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Hybrid toric varieties and the non-archimedean SYZ fibration on Calabi-Yau hypersurfaces

Léonard Pille-Schneider

TL;DR

The paper develops a non-archimedean SYZ framework for maximally degenerate Calabi–Yau hypersurfaces by combining Yamamoto’s tropical contraction with a robust theory of hybrid toric metrics. It builds a bridge between NA Monge–Ampère equations on the analytification and real Monge–Ampère equations on the essential skeleton, via affinoid torus fibrations over Sk(X) minus a discriminant. In the Fermat symmetric case, it proves a strong comparison property and constructs a canonical NA Calabi–Yau metric solving MA(φ)=μ0, with explicit toric potentials controlling local behavior. The work also shows that the archimedean SYZ fibrations on nearby fibers converge, in the hybrid topology, to the non-archimedean retraction, providing a concrete realization of the SYZ picture in the degenerate CY hypersurface setting. Collectively, these results highlight how tropical geometry, hybrid pluripotential theory, and toric techniques yield a coherent NA SYZ fibration, with concrete convergence statements and explicit models for both the base and the fibers.

Abstract

Using a construction by Yamamoto of tropical contractions, we construct a non-archimedean SYZ fibration on the Berkovich analytification of a class of maximally degenerate hypersurfaces in projective space. We furthermore prove that under a discrete symmetry assumption, the potential for the non-archimedean Calabi-Yau metric is constant along the fibers of the retraction. The proof uses the work of Li on the Fermat degeneration of hypersurfaces, and an explicit description of toric plurisubharmonic metrics on the hybrid space associated to a complex toric variety.

Hybrid toric varieties and the non-archimedean SYZ fibration on Calabi-Yau hypersurfaces

TL;DR

The paper develops a non-archimedean SYZ framework for maximally degenerate Calabi–Yau hypersurfaces by combining Yamamoto’s tropical contraction with a robust theory of hybrid toric metrics. It builds a bridge between NA Monge–Ampère equations on the analytification and real Monge–Ampère equations on the essential skeleton, via affinoid torus fibrations over Sk(X) minus a discriminant. In the Fermat symmetric case, it proves a strong comparison property and constructs a canonical NA Calabi–Yau metric solving MA(φ)=μ0, with explicit toric potentials controlling local behavior. The work also shows that the archimedean SYZ fibrations on nearby fibers converge, in the hybrid topology, to the non-archimedean retraction, providing a concrete realization of the SYZ picture in the degenerate CY hypersurface setting. Collectively, these results highlight how tropical geometry, hybrid pluripotential theory, and toric techniques yield a coherent NA SYZ fibration, with concrete convergence statements and explicit models for both the base and the fibers.

Abstract

Using a construction by Yamamoto of tropical contractions, we construct a non-archimedean SYZ fibration on the Berkovich analytification of a class of maximally degenerate hypersurfaces in projective space. We furthermore prove that under a discrete symmetry assumption, the potential for the non-archimedean Calabi-Yau metric is constant along the fibers of the retraction. The proof uses the work of Li on the Fermat degeneration of hypersurfaces, and an explicit description of toric plurisubharmonic metrics on the hybrid space associated to a complex toric variety.
Paper Structure (24 sections, 50 theorems, 220 equations, 3 figures)

This paper contains 24 sections, 50 theorems, 220 equations, 3 figures.

Table of Contents

  1. Non-archimedean geometry
  2. Models and Berkovich skeleta
  3. Hybrid spaces
  4. Pluripotential theory
  5. Hybrid toric metrics
  6. Convex geometry
  7. Toric varieties
  8. Analytification of toric varieties
  9. Fiberwise-toric metrics
  10. Hybrid toric metrics
  11. Hybrid family of Haar measures
  12. The non-archimedean SYZ picture
  13. The classical SYZ picture
  14. The essential skeleton
  15. Affinoid torus fibrations
  16. ...and 9 more sections

Key Result

Proposition 1.3

MN Let $Y \subset \mathscr{X}_0$ be a stratum which is a connected component of $D_J$ for a $J \subset I$, with generic point $\eta$, and furthermore fix a local equation $z_j \in \mathcal{O}_{\mathscr{X},\eta}$ for $D_j$, for any $j \in J$. For any $w \in \tau_Y = \{ w \in \mathbb R^{|J|}_{\geqslan where $\langle \; , \; \rangle$ is the usual scalar product on $\mathbb R^{|J|}$. The above valuati

Figures (3)

  • Figure 1: The tropicalization of a maximally degenerate K3 surface
  • Figure 2: The tropical contraction in dimension 2
  • Figure :

Theorems & Definitions (112)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Lemma 1.9
  • Proposition 1.10
  • ...and 102 more