On the anisotropy and Lefschetz property for PL-spheres

TL;DR

This paper explores the anisotropy property of PL-spheres, providing a simplified proof of a recent theorem and establishing that 2-dimensional spheres are generically anisotropic over any field, which relates to the Lefschetz property.

Contribution

It offers an equivalent condition for generic anisotropy, simplifies the proof of the Papadakis-Petrotou theorem for PL-spheres, and proves 2D spheres are generically anisotropic over any field.

Findings

Simplified proof of Papadakis-Petrotou theorem for PL-spheres

Equivalent condition for generic anisotropy

2-dimensional spheres are generically anisotropic over any field

Abstract

A simplicial sphere $\Delta$ is said to be generically anisotropic over a field $\mathbb{F}$ if, for a certain purely transcendental field extension $\mathbf{k}$ of $\mathbb{F}$, a certain Artinian reduction $A$ of the face ring $\mathbf{k}[\Delta]$ has the following property: For every nonzero homogeneous element $\alpha\in A$ of degree at most $(\dim\Delta+1)/2$, its square $\alpha^2$ is also nonzero. The importance of this property is that the hard Lefschetz property for simplicial spheres can be derived from it. A recent result of Papadakis and Petrotou shows that every simplicial sphere is generically anisotropic over any field of characteristic $2$. In this paper, we give an equivalent condition of being generically anisotropic, and use it to present a simplified proof of Papadakis-Petrotou theorem for PL-spheres. We also prove that the simplicial spheres of dimension $2$ are generically anisotropic over any field $\mathbb{F}$.