Determinants of Hodge-Riemann forms

TL;DR

This paper computes the determinant of a bilinear form in Stanley-Reisner rings of odd-dimensional simplicial spheres, proving a conjecture and establishing invariants for these topological objects.

Contribution

It proves the odd multiplicity conjecture for simplicial spheres and extends the results to simplicial homology manifolds, including characteristic 2 cases.

Findings

Determinant of the bilinear form is a complete invariant of the simplicial sphere.

Proves the odd multiplicity conjecture of Papadakis and Petrotou.

Establishes the strong Lefschetz property for certain Stanley-Reisner ring quotients.

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.