master doc.toc

\contentsline {chapter}{Declaration}{7}{chapter*.4}
\contentsline {chapter}{Copyright}{8}{chapter*.5}
\contentsline {chapter}{Acknowledgements}{9}{chapter*.6}
\contentsline {chapter}{\numberline {1}Introduction}{10}{chapter.1}
\contentsline {section}{\numberline {1.1}Content of the Thesis}{14}{section.1.1}
\contentsline {chapter}{\numberline {2}Preliminaries}{16}{chapter.2}
\contentsline {section}{\numberline {2.1}K\"ahler geometry}{16}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Basic definitions}{16}{subsection.2.1.1}
\contentsline {subsubsection}{K\"ahler manifolds}{16}{section*.12}
\contentsline {subsubsection}{Hermitian and K\"ahler metrics}{17}{section*.13}
\contentsline {subsection}{\numberline {2.1.2}Line bundles and Kodaira Embedding}{19}{subsection.2.1.2}
\contentsline {subsection}{\numberline {2.1.3}Canonical metrics on K\"ahler manifolds}{21}{subsection.2.1.3}
\contentsline {section}{\numberline {2.2}Algebraic and symplectic tools}{22}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}The algebraic torus}{22}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Linearizations}{23}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}Hamiltonian actions and moment maps}{24}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}Chow and GIT quotients}{25}{subsection.2.2.4}
\contentsline {subsubsection}{GIT quotients}{25}{section*.14}
\contentsline {subsubsection}{Kempf-Ness approach to GIT quotients}{26}{section*.15}
\contentsline {subsubsection}{GIT quotients under smooth blowup}{28}{section*.16}
\contentsline {subsubsection}{Chow and limit quotients}{29}{section*.17}
\contentsline {section}{\numberline {2.3}$T$-varieties}{29}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}Toric varieties}{30}{subsection.2.3.1}
\contentsline {subsection}{\numberline {2.3.2}Higher complexity $T$-varieties}{31}{subsection.2.3.2}
\contentsline {subsection}{\numberline {2.3.3}\(f\)-divisors and divisorial polytopes}{35}{subsection.2.3.3}
\contentsline {section}{\numberline {2.4}Equivariant $K$-stability}{40}{section.2.4}
\contentsline {subsection}{\numberline {2.4.1}Twisted equivariant $K$-stability}{40}{subsection.2.4.1}
\contentsline {subsection}{\numberline {2.4.2}$K$-stability of $T$-varieties}{42}{subsection.2.4.2}
\contentsline {chapter}{\numberline {3}K\"ahler-Ricci solitons on Fano threefolds}{44}{chapter.3}
\contentsline {section}{\numberline {3.1}The method of proof}{44}{section.3.1}
\contentsline {section}{\numberline {3.2}Two examples in detail}{48}{section.3.2}
\contentsline {section}{\numberline {3.3}Barvinok Integration}{52}{section.3.3}
\contentsline {chapter}{\numberline {4}$R(X)$ in complexity one}{56}{chapter.4}
\contentsline {section}{\numberline {4.1}A short digression into convex geometry}{60}{section.4.1}
\contentsline {section}{\numberline {4.2}Proof of Theorem 4}{63}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}Product Configurations}{63}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Non-Product Configurations}{63}{subsection.4.2.2}
\contentsline {chapter}{\numberline {5}K\"ahler-Einstein metrics in complexity two}{65}{chapter.5}
\contentsline {section}{\numberline {5.1}Chow quotient calculations}{66}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}Bidegree $(a,b)$ hypersurfaces}{66}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}A wonderful compactification on the quadric}{69}{subsection.5.1.2}
\contentsline {section}{\numberline {5.2}Log canonical thresholds and Tian's criterion}{72}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}Log canonical thresholds}{72}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}Tian's alpha invariant and criterion}{73}{subsection.5.2.2}
\contentsline {subsection}{\numberline {5.2.3}Log canonical threshold bounds}{76}{subsection.5.2.3}
\contentsline {chapter}{\numberline {6}Conclusions and further work}{78}{chapter.6}
\contentsline {chapter}{\numberline {A}Threefold Data}{80}{appendix.A}
\contentsline {chapter}{\numberline {B}SageMath Code}{89}{appendix.B}
\contentsline {chapter}{Bibliography}{93}{appendix*.50}