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

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