Lefschetz classes on projective varieties

TL;DR

This paper constructs examples of smooth complex projective varieties where the Lefschetz algebra fails to satisfy the hard Lefschetz theorem and Poincaré duality, challenging classical assumptions in algebraic geometry.

Contribution

It provides explicit examples of varieties with Lefschetz algebras that do not adhere to expected duality and symmetry properties, highlighting limitations of classical theorems.

Findings

Lefschetz algebra can violate hard Lefschetz theorem

Lefschetz algebra can violate Poincaré duality

Examples challenge classical geometric assumptions

Abstract

The Lefschetz algebra $L^*(X)$ of a smooth complex projective variety $X$ is the subalgebra of the cohomology algebra of $X$ generated by divisor classes. We construct smooth complex projective varieties whose Lefschetz algebras do not satisfy analogues of the hard Lefschetz theorem and Poincar\'e duality.