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Poincaré duality for singular tropical hypersurfaces

Samuel Dentan

TL;DR

The paper addresses tropical Poincaré duality for singular hypersurfaces arising from non-primitive patchworking. It introduces an $R$-primitivity framework and develops tropical cosheaves $\mathcal{F}^R_p$ and dual sheaves, together with a cap-product and a fundamental class, to relate tropical cohomology and homology via spectral sequences. The main contributions are a complete Poincaré duality over any integral domain $R$ for $R$-primitive triangulations and a partial duality for $(k,R)$-primitive triangulations, with a corollary that simple polytopes satisfy complete PD over $\mathbb{Q}$; this extends RS 2023 and BMR 2024 to non-primitive patchworkings. The methods provide a robust framework for bounding tropical Betti numbers and clarifying the interaction between tropical and classical dualities in patchworking, with potential applications to real algebraic hypersurfaces constructed via Viro patchworking.

Abstract

We find a partial extension of Poincaré duality theorem of Jell-Rau-Shaw to hypersurfaces obtained by non-primitive Viro's combinatorial patchworking. We define a classification of the triangulations of a lattice polytope by a level of primitivity and we find a partial Poincaré duality for patchworkings depending on the level of primitivity of the triangulation. Our notion of primitivity is defined modulo a certain integral domain, it is weaker than the classical definition of primitivity. We obtain also a generalization of the complete Poincaré duality over a certain integral domain to hypersurfaces obtained by patchworkings which are primitive modulo this integral domain. In particular, our corollary is that any tropical hypersurface obtained by patchworking from a triangulation of a simple lattice polytope satisfies complete Poincaré duality over the field of rationals, which is a converse of a theorem of Aksnes.

Poincaré duality for singular tropical hypersurfaces

TL;DR

The paper addresses tropical Poincaré duality for singular hypersurfaces arising from non-primitive patchworking. It introduces an -primitivity framework and develops tropical cosheaves and dual sheaves, together with a cap-product and a fundamental class, to relate tropical cohomology and homology via spectral sequences. The main contributions are a complete Poincaré duality over any integral domain for -primitive triangulations and a partial duality for -primitive triangulations, with a corollary that simple polytopes satisfy complete PD over ; this extends RS 2023 and BMR 2024 to non-primitive patchworkings. The methods provide a robust framework for bounding tropical Betti numbers and clarifying the interaction between tropical and classical dualities in patchworking, with potential applications to real algebraic hypersurfaces constructed via Viro patchworking.

Abstract

We find a partial extension of Poincaré duality theorem of Jell-Rau-Shaw to hypersurfaces obtained by non-primitive Viro's combinatorial patchworking. We define a classification of the triangulations of a lattice polytope by a level of primitivity and we find a partial Poincaré duality for patchworkings depending on the level of primitivity of the triangulation. Our notion of primitivity is defined modulo a certain integral domain, it is weaker than the classical definition of primitivity. We obtain also a generalization of the complete Poincaré duality over a certain integral domain to hypersurfaces obtained by patchworkings which are primitive modulo this integral domain. In particular, our corollary is that any tropical hypersurface obtained by patchworking from a triangulation of a simple lattice polytope satisfies complete Poincaré duality over the field of rationals, which is a converse of a theorem of Aksnes.
Paper Structure (19 sections, 42 theorems, 182 equations, 5 figures)

This paper contains 19 sections, 42 theorems, 182 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Context and Motivations
  3. Poincaré Duality
  4. Acknowledgment
  5. Preliminaries
  6. Homology and Cohomology from a lattice polytope and an integral triangulation
  7. Around $R$-primitivity and $R$-non-singularity
  8. Cap-product
  9. Fundamental Class
  10. Cap-product with the fundamental class
  11. Proof of the partial and complete Poincaré duality
  12. The lemmas and their consequences
  13. Proof of the complete Poincaré duality
  14. Proof of the partial Poincaré duality
  15. Proof of the lemmas
  16. ...and 4 more sections

Key Result

Theorem 1

(Combinatorial Viro's patchworking theorem) If the triangulation $\Gamma$ is convex there exists a real algebraic hypersurface of degree $d$ of $\mathbb{R}P^n$ satisfying the following homeorphism of pair:

Figures (5)

  • Figure 1: A real algebraic curve of $\mathbb{R}P^2$ of degree $3$ obtained by Viro's patchworking.
  • Figure 2: R-non-singularity of polytopes
  • Figure 3: $R$-primitivity of triangles
  • Figure 4: R-primitivity for $3$-simplices
  • Figure 5: Different triangulation of a lattice polygon

Theorems & Definitions (83)

  • Theorem 1
  • Example 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Corollary 1
  • Definition 1
  • Definition 2
  • ...and 73 more