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\title{Stability of varieties with a torus action}
\author{Jacob Cable}
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\school{Mathematics}
\faculty{Science and Engineering}
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In this thesis we study several problems related to the existence problem of invariant canonical metrics on Fano orbifolds 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 find new K\"ahler-Einstein metrics on some general arrangement varieties. 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 discuss joint work with my supervisor to find new K\"ahler-Ricci solitons on smooth Fano threefolds admitting a complexity one torus action. 

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\prefacesection{Acknowledgements}


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