The Lefschetz defect of Fano varieties

TL;DR

This survey explores the Lefschetz defect, an invariant of smooth Fano varieties, highlighting its definition, known results, and implications for classifying Fano 3- and 4-folds based on their Picard number.

Contribution

It provides a comprehensive overview of the Lefschetz defect, including its origin, properties, and applications to classify Fano varieties with high Picard numbers, especially in dimensions 3 and 4.

Findings

Lefschetz defect relates to Picard number and prime divisors.

Classification of Fano 3-folds with rho > 4 is recovered.

Few examples of Fano 4-folds with rho > 5 are known.

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}.