Lefschetz Defect in Families

TL;DR

This paper offers a cohomological approach to understanding the Lefschetz defect in smooth complex projective varieties, demonstrating its invariance under deformation for Fano varieties and relating it to isogeny factors in abelian varieties.

Contribution

It introduces a cohomological characterization of the Lefschetz defect and explores its behavior in families and under deformation, providing new insights into its properties.

Findings

Lefschetz defect of a smooth Fano variety is deformation invariant.

Lefschetz defect of an abelian variety relates to its isogeny factors.

Cohomological methods effectively characterize the Lefschetz defect.

Abstract

We provide a cohomological characterization of the Lefschetz defect of smooth complex projective varieties. As a consequence, we deduce that the Lefschetz defect of a smooth Fano variety is invariant under smooth deformation. We also characterize the Lefschetz defect of an abelian variety in terms of its isogeny factors, and study it in families.