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arXiv · 2204.07758

On the anisotropy theorem of Papadakis and Petrotou

Kalle Karu·Elizabeth Xiao
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Abstract

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen's g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field $\mathbb{Q}$. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.

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Kalle Karu, Elizabeth Xiao. 2022-04-16. On the anisotropy theorem of Papadakis and Petrotou. https://arxiv.org/abs/2204.07758

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