit computation of the Levi-Civita connection of the Kähler metric. Hitchin’s methods are also generalized to give a construction of quaternionic complex manifolds from quaternionic manifolds without a metric.