Let $M$ be a Ricci-flat Kähler manifold, i.e. a compact Kähler manifold with trivial first Chern class $c_{1}(M)\in H^{2}(M,{\mathbb{C}})$. Then $M$ carries a canonical probability measure $\mu$, given by $\mu=e^{2\psi}/\int_{M}e^{2\psi}$ where $\psi$ is a Hermitian metric on $K_{M}$ with curvature $0$ (and hence unique up to a constant).

By the Calabi-Yau theorem, each Kähler $(1,1)$-class on $M$ further contains a unique Ricci-flat Kähler metric $\omega$, characterized by

$$\frac{\omega^{n}}{\int_{M}\omega^{n}}=\mu.$$

Recall also that $K_{M}$ is torsion, i.e. $rK_{M}\simeq{\mathcal{O}}_{M}$ for some positive integer $r$. Indeed, this is a consequence of the Beauville-Bogomolov theorem [Beau83, Bog74], which implies that $M$ admits a finite étale cover $p:M^{\prime}\to M$ with $K_{M^{\prime}}=p^{*}K_{M}$ trivial. A trivializing section $\eta$ of $rK_{M}$ defines a metric $\psi=\tfrac{1}{r}\log|\eta|$ on $K_{M}$ as above, and hence $\mu=|\eta|^{2/r}/\int|\eta|^{2/r}$.

As a consequence of Theorem A, we shall prove: