ScalingStacks

Theorem 1.1. Let $X$ be a compact projective Calabi-Yau manifold, and let $\alpha\in N^{1}(X)_{\mathbb{R}}$ be a big and nef class that is not ample. Then there exist a proper analytic subvariety $E\subset X$ and a smooth incomplete Ricci-flat Kähler metric $\omega_{1}$ on $X\backslash E$, that depend only on $\alpha$, such that for any smooth path $\alpha_{t}\in\mathcal{K}_{NS}$ with $\alpha_{1}=\alpha$, the Ricci-flat metrics $\omega_{t}\in\alpha_{t}$ converge to $\omega_{1}$ in the $C^{\infty}$ topology on compact sets of $X\backslash E$. Moreover $\omega_{1}$ extends to a closed positive current with continuous potentials on the whole of $X$, which lies in $\alpha$. If $\alpha\in N^{1}(X)_{\mathbb{Z}}$, that is if $\alpha=c_{1}(L)$ for some line bundle $L$, then $E$ is the null locus of $L$ and $\omega_{1}$ is the pullback of a singular Ricci-flat Kähler metric on a Calabi-Yau model of $X$ obtained from the contraction map of $L$.