Extremal Kähler metrics and separable toric geometries

Abstract

In this thesis we unify and find new, and recover all known, explicit local examples of extremal toric Kähler metrics and describe how to compactify them. To do so, we define explicitly a class of toric geometries of Sasaki type with toric Kähler quotients, both called \textit{separable} geometries, using \textit{factorization structures}. We conjecture factorization structures to be decomposable in which case we find their explicit description to be of Segre-Veronese type. A compatible factorization structure gives rise to \textit{separable coordinates} on the image of the momentum map of a given separable geometry. In such coordinates the extremality equation for separable Kähler geometries becomes a functional system of ODEs which, in our case, is a system obtained from a generalisation of the method for separation of variables for PDEs. We derive necessary conditions for its solutions and find a complete set of solutions in the case of the product Segre-Veronese factorization structure with a decomposable Sasaki structure. We use generalised equipoised condition for extremal affine functions to geometrically characterise some compactifications of such extremal metrics.

Date of Award	14 Feb 2022
Original language	English
Awarding Institution	University of Bath
Supervisor	David Calderbank (Supervisor) & Karim Anaya-Izquierdo (Supervisor)