Fano manifolds with Lefschetz defect 3

TL;DR

This paper characterizes the structure of smooth Fano varieties with Lefschetz defect 3, showing they can be constructed via explicit blow-up procedures from lower-dimensional Fano varieties, and applies this to classify certain Fano 4-folds.

Contribution

It provides a detailed structure theorem for Fano varieties with Lefschetz defect 3, including explicit constructions and classification results for Fano 4-folds.

Findings

Fano varieties with delta=3 are obtained from a lower-dimensional Fano T via blow-ups.

Complete classification of Fano 4-folds with delta=3 is achieved.

Explicit construction methods involve P^2-bundles and blow-ups along disjoint subvarieties.

Abstract

Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the case where delta(X)=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 P^2-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 delta(X)=3.