ScalingStacks

Proof. First we observe that the choice of the Fubini-Study metric $\omega_{X}$ is immaterial. Given any two choices, the relative Kähler potential between them is bounded by $O(|\log|t||)$ for $0<|t|\ll 1$, because the pole order of a section near $t=0$ must be finite. Thus the relative Kähler potential between two choices of $\omega_{t}$ is bounded by $O(1)$ independent of $t$, which affects the Skoda constant $A$ but not its uniform nature.

We now pass to a finite base change and find a semistable reduction. The Calabi-Yau measure $d\mu_{t}$ on $X_{t}$ is independent of the parametrisation of the base, and is preserved under finite base change. Thus it is enough to prove it assuming $\omega_{X}$ agrees with a smooth Kähler metric on a semistable snc model $\mathcal{X}$; this is a special case of Theorem 2.9. ∎