On some Fano 4-folds with Lefschetz defect 3

TL;DR

This paper characterizes Fano 4-folds with Picard number 5 and Lefschetz defect 3, linking their structure to toric varieties and prime divisor Picard numbers, and discusses classification of certain fiber type contractions.

Contribution

It provides a complete characterization of Fano 4-folds with Lefschetz defect 3 and Picard number 5, including their toric nature and divisor properties.

Findings

Fano 4-folds with Picard number 5 have Lefschetz defect 3 iff they are toric of type K.

Characterization of these varieties via Picard number of prime divisors.

Classification results for 4-folds with specific fiber type contractions.

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.