Fibrations by Lagrangian tori for maximal Calabi-Yau degenerations and beyond

Technical Summary: Fibrations by Lagrangian Tori for Maximal Calabi–Yau Degenerations and Beyond

Problem Statement
The paper addresses the Strominger–Yau–Zaslow (SYZ) conjecture, which posits that mirror symmetry for Calabi–Yau manifolds arises from dual fibrations by special Lagrangian tori. Specifically, the authors focus on the "large complex structure limit," which corresponds to a maximal Calabi–Yau degeneration. In this setting, Kontsevich and Soibelman formulated precise conjectures regarding the existence of special Lagrangian torus fibrations on the fibers of such degenerations and their convergence to a metric space (the essential skeleton) satisfying a real Monge–Ampère equation. While partial results exist for specific families (e.g., Fermat families or hypersurfaces), a general construction for arbitrary maximal Calabi–Yau degenerations has remained elusive. The paper aims to provide a general technique to construct such fibrations and prove their asymptotic properties, specifically that they become special Lagrangian and volume-minimizing as the degeneration parameter approaches zero.

Methodology
The authors introduce a novel geometric framework centered on the A'Campo space, a "hybrid" space that compactifies the degeneration by combining the real oriented blowup of the total space with the dual complex of the central fiber. The methodology proceeds through the following steps:

1. Construction of the A'Campo Space ($A$):
   The authors utilize the A'Campo space $A$ associated with a holomorphic map $f: X \to \mathbb{C}$ with a simple normal crossing (snc) central fiber. This space is constructed to resolve the indeterminacy of the degeneration at "radius zero" (the central fiber). It is equipped with a smooth structure (specifically a $C^\infty$ atlas) and a stratification lifting the strata of the central fiber.

2. Fiberwise Kähler Forms:
   A critical technical innovation is the construction of a family of exact modifications of a given Kähler form $\omega_X$ on $X$. Defined on the A'Campo space, these forms are denoted $\omega^\varepsilon_q$ for parameters $q \in [0, 1]$ and a small $\varepsilon > 0$.

   • The form is constructed as $\omega^\varepsilon_q = \pi^*\omega_X + \varepsilon \cdot (q \cdot \omega^\sharp + (1-q) \cdot \omega^\flat)$.
   • $\omega^\sharp$ is designed to be compatible with the complex structure (Kähler) at positive radius.
   • $\omega^\flat$ is inspired by "semi-flat" models and is designed to degenerate to a Euclidean metric on the dual complex at radius zero.
   • The authors prove that for sufficiently small $\varepsilon$, these forms are fiberwise Kähler at positive radius and fiberwise symplectic at radius zero (specifically on the "generic" region of the skeleton).

3. Lagrangian Fibration at Radius Zero:
   The construction of the fibration is first performed on the boundary of the A'Campo space (radius zero).

   • The base of the fibration is an expanded skeleton ($ES$), a smooth manifold (or "ivy-like" manifold in the general case) constructed by expanding the faces of the essential skeleton $\Delta_S$.
   • On the "radius-zero" fibers, the authors explicitly construct a Lagrangian torus fibration. For maximal faces of the skeleton, the fibers are products of tori and simplices. For submaximal faces, they utilize Morse functions on the strata to define the fibration.
   • In the specific case of maximal Calabi–Yau degenerations, the authors select a model where the strata are isomorphic to $\mathbb{C}^*$, allowing for a canonical choice of Morse functions (absolute value of a coordinate) that eliminates singularities in the fibration.

4. Transport to Positive Radius:
   Using the symplectic connection associated with the family of forms $\omega^\varepsilon_q$, the authors transport the Lagrangian tori from the radius-zero fiber to the nearby fibers $X_z$ of the original degeneration. This transport preserves the Lagrangian property.

5. Asymptotic Analysis:
   The authors analyze the behavior of these transported fibrations as the parameter $z \to 0$ along specific paths in the $(z, q)$-plane. They study the Gromov–Hausdorff limit of the rescaled metrics and the convergence of the volume forms.

Key Contributions and Results

• General Construction (Theorem 6.14): The paper establishes a general technique to construct Lagrangian torus fibrations for any Kähler degeneration with an snc central fiber, provided a subset of components is chosen to define an "essential skeleton."
• Maximal Calabi–Yau Degenerations (Theorem 1.1): For any maximal Calabi–Yau degeneration admitting a semi-stable model, the authors prove the existence of a family of Kähler forms $\omega_q$ and a smooth family of maps $l_{z,q}$ defining a Lagrangian torus fibration on a generic region of the fiber.
  • The base of this fibration is the expanded skeleton $ES$.
  • The Gromov–Hausdorff limit of the rescaled fibers is the essential skeleton $\Delta_S$ equipped with a Euclidean metric.
• Asymptotic Specialness (Theorem 1.2): The paper proves that for "tame" paths approaching the degeneration, the Lagrangian tori in the generic region become asymptotically special.
  • Volume Filling: The volume of the generic region (fibred by tori) converges to the total volume of the fiber.
  • Asymptotic Ricci-Flatness: The constructed forms $\omega_q$ approximate the Ricci-flat metric in the sense that the ratio of their volume forms to the holomorphic volume form converges to a constant.
  • Special Condition: The imaginary part of the holomorphic volume form restricted to the tori vanishes asymptotically (up to a phase), implying the tori are asymptotically special Lagrangian.
  • Maslov Class: The fibers have trivial Maslov class.
• Resolution of Singularities in the Base: The authors construct a base $ES$ that is a smooth manifold (in the Calabi–Yau case), avoiding the singularities often associated with the discriminant locus in other constructions. The discriminant is pushed to the "non-generic" region, which has vanishing volume in the limit.

Significance and Claims
The authors claim that their work provides an asymptotic version of the Kontsevich–Soibelman prediction for the general case of maximal Calabi–Yau degenerations.

• Existence: They prove the existence of Lagrangian torus fibrations with the expected asymptotic properties (specialness, volume minimization, Gromov–Hausdorff convergence to the essential skeleton) without restricting to specific families like Fermat hypersurfaces.
• Archimedean Analogue: The construction is presented as an Archimedean analogue of the non-Archimedean affinoid torus fibrations constructed by Nicaise, Xu, and Yu. The base $ES$ is shown to be homeomorphic to the interior of the non-Archimedean base.
• Technical Novelty: The paper emphasizes the novelty of the "hybrid" A'Campo space construction combined with the specific deformation of Kähler forms ($\omega^\varepsilon_q$) that allows for a smooth transition between the complex geometry of the fiber and the tropical geometry of the skeleton.
• Modesty: The authors explicitly state that their results are "asymptotic." They do not claim to construct special Lagrangian fibrations for every fiber at a fixed positive radius, nor do they claim the constructed forms are exactly the Ricci-flat metrics. Instead, they show that the constructed forms and fibrations approximate the conjectural SYZ behavior as the degeneration parameter vanishes. They note that the volume of the "non-generic" region does not vanish with respect to their constructed forms, distinguishing their approach from the exact Ricci-flat metric where the volume concentrates entirely on the generic region.

In summary, the paper offers a robust, general framework for realizing the SYZ picture in the context of maximal Calabi–Yau degenerations, bridging the gap between the complex geometry of the fibers and the tropical geometry of the dual complex via a carefully engineered symplectic deformation.