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.
Deligne,Th´ eor` eme de Lefschetz et crit` eres de d´ eg´ en´ erescence de suites spectrales, Inst
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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.