Poincaré duality, degeneracy, and real Lefschetz property for T-Hypersurfaces

In this article, we present two structural results about the Renaudineau–Shaw spectralsequence that computes the cohomology of T-hypersurfaces. The first is a Poincaré duality satisfied by all its pages of positive index. The second is a vanishing criterion. It reformulates the vanishing of the boundary operators of the spectral sequence as the injectivity of some morphisms induced in cohomology by the inclusion of the T-hypersurface in its surrounding toric variety. It implies that the Renaudineau–Shaw spectral sequence of a T-hypersurface degenerates at the second page if and only if the T-hypersurface satisfies a real version of the Lefschetz hyperplane section theorem.