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Mirror symmetry for tropical hypersurfaces and patchworking

Diego Matessi, Arthur Renaudineau

TL;DR

This work develops a robust bridge between tropical mirror symmetry and patchworking by proving canonical isomorphisms between integral tropical homology groups of mirror tropical Calabi–Yau hypersurfaces. It introduces a detailed combinatorial framework that refines posets and defines a mirror cosheaf, yielding a rigorous mirror correspondence for tropical homology that recovers Hodge dualities in convex cases. The authors then connect these tropical constructions to real Calabi–Yau hypersurfaces obtained from primitive patchworking, deriving a precise criterion: under a tropical-homology vanishing hypothesis, the real patchworked hypersurface is connected if and only if the tropical divisor class on the mirror vanishes; otherwise it has at most two components. The results illuminate how real topological features of patchworked varieties are governed by combinatorial and tropical data on the mirror, with potential extensions to higher pages of the patchworking spectral sequence and to non-convex unimodular subdivisions.

Abstract

In the first part of the paper, we prove a mirror symmetry isomorphism between integral tropical homology groups of a pair of mirror tropical Calabi-Yau hypersurfaces. We then apply this isomorphism to prove that a primitive patchworking of a central triangulation of a reflexive polytope gives a connected real Calabi-Yau hypersurface if and only if the corresponding divisor class on the mirror is not zero.

Mirror symmetry for tropical hypersurfaces and patchworking

TL;DR

This work develops a robust bridge between tropical mirror symmetry and patchworking by proving canonical isomorphisms between integral tropical homology groups of mirror tropical Calabi–Yau hypersurfaces. It introduces a detailed combinatorial framework that refines posets and defines a mirror cosheaf, yielding a rigorous mirror correspondence for tropical homology that recovers Hodge dualities in convex cases. The authors then connect these tropical constructions to real Calabi–Yau hypersurfaces obtained from primitive patchworking, deriving a precise criterion: under a tropical-homology vanishing hypothesis, the real patchworked hypersurface is connected if and only if the tropical divisor class on the mirror vanishes; otherwise it has at most two components. The results illuminate how real topological features of patchworked varieties are governed by combinatorial and tropical data on the mirror, with potential extensions to higher pages of the patchworking spectral sequence and to non-convex unimodular subdivisions.

Abstract

In the first part of the paper, we prove a mirror symmetry isomorphism between integral tropical homology groups of a pair of mirror tropical Calabi-Yau hypersurfaces. We then apply this isomorphism to prove that a primitive patchworking of a central triangulation of a reflexive polytope gives a connected real Calabi-Yau hypersurface if and only if the corresponding divisor class on the mirror is not zero.
Paper Structure (20 sections, 19 theorems, 167 equations, 8 figures)

This paper contains 20 sections, 19 theorems, 167 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Tropical mirror symmetry
  3. Patchworking and mirror symmetry
  4. Related works
  5. Outlook
  6. Toric, tropical and combinatorial geometry
  7. Polytopes and toric varieties
  8. Reflexive polytopes and mirror symmetry
  9. Combinatorial mirror pairs
  10. Poset homology
  11. Tropical homology
  12. Combinatorial mirror symmetry
  13. A refinement of the poset $\mathcal{P}(T^\circ,T)$
  14. The mirror cosheaf
  15. Proof of Theorem \ref{['thm:main1']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $A$ be a commutative ring. Given central unimodular subdivisions of both $\Delta$ and $\Delta^{\circ}$ with associated mirror tropical hypersurfaces $X_{\text{trop}}$ and $X^{\circ}_{\text{trop}}$, we have canonical isomorphisms

Figures (8)

  • Figure 1: The subdivision of $\mathbb{T}\Sigma_{T^\circ}$ induced by a tropical curve dual to $T$.
  • Figure 2: The subdivision $\mathcal{P}(T,T^\circ)$
  • Figure 3: The subdivision $\mathcal{J}(T,T^\circ)$
  • Figure 4: The subdivision $\mathcal{J}(T^\circ, T)$
  • Figure 5: The signs corresponding to $D = D_7 + D_8$ and to $D=D_8$ and the corresponding real cubics.
  • ...and 3 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • ...and 43 more