Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles

TL;DR

This paper proves the hard Lefschetz property for pseudomanifolds and cycles across all characteristics, extending previous results and solving a generalized g-conjecture for Cohen Macaulay complexes.

Contribution

It introduces a new proof combining biased pairing theory and a generalized formula, extending Lefschetz properties to broader classes of complexes.

Findings

Proves the Lefschetz theorem for doubly Cohen Macaulay complexes.

Extends the Lefschetz property to pseudomanifolds and cycles in any characteristic.

Provides a simplified approach for characteristic 2 cases.

Abstract

We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles.