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

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