Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Poincar{é} Duality, Degeneracy, and Real Lefschetz Property for T-Hypersurfaces

Jules Chenal

TL;DR

This work establishes a Poincaré duality framework for the Renaudineau–Shaw spectral sequence computing the cohomology of T-hypersurfaces and shows that its pages satisfy a robust duality symmetry. It introduces a concrete degeneracy criterion linked to a real analogue of the Lefschetz Hyperplane Theorem, via injectivity of the inclusion maps on real loci and new invariants $r(\mathbb{R} X_\varepsilon)$ and $\ell(\mathbb{R} X_\varepsilon)$. The paper further connects these degeneracy properties to tropical and Kalinin invariants, providing both general results and specific instances where degeneracy occurs (notably for Viro triangulations and certain Itenberg-like triangulations). These findings yield degeneracy of the RS spectral sequence on the second page in broad settings, with implications for real locus topology and tropical–algebraic correspondence. Overall, the work deepens the understanding of how real and tropical geometry govern the cohomology of $T$-hypersurfaces and their spectral sequences, offering new tools to bound Betti numbers and to detect real Lefschetz-type behavior.

Abstract

In this article, we present two structural results about the Renaudineau-Shaw spectral sequence that computes the cohomology of T-hypersurfaces. The first is a Poincar{é} 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.

Poincar{é} Duality, Degeneracy, and Real Lefschetz Property for T-Hypersurfaces

TL;DR

This work establishes a Poincaré duality framework for the Renaudineau–Shaw spectral sequence computing the cohomology of T-hypersurfaces and shows that its pages satisfy a robust duality symmetry. It introduces a concrete degeneracy criterion linked to a real analogue of the Lefschetz Hyperplane Theorem, via injectivity of the inclusion maps on real loci and new invariants and . The paper further connects these degeneracy properties to tropical and Kalinin invariants, providing both general results and specific instances where degeneracy occurs (notably for Viro triangulations and certain Itenberg-like triangulations). These findings yield degeneracy of the RS spectral sequence on the second page in broad settings, with implications for real locus topology and tropical–algebraic correspondence. Overall, the work deepens the understanding of how real and tropical geometry govern the cohomology of -hypersurfaces and their spectral sequences, offering new tools to bound Betti numbers and to detect real Lefschetz-type behavior.

Abstract

In this article, we present two structural results about the Renaudineau-Shaw spectral sequence that computes the cohomology of T-hypersurfaces. The first is a Poincar{é} 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.
Paper Structure (13 sections, 40 theorems, 119 equations, 10 figures, 1 table)

This paper contains 13 sections, 40 theorems, 119 equations, 10 figures, 1 table.

Table of Contents

  1. Poincaré Duality.
  2. Degeneracy and Real Lefschetz Property.
  3. Rank and Degree.
  4. Viro Triangulations.
  5. Duality in Spectral Sequences
  6. Basic Objects and Notations
  7. Construction of T-Hypersurfaces
  8. Renaudineau-Shaw Spectral Sequences
  9. The Augmentation Filtration in Group Algebras.
  10. The Induced Filtrations.
  11. Properties of the Spectral Sequences.
  12. Degeneracy and Real Lefschetz Property
  13. Degeneracy for a Family of Triangulations

Key Result

Proposition 1.4

The spectral sequences $(E^r_{p,q}(A^*))_{p,q,r\geq 0}$ and $(E_r^{p,q}(A))_{p,q,r\geq0}$ are dual to each other through a collection of duality pairings: defined for all integers $p,q,r\geq 0$. Moreover, $\textnormal{d}_r^{p,q}$ is the adjoint of $\partial^r_{p-r,q+1}$ relatively to the pairing.

Figures (10)

  • Figure 1: The cubical subdivision of the triangle. Some cubical cells are marked by the pair of cells that represents them.
  • Figure 2: Two examples of dual hypersurfaces.
  • Figure 3: A triangle and the groups associated by $F_1^{P}$ to some of its cubical cells. We denote by $(e_1^*;e_2^*)$ the canonical basis of $M\cong\mathbb{Z}^2$ and by $(e_1;e_2)$ the dual basis of $N$.
  • Figure 4: An example of T-curve in $\mathbb{P}^1(\mathbb{R})\times \mathbb{P}^1(\mathbb{R})$.
  • Figure 5: Some page of the spectral sequence of a T-hypersurface of dimension 3.
  • ...and 5 more figures

Theorems & Definitions (118)

  • Conjecture : Conjecture 1.10 in Ren-Sha_bou_bet
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: (1.1.6) in Del_the_hod
  • Proposition 1.4
  • proof
  • Lemma 1.5: Propagation of Poincaré Duality
  • proof
  • Proposition 1.6
  • Definition 2.1
  • ...and 108 more