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.
Birational complexity and dual complexes
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abstract
We introduce the notion of birational complexity of a log Calabi-Yau pair. This invariant measures how far the log Calabi-Yau pair is to being birational to a toric pair. We study fundamental properties of the new invariant, with a particular focus on the geometry of dual complexes.
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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.