The Bott-Chern complexity of a non-projective Calabi-Yau Kähler pair is at least 3, and this bound is attained by singular non-projective K3 surfaces with Picard rank zero.
On the Existence of Good Minimal Models for K\"ahler Varieties with Projective Albanese Map
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In this article, we establish the existence of a good minimal model for a compact K\"ahler klt pair $(X, B)$ when the Albanese map of $X$ is a projective morphism and the general fiber of $(X, B)$ has a good minimal model.
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Bott-Chern complexity of K\"ahler pairs
The Bott-Chern complexity of a non-projective Calabi-Yau Kähler pair is at least 3, and this bound is attained by singular non-projective K3 surfaces with Picard rank zero.