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

TL;DR

This paper classifies K-polystability of certain Fano 4-folds with high Lefschetz defect, identifying specific families that are K-polystable or K-unstable.

Contribution

It provides a detailed classification of K-polystability for Fano 4-folds with Lefschetz defect 2 and 3, focusing on specific families and their stability status.

Findings

5 of 19 Fano 4-fold families with Lefschetz defect 3 are K-polystable.

Among 175 Casagrande-Druel Fano 4-fold families with Lefschetz defect 2, 5 are K-polystable.

132 of these families are K-unstable.

Abstract

In this paper, we investigate K-polystability on smooth complex Fano 4-folds with Lefschetz defect at least 2, focusing on the case of Lefschetz defect 3 and on Casagrande-Druel Fano 4-folds with Lefschetz defect 2. We show that exactly 5 of the 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable. Moreover, among 175 families of Casagrande-Druel Fano 4-folds with Lefschetz defect 2, we prove that 5 are K-polystable, while 132 are K-unstable.