Determinants of Hodge-Riemann forms and simplicial manifolds

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, and we conjecture a generalization to the Hodge-Riemann forms of any connected oriented simplicial homology manifold. We show that our conjecture follows from the strong Lefschetz property for certain quotients of the Stanley-Reisner rings.