ScalingStacks

We say $\pi$ is a degeneration family of Calabi-Yau manifolds if there is a trivialising section $\Omega$ of the canonical bundle $K_{X}$. Over a small disc $\mathbb{D}_{t}$ around $0\in S$, this induces holomorphic volume forms $\Omega_{t}$ on $X_{t}$ via $\Omega=dt\wedge\Omega_{t}$. The normalised Calabi-Yau measure on $X_{t}$ is the probability measure

$$d\mu_{t}=\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}.$$

(5)

The Calabi-Yau metrics $\omega_{CY,t}$ on $X_{t}$ are the unique Kähler metrics in the class $\frac{1}{|\log|t||}c_{1}(L)$ such that

$$\frac{\omega_{CY,t}^{n}}{\int_{X_{t}}\omega_{CY,t}^{n}}=d\mu_{t}.$$

(6)