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First Betti number of real Calabi-Yau hypersurfaces: examples

Diego Matessi, Arthur Renaudineau

TL;DR

The work develops a cohomological patchworking framework for real Calabi–Yau hypersurfaces in toric varieties, tying tropical mirror symmetry to the topology of patched real loci via a spectral sequence. It proves a divisor-dependent criterion for the first Betti number and shows how to compute or maximize b1 in concrete settings, notably for the dual cube and quintics. By combining tropical cohomology, mirror maps, and local-to-global parity constructions, the paper provides explicit maximality results in dimension three and outlines obstructions in higher dimensions, with broader implications for the topology of real Calabi–Yau hypersurfaces arising from patchworking. These insights enrich the toolbox for understanding real locus topology through tropical and mirror-symmetric data, with potential applications to explicit Calabi–Yau constructions in toric settings.

Abstract

Continuing the investigation of real Calabi-Yau hypersurfaces in toric varieties obtained by patchworking, we present a new theorem concerning the computation of their first Betti number using mirror symmetry. Although the proof of this result will appear elsewhere, we focus here on its consequences and applications to the topology of real Calabi-Yau hypersurfaces.

First Betti number of real Calabi-Yau hypersurfaces: examples

TL;DR

The work develops a cohomological patchworking framework for real Calabi–Yau hypersurfaces in toric varieties, tying tropical mirror symmetry to the topology of patched real loci via a spectral sequence. It proves a divisor-dependent criterion for the first Betti number and shows how to compute or maximize b1 in concrete settings, notably for the dual cube and quintics. By combining tropical cohomology, mirror maps, and local-to-global parity constructions, the paper provides explicit maximality results in dimension three and outlines obstructions in higher dimensions, with broader implications for the topology of real Calabi–Yau hypersurfaces arising from patchworking. These insights enrich the toolbox for understanding real locus topology through tropical and mirror-symmetric data, with potential applications to explicit Calabi–Yau constructions in toric settings.

Abstract

Continuing the investigation of real Calabi-Yau hypersurfaces in toric varieties obtained by patchworking, we present a new theorem concerning the computation of their first Betti number using mirror symmetry. Although the proof of this result will appear elsewhere, we focus here on its consequences and applications to the topology of real Calabi-Yau hypersurfaces.

Paper Structure

This paper contains 23 sections, 20 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tropical mirror symmetry and connectedness
  3. Cohomological patchworking spectral sequence
  4. First Betti number of real Calabi-Yau patchworking
  5. Applications
  6. Tropical mirror pairs and cohomology
  7. Reflexive Polytopes and Mirror Symmetry
  8. Tropical mirror pairs
  9. Tropical cohomology
  10. Picard group of the mirror
  11. First Betti number from the dual of the cube
  12. Real Calabi-Yau hypersufaces in the dual of the cube
  13. The zero divisor
  14. Non-zero divisors
  15. Maximizing the first Betti number
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $A$ be a commutative ring. Given central primitive 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 (4)

  • Figure 1: The simple vertex: the chosen parity (in red) in any face is the parity of the vertex.
  • Figure 2: The pointed vertex for $E_4$. The chosen parity is drawn in red in any face.
  • Figure 3: The simple edge: the chosen parity (in red) in any face is a parity of the edge.
  • Figure 4: The empty edge: the chosen parity is drawn in red in any face.

Theorems & Definitions (50)

  • Theorem 1.1: MR24
  • Theorem 1.2: MR24
  • Theorem 1.3
  • Theorem 1.4: MR26
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • proof
  • Theorem 1.7
  • Corollary 1.8
  • ...and 40 more