Small deformations of fibered smooth K-torsion Calabi-Yau varieties with H²(X, O_X)=0 remain fibered, and semiample line bundles remain semiample up to homological equivalence even without the vanishing condition.
Douady,Le probl` eme des modules locaux pour les espacesC-analytiques compacts, Ann
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Deformations of fibered Calabi--Yau varieties
Small deformations of fibered smooth K-torsion Calabi-Yau varieties with H²(X, O_X)=0 remain fibered, and semiample line bundles remain semiample up to homological equivalence even without the vanishing condition.