K-Stability of Complexity One G-Varieties

Abstract

This thesis concerns the problem of existence of Kaehler-Einstein metrics on Fano manifolds equipped with actions by reductive algebraic groups. We describe the basic ideas of Kaehler geometry and the significance of Kaehler-Einstein metrics before explaining how the solution of the Yau-Tian-Donaldson conjecture by Chen-Donaldson-Sun allows these metrics to be studied via the algebro-geometric concept of K-stability. The equivariant K-stability of Datar-Szekelyhidi, which allows concrete criteria for K-stability to be found for varieties with group actions, is then described. After discussing various aspects of the theory of algebraic group actions on varieties, and the combinatorial description of varieties of complexity one due to Timashev in particular, we apply this theory to the specific case of smooth Fano threefolds admitting SL2- actions. We give a detailed combinatorial description of these varieties, which we then use to prove the K-stability of several examples, via the beta-invariant of Fujita and Li.

Date of Award	1 Aug 2022
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Peter Rowley (Supervisor) & Hendrik Süß (Supervisor)