Almost Toric Manifolds in Four Dimensions

• Almost toric manifolds are four-dimensional symplectic spaces equipped with singular Lagrangian fibrations that permit only elliptic and focus–focus singularities.
• They integrate techniques from symplectic topology, tropical geometry, and mirror symmetry, revealing rich monodromy phenomena and combinatorial structures.
• Their study enhances our understanding of geometric quantization, integrable systems, and topological classification, with applications ranging from K3 surfaces to del Pezzo surfaces.

An almost toric manifold, in dimension four, is a compact symplectic manifold equipped with a Lagrangian fibration whose only singular fibres are of elliptic and focus–focus type; that is, it generalizes the notion of a toric symplectic manifold by permitting certain non-toric singular fibres but prohibits hyperbolic singularities. The study of these objects intersects symplectic topology, toric and semitoric geometry, tropical and mirror symmetry, and topological classification of manifolds with torus actions. Almost toric four-manifolds encode rich monodromy phenomena, allow for combinatorial and tropical techniques, and underlie modern approaches to geometric quantization and open Gromov–Witten theory.

1. Definitions and Structural Properties

A toric $2m$-dimensional symplectic manifold $(M,\omega)$ admits an effective Hamiltonian $T^m$-action, and its geometry is classified by a Delzant polytope in $\mathbb{R}^m$. In contrast, an almost toric four-manifold $(M^4,\omega)$ is a symplectic four-manifold endowed with a singular Lagrangian fibration $F: M^4 \to B$, where the only allowed singularities are elliptic (corresponding locally to toric fixed points) and focus–focus (giving nodal torus fibers) (Miranda et al., 2017, Venugopalan et al., 3 Apr 2026).

On the set $B^0 = B \setminus \{ \text{focus–focus values} \}$, the map $F$ restricts to an ordinary $T^2$-moment map, and $B^0$ inherits an integral affine structure. At each focus–focus value $(M,\omega)$0, the affine monodromy is given by

$(M,\omega)$1

for some $(M,\omega)$2, and the base $(M,\omega)$3 acquires the characteristic structure of rays ("branch cuts") that encode this monodromy. This permits only a global $(M,\omega)$4-action, in contrast to the $(M,\omega)$5-action on toric manifolds (Miranda et al., 2017).

2. Integrable Systems, Monodromy, and Base Diagrams

An almost toric fibration can be constructed from a semitoric integrable system, defined on $(M,\omega)$6 by two Poisson-commuting functions $(M,\omega)$7 and $(M,\omega)$8, where $(M,\omega)$9 generates a global Hamiltonian $T^m$0-action, and all singularities are of elliptic–elliptic or focus–focus type (with no hyperbolic or degenerate blocks) (Miranda et al., 2017).

Near a focus–focus singularity, in Eliasson coordinates $T^m$1, the system has local first integrals

$T^m$2

The corresponding fiber is a pinched torus, and associated monodromy in the affine structure is realized by the shear matrix above. Multiple focus–focus points introduce powers of this monodromy, yielding more elaborate branch structures in $T^m$3 and reflecting the topological and symplectic complexity of the total space (Miranda et al., 2017, Venugopalan et al., 3 Apr 2026).

Base diagrams for almost toric manifolds extend the classical Delzant polygons by incorporating branch cuts at each focus–focus value, along which the affine structure is singular. These diagrams, together with the combinatorial data of nodal trades and monodromy directions, determine the global topology and Hamiltonian geometry.

3. Geometric Quantization and Sheaf Cohomology

Geometric quantization of toric manifolds is governed by the counting of integral interior points in the associated Delzant polytope, with real polarization given by torus orbits. Here, the quantization space

$T^m$4

where $T^m$5 is the sheaf of flat sections of a prequantum line bundle (Miranda et al., 2017).

In the almost toric case, the real polarization is still given by the invariant Lagrangian foliation, but singularities in the foliation (especially focus–focus points) alter the cohomology of the sheaf $T^m$6. The global quantization space decomposes as

$T^m$7

where $T^m$8 are regular Bohr–Sommerfeld fibers; $T^m$9 are Bohr–Sommerfeld focus–focus fibers with $\mathbb{R}^m$0 nodes. Each contributing focus–focus fiber produces an infinite-dimensional component, a phenomenon absent in the toric case and responsible for the divergence between real and Kähler quantization (as in the quantization of K3 surfaces or the spin–spin system) (Miranda et al., 2017). This infinite degeneracy at singular Bohr–Sommerfeld fibers motivates refinements of quantization in singular settings.

4. Tropical Geometry and Disk Potentials

Almost toric four-manifolds furnish a natural stage for tropical computations of open Gromov–Witten disk invariants. Let $\mathbb{R}^m$1 be a Lagrangian torus fiber equipped with a rank-1 local system $\mathbb{R}^m$2. For monotone symplectic forms, the disk (Fukaya) potential is

$\mathbb{R}^m$3

where $\mathbb{R}^m$4 is the Maslov index, $\mathbb{R}^m$5 is the (signed, weighted) count of rigid holomorphic disks in class $\mathbb{R}^m$6, and $\mathbb{R}^m$7 records boundary holonomy (Venugopalan et al., 3 Apr 2026).

A tropical formula expresses $\mathbb{R}^m$8 as a sum over rigid Maslov-2 tropical graphs in the dual affine complex $\mathbb{R}^m$9:

$(M^4,\omega)$0

where $(M^4,\omega)$1 denotes local vertex multiplicities, explicitly computable via:

• Mikhalkin's determinant for trivalent (pants) vertices,
• Cho–Oh's results for toric cylinders/disks,
• Bryan–Pandharipande formula $(M^4,\omega)$2 for multiple covers at focus–focus nodes.

Wall-crossing—motion of focus–focus nodes across the fiber—induces mutations of the potential, aligning with the algebraic mutations of Pascaleff–Tonkonog and the structure of mirror symmetry via the Gross–Siebert program (Venugopalan et al., 3 Apr 2026).

Example Disk Potentials for Monotone del Pezzo Surfaces

Manifold	Potential $(M^4,\omega)$3
$(M^4,\omega)$4	$(M^4,\omega)$5
$(M^4,\omega)$6	$(M^4,\omega)$7
$(M^4,\omega)$8 ($(M^4,\omega)$9)	See explicit forms in (Venugopalan et al., 3 Apr 2026) Table 1

These tropical disk potentials coincide with the "maximally mutable" Laurent polynomials of Akhtar–Coates–Corti–et al., and their critical values match quantum cup-product eigenvalues for $F: M^4 \to B$0 (Venugopalan et al., 3 Apr 2026).

5. Combinatorics of Four-Dimensional Almost Complex Torus Manifolds

A 4-dimensional almost complex torus manifold is a compact connected 4-manifold with an almost complex structure and an effective $F: M^4 \to B$1-action with isolated fixed points (Jang, 2023). At each fixed point the tangent representation decomposes as $F: M^4 \to B$2 with integer weight pairs forming a $F: M^4 \to B$3-basis of the lattice.

These manifolds admit two equivalent combinatorial descriptions:

• Families of multi-fans: Cyclically ordered collections of nonzero vectors in $F: M^4 \to B$4 such that each adjacent pair forms a basis, with specific recurrence relations.
• Labeled 2-graphs: Directed graphs with vertices labeled by fixed points and edges by weights, subject to admissibility conditions on labelings (pairwise bases and recurrence) (Jang, 2023).

A key result is that the existence of such a manifold is equivalent to the existence of an admissible multi-fan family or labeled 2-graph describing its fixed point data (Theorem 1.9 in (Jang, 2023)). Every such manifold can be reduced, via equivariant blow-ups and blow-downs (described combinatorially as operations on fans and graphs), to a minimal model with weight data given by the standard four unit vectors; the unique minimal complex torus manifold is $F: M^4 \to B$5.

6. Examples and Applications

• K3 Surfaces: Constructed by symplectically summing toric blow-ups of $F: M^4 \to B$6, acquiring up to 24 focus–focus fibers. Real geometric quantization yields

$F: M^4 \to B$7

with $F: M^4 \to B$8 counting regular Bohr–Sommerfeld fibers (Miranda et al., 2017).

• Spin–Spin Systems: Generation of a focus–focus fiber via a nodal trade in $F: M^4 \to B$9; quantization

$B^0 = B \setminus \{ \text{focus–focus values} \}$0

(Miranda et al., 2017).

• Spherical Pendulum and Spin–Oscillator: Semitoric systems with a single focus–focus (possibly Bohr–Sommerfeld) fiber, producing infinite-dimensional quantization in relevant cases.
• del Pezzo Surfaces: Explicit computation of tropical disk potentials for Lagrangian tori in $B^0 = B \setminus \{ \text{focus–focus values} \}$1, matching Laurent polynomial mirror symmetry predictions (Venugopalan et al., 3 Apr 2026).

7. Connections to Mirror Symmetry and Topological Classification

Almost toric manifolds constitute a unifying context for several major themes in symplectic and algebraic geometry:

• Mirror Symmetry: Their base affine structure, focus–focus monodromy, and disk potential computations directly realize mirror Landau–Ginzburg models via tropical and combinatorial data—extending from toric to almost toric settings and aligning with the Gross–Siebert program (Venugopalan et al., 3 Apr 2026).
• Topological Classification: The combinatorial models of multi-fans and labeled graphs, together with the operations of blow-up and blow-down, give a purely equivariant method to classify all 4-dimensional almost complex torus manifolds, generalizing the classical result for toric surfaces (Jang, 2023). In the integrable case, the fan uniquely determines the manifold up to equivariant biholomorphism; for almost complex cases, the multi-fan family encodes all topological data up to modification.

A plausible implication is that almost toric techniques will remain central in ongoing advances in singular symplectic topology, quantum invariants, and explicit constructions of mirrors for non-toric Fano and Calabi–Yau four-manifolds.