Lagrangian Torus Fibration Concepts

• Lagrangian torus fibrations are maps from 2n-dimensional symplectic manifolds to n-dimensional bases where regular fibers are Lagrangian n-tori.
• They utilize action–angle coordinates that induce an integral affine structure on the base while singular fibers signal critical geometric transitions.
• These fibrations underpin key frameworks in mirror symmetry, integrable systems, and Floer theory, offering practical tools for manifold classification.

A Lagrangian torus fibration is a surjective map from a $2n$-dimensional symplectic manifold $(X^{2n},\omega)$ (or complex, Kähler, or algebraic geometric analogues) to an $n$-dimensional base $B$, such that the regular fibers are Lagrangian $n$-tori—i.e., each fiber is a real submanifold of maximal dimension for which the symplectic form restricts to zero. These fibrations appear in symplectic geometry, algebraic geometry, and mathematical physics, underpinning major frameworks such as the Strominger–Yau–Zaslow (SYZ) conjecture in mirror symmetry, the theory of integrable systems, the structure of hyperkähler and Calabi–Yau manifolds, toric and almost-toric symplectic topology, and the construction of Floer-theoretic invariants. Their bases inherit canonical integral affine structures, and their singular fibers encode rich geometry and symplectic topology.

1. Definitions and Basic Structure

A Lagrangian torus fibration on a $2n$-dimensional symplectic manifold $(X,\omega)$ is a continuous, proper map $F: X \to B$ such that:

• Over the regular locus $B^{\mathrm{reg}}\subset B$, $F$ is a smooth submersion with connected fibers $(X^{2n},\omega)$0 that are Lagrangian $(X^{2n},\omega)$1-tori: $(X^{2n},\omega)$2, $(X^{2n},\omega)$3.
• The base $(X^{2n},\omega)$4 is typically a stratified topological space whose top stratum is an $(X^{2n},\omega)$5-manifold; lower strata correspond to singular or degenerate fibers.

Integral affine structures naturally arise on $(X^{2n},\omega)$6, with transition maps in $(X^{2n},\omega)$7, induced by the geometry of action-angle coordinates in the total space (Evans, 2021).

In complex or Kähler contexts, one often considers holomorphic Lagrangian torus fibrations or special Lagrangian torus fibrations, as in Calabi–Yau or hyperkähler manifolds (Hwang et al., 2012, Morrison et al., 2015).

2. Action–Angle Coordinates and the Base Affine Structure

In a neighborhood of a regular fiber, the classical action-angle theorem provides local Darboux coordinates: $(X^{2n},\omega)$8 such that $(X^{2n},\omega)$9, where $n$0 are commuting Hamiltonians generating the torus action (Evans, 2021). The period lattice $n$1 defines an integral affine structure on $n$2. Singular loci correspond to monodromy defects in this structure, and are responsible for the global geometry of the fibration.

In complex or algebraic cases, such as projective symplectic or hyperkähler manifolds, holomorphic analogues of action-angle coordinates exist (Hwang et al., 2012): in local charts, one has a symplectomorphism $n$3 with

$n$4

and the projection to $n$5 yields the local Lagrangian fibration.

3. Singular Fibers, Discriminant Locus, and Monodromy

The discriminant locus $n$6 is the set of critical values where the fibers of $n$7 become singular. Its structure and codimension depend on the geometric context:

• In almost-toric and toric settings, $n$8 is typically real codimension-one or pointwise, and monodromy around $n$9 is encoded by matrices in $B$0 (Evans, 2021, Prince, 2018).
• In the SYZ scenario for Calabi–Yau threefolds, the discriminant is conjectured to be a trivalent graph of codimension two, with distinct monodromy and local models at positive and negative vertices (Morrison et al., 2015).

Local models for singularities include:

• Focus–focus type: a pinched torus singularity with elliptic monodromy, prevalent in 2D and 4D almost-toric systems (Evans, 2021, Prince, 2018).
• Triple-node and negative-node: in Calabi–Yau SYZ fibrations, with Euler characteristics +1 and –1 respectively, determined by local monodromy matrices (Morrison et al., 2015).

Monodromy encodes transitions in the integral affine structure and is fundamentally related to wall-crossing and mirror symmetry phenomena.

4. Existence, Classification, and Construction Methods

Lagrangian torus fibrations arise in a spectrum of geometric settings:

• Toric manifolds: The moment map for a $B$1-action is a Lagrangian torus fibration, with base a Delzant polytope (Fernandes et al., 2024). Singularities occur over the boundary.
• Hyperkähler and projective symplectic manifolds: The presence of a single Lagrangian torus yields (almost holomorphic) global Lagrangian torus fibrations under strong hypotheses (Hwang et al., 2012, Greb et al., 2011, Amerik et al., 2013, Greb et al., 2011). In the non-projective hyperkähler case, the fibration is holomorphic (Greb et al., 2011). In the projective case, it can be birationally modified to a holomorphic fibration after flops (Greb et al., 2011, Greb et al., 2011).
• Degenerations and cluster algebras: Toric degenerations via Newton–Okounkov bodies and tropical cluster mutations yield families of Lagrangian torus fibrations, often with infinitely many pairwise non-symplectomorphic monotone fibers, especially on flag varieties (Cho et al., 2023).
• Symplectic topology and almost-toric/affine models: Explicit local models with prescribed discriminant loci provide local building blocks for global constructions, e.g., negative vertices, pair-of-pants models, and neighborhood models via the cone over dual complexes (Evans et al., 2019, Bobadilla et al., 2023, Achig-Andrango, 2022).
• Quantum and dynamical settings: Invariant Lagrangian tori in (quasi-periodic) Hamiltonian systems are produced by parameterization-KAM algorithms, with super-exponential convergence and explicit small-divisor analysis (Calleja et al., 1 Apr 2025).
• Pseudotoric and non-toric moduli: Generalizations to pseudotoric structures produce special Lagrangian torus fibrations, key for SYZ mirrors of flag varieties and quadrics (Chan et al., 2018).

5. Symplectic, Complex, and Floer-Theoretic Invariants

Lagrangian torus fibrations form the geometric backbone for key invariants:

• Floer cohomology and disk potential: For torus fibers, Floer cohomology is computed via counts of holomorphic disks; the disk potential determines non-displaceability and monotonicity, and governs the Landau–Ginzburg mirror superpotential (Xiao, 2023, Achig-Andrango, 2022, Cho et al., 2023).
• SYZ mirror symmetry: The SYZ framework posits that a Calabi–Yau manifold near a large complex structure limit admits a special Lagrangian torus fibration whose dual fibration (over the same affine base, with dual tori) should produce the mirror (Morrison et al., 2015). Wall crossing and singularities in the fibration are responsible for nontrivial corrections in the mirror (Chan et al., 2018).
• Cohomological consequences: In Lagrangian fibrations of hyperkähler manifolds, genericity of sections implies density of torsion points and constrains the Chow ring, supporting the splitting conjecture for algebraic cycles (Voisin, 2016).
• Birational invariants and Torelli-type results: On compact hyperkähler manifolds, dual Lagrangian torus fibrations are constructed by passing to quotient orbifolds, with period morphisms and Beauville–Bogomolov forms preserved, leading to "mirror" orbifolds (Kim, 2021).

6. Examples and Geometric Constructions

• Projective space $B$2: The standard moment map produces a Lagrangian torus fibration over the simplex; fibers degenerate to lower-dimensional tori on the boundary (Fernandes et al., 2024, Prince, 2018).
• Fano threefolds: For every smooth Fano threefold with $B$3, a Lagrangian torus fibration can be constructed from suitable degeneration data and affine manifolds with singularities, matching all topological and numerical invariants (Prince, 2018).
• Conic bundles and SYZ transforms: In conic bundles, explicit Lagrangian sections admit SYZ transforms to holomorphic line bundles on the mirror, matching the wrapped Floer cohomology ring with the coordinate ring of the mirror (Chan et al., 2016).
• Complete intersection Calabi–Yau's in toric varieties: Geometric constructions supply (conjectural) special Lagrangian fibrations whose bases and discriminant loci match combinatorial predictions, including monodromy and Euler characteristic formulas (Morrison et al., 2015).

7. Open Problems and Future Directions

• Classification: A central problem is the global classification of integral affine structures admitting singularities compatible with Lagrangian torus fibrations, especially in higher dimensions (Evans, 2021).
• SYZ conjecture: The existence and global structure of special Lagrangian torus fibrations on generic Calabi–Yau manifolds remain open, especially with codimension-two discriminant loci as predicted by mirror symmetry (Morrison et al., 2015, Bobadilla et al., 2023).
• Wall-crossing and enumerative meaning: Understanding the enumerative implications of wall-crossing, disk instantons, and discrimination loci for mirror symmetry and Floer-theoretic invariants is ongoing.
• Birational geometry and moduli: The interplay between birational modifications (e.g., flops), period maps, and Lagrangian fibrations in hyperkähler geometry continues to play an essential role in the structure of moduli spaces (Kim, 2021, Greb et al., 2011, Greb et al., 2011).
• Floer-theoretic uniqueness: The emergence of infinitely many monotone Lagrangian tori on complex flag manifolds and the distinction of their Floer-theoretic types is an area of active research, especially in the context of cluster algebras and Newton–Okounkov bodies (Cho et al., 2023).

Lagrangian torus fibrations thus constitute a central unifying structure in symplectic, complex, and mirror symmetry geometry, intertwining topology, analysis, and algebraic invariants across diverse mathematical settings.