Fano manifolds with Lefschetz defect 3

Abstract

Let X be a smooth, complex Fano variety, and δX its Lefschetz defect. By [4], if δX≥4, then X≅S×T, where dim⁡T=dim⁡X−2. In this paper we prove a structure theorem for the case where δX=3. We show that there exists a smooth Fano variety T with dim⁡T=dim⁡X−2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. Then we apply the structure theorem to Fano 4-folds, to the case where X has Picard number 5, and to Fano varieties having an elementary divisorial contraction sending a divisor to a curve. In particular we complete the classification of Fano 4-folds with δX=3, started in [6].