A counterexample to Tian's Stabilization Conjecture

TL;DR

This paper presents a counterexample disproving Tian's conjecture that the global log canonical threshold equals the level $k$ threshold for large $k$, and also refutes the monotonicity folklore conjecture.

Contribution

It provides the first known counterexample to both Tian's stabilization conjecture and the monotonicity conjecture of the $eta_k$ invariants.

Findings

Counterexample disproves Tian's conjecture.

Counterexample refutes the folklore monotonicity conjecture.

Results impact the understanding of log canonical thresholds.

Abstract

It was conjectured by Tian that the global log canonical threshold (known as the $\alpha$-invariant) is equal to the level $k$ log canonical threshold (known as the $\alpha_k$-invariant) for all sufficiently large $k$. A weaker folklore conjecture has been that the invariants $\alpha_k$ are eventually monotone. We provide a counterexample to both conjectures.