Do all complete intersections have the Lefschetz properties?

Determine whether every standard graded Artinian complete intersection A = k[x_1, ..., x_n]/(f_1, ..., f_n) over a field of characteristic zero has the Strong Lefschetz Property; if this fails in some cases, ascertain whether at least the Weak Lefschetz Property always holds for such complete intersections.

Background

The Weak and Strong Lefschetz Properties (WLP/SLP) concern whether multiplication by a general linear form, or powers thereof, has maximal rank in every degree. It is known that every Artinian monomial complete intersection over a field of characteristic zero has the SLP, and as a consequence a general complete intersection with fixed generator degrees has the SLP. Moreover, for complete intersections in three variables (height 3), the WLP is proven using vector bundle techniques (Grauert–Mülich applied to the syzygy bundle).

Despite these advances, a comprehensive resolution across all complete intersections remains elusive. The authors highlight that the question of whether every complete intersection possesses SLP (or even WLP) is still open, underscoring a central problem in the Lefschetz property literature.