The Lefschetz defect of Fano varieties

Abstract

This note is a short survey on the Lefschetz defect, an invariant of smooth Fano varieties that has been recently introduced; it is related to the Picard number rho(X) of X, and to the Picard number of prime divisors in X. We explain the definition of the Lefschetz defect delta(X) and its origin, the known results - mainly on the case delta(X)>1, and several examples, especially in dimensions 3 and 4. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with rho(X)>4, due to Mori and Mukai. We also review the known examples of Fano 4-folds with rho(X)>5, that are remarkably few: apart products and toric examples, there are only 6 known families, with rho(X) in {6,7,8,9}.