Convex geometry and asymptotics of stability thresholds in algebraic geometry

Abstract

This thesis investigates stability thresholds in algebraic geometry, particularly the $\alpha$- and $\delta$-invariants, which play a central role in various areas, including the existence of K"ahler-Einstein metrics on Fano manifolds. The work is divided into four main parts. First, we provide a counterexample to Tian's stability conjecture for $\alpha_k$-invariants, demonstrating that these invariants do not always stabilize or become monotone for large $k$. Second, we resolve the Cheltsov--Rubinstein problem for strongly asymptotically log del Pezzo surfaces by studying the last remaining pair $\mathrm{(II.6A.n.m)}$ and establishing necessary and sufficient conditions for the existence of K"ahler-Einstein edge metrics. In the last two parts, we study the stabilization and asymptotics of toric $\alpha_k$- and $\delta_k$-invariants, showing that while $\alpha_k$-invariants stabilize from $k=1$ in the toric setting, $\delta_k$-invariants rarely do, and we derive their asymptotic expansions. The results combine techniques from convex geometry, Ehrhart theory, and birational geometry, offering new insights into the interplay between convex geometry and algebraic geometry.