On some Fano 4-folds with Lefschetz defect 3

Abstract

We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we discuss classification results for 4-dimensional complex smooth projective varieties admitting some particular fiber type contractions.