On the quasi-monomiality of the $\alpha$- and $\delta$-invariants

TL;DR

This paper proves that for certain algebraic pairs and divisors, the invariants alpha and delta are determined by quasi-monomial valuations, extending previous results without uncountability assumptions.

Contribution

It establishes that alpha and delta invariants are computed by quasi-monomial valuations for projective klt pairs over algebraically closed fields of characteristic zero.

Findings

Alpha and delta invariants are computed by quasi-monomial valuations.

No uncountability assumption on the base field is needed.

Results apply to any big Q-Cartier Q-divisor on the pair.

Abstract

In this paper, we show that for any projective klt pair $(X,\Delta)$ over an algebraically closed field of characteristic \(0\) and any big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L$ on $X$, the invariants $\alpha(X,\Delta,L)$ and $\delta(X,\Delta,L)$ are computed by quasi-monomial valuations, without any uncountability assumption on the base field.