For log Calabi-Yau pairs, a smooth boundary gives holonomy SU(n), Bochner principle, and stability, while two boundary components break all three and make the universal cover non-compactifiable.
Yau -- On the R icci curvature of a compact K \"a hler manifold and the complex M onge- A mp \`e re equation
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Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover
For log Calabi-Yau pairs, a smooth boundary gives holonomy SU(n), Bochner principle, and stability, while two boundary components break all three and make the universal cover non-compactifiable.