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arXiv · 2408.02737

Determinants of Hodge-Riemann forms

Matt Larson·Isabella Novik·Alan Stapledon
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Abstract

We calculate the determinant of the bilinear form in middle degree of the generic artinian reduction of the Stanley-Reisner ring of an odd-dimensional simplicial sphere. This proves the odd multiplicity conjecture of Papadakis and Petrotou and implies that this determinant is a complete invariant of the simplicial sphere. We extend this result to odd-dimensional connected oriented simplicial homology manifolds. In characteristic 2, we prove a generalization to the Hodge-Riemann forms of any connected simplicial homology manifold. To prove the latter theorem we establish the strong Lefschetz property for certain quotients of the Stanley-Reisner rings of connected simplicial pseudomanifolds.

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Matt Larson, Isabella Novik, Alan Stapledon. 2024-08-05. Determinants of Hodge-Riemann forms. https://arxiv.org/abs/2408.02737

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