K-polystability of Fano 4-folds with large Lefschetz defect

Abstract

We study K-stability on smooth complex Fano 4-folds having large Lefschetz defect, that is greater or equal then 3, with a special focus on the case of Lefschetz defect 3. In particular, we determine whether these Fano 4-folds are K-polystable or not, and show that there are 5 families (out of 19) of K-polystable smooth Fano 4-folds with Lefschetz defect 3.