In this thesis we study several problems related to the existence problem of invariant canonical metrics on Fano manifolds in the presence of an effective algebraic torus action. The first chapter gives an introduction. The second chapter reviews the existing theory of T-varieties and reviews various stability thresholds and K--stability constructions which we make use of to obtain new results. In the third chapter we discuss some joint work with my supervisor to find new Kähler-Ricci solitons on smooth Fano threefolds admitting a complexity one torus action. In the fourth chapter we present a new formula for the greatest lower bound on Ricci curvature, an invariant which is now known to coincide with Tian's delta invariant. In the fifth chapter we find new Kähler-Einstein metrics on some general arrangement varieties.
| Date of Award | 31 Dec 2020 |
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| Original language | English |
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| Awarding Institution | - The University of Manchester
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| Supervisor | Alexandre Borovik (Supervisor) & Hendrik Süß (Supervisor) |
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- complexity one
- alpha invariant
- complex geometry
- algebraic geometry
- Kähler-Einstein metrics
- Kähler-Ricci solitons
- K-stability
Stability of varieties with a torus action
Cable, J. (Author). 31 Dec 2020
Student thesis: Phd