Existence of non-stabilizing αk-invariants for Fano manifolds with anticanonical polarization

Determine whether there exists a Fano manifold X such that, for the anticanonical polarization L = −K_X, the sequence of quantized α-invariants α_k(L) does not stabilize to α(L) and is not eventually monotone; equivalently, establish whether there is a Fano manifold X with L = −K_X for which α_k(L) fails to equal α(L) for arbitrarily large k and the sequence {α_k(L)} does not become monotone for large k.

Background

Tian introduced the global log canonical threshold α(L) and its quantized versions α_k(L), conjecturing that α_k(L) stabilizes to α(L) for sufficiently large k (Conjecture 1.1) and, more weakly, that {α_k(L)} is eventually monotone (Conjecture 1.2).

This paper constructs an ample line bundle L on a smooth surface S for which α_k(L) neither stabilizes nor is eventually monotone, providing counterexamples to Conjectures 1.1 and 1.2 in general. However, the example does not use the anticanonical polarization on a Fano manifold, leaving unresolved whether similar non-stabilization occurs when L = −K_X on Fano X, which is the most relevant case for Kähler–Einstein metrics.