Constructing local models for Lagrangian torus fibrations

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on the link of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is the dual complex of the divisor and the discriminant locus is the union of codimension 2 strata in the dual complex. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.


Introduction
The Strominger-Yau-Zaslow conjecture [42,18,28] asserts that a Calabi-Yau manifold admitting a degeneration to a large complex structure limit also admits a Lagrangian torus fibration. Indeed, the fibres are special Lagrangian and the Calabi-Yau metric should undergo Gromov-Hausdorff collapse along the fibration as one approaches the limit. If we take a semistable reduction of the degeneration, so that the singular fibre is a reduced simple normal crossing variety, the base of the SYZ torus fibration should be the dual complex 1 of the singular fibre [28,Section 3.3].
In this paper, we focus on the problem of finding Lagrangian torus fibrations (not necessarily special) on Weinstein manifolds W arising as X \ Y where X is a complex projective variety and Y is a simple normal crossing divisor. These should be thought of as local models for Lagrangian torus fibrations on compact Calabi-Yaus or other varieties (for example surfaces of general type). The idea is very simple: construct a Lagrangian torus fibration on the contact "link at infinity" of W , extend it to the interior of W using Liouville flow, and add a central fibre corresponding to the Lagrangian skeleton of W . If B is the base of the Lagrangian torus fibration on the link then we are careful to identify B with the dual complex of Y . The base of the Lagrangian torus fibration on W is then Cone(B).
The singularities of the Lagrangian torus fibrations constructed in this way are usually too severe to be of much use. However, in some cases, this construction yields Lagrangian fibrations with mild singularities which are difficult to construct any other way. We demonstrate this with two examples: • Mikhalkin [31] introduced the tropicalisation map which, amongst other things, gives a topological torus fibration from a variety to its tropicalisation. In particular, one obtains a map from the 4-dimensional pair-of-pants to the cell complex which is the cone on the 1-skeleton of a tetrahedron. In Section 7.2, we give a version of this map with Lagrangian fibres. We anticipate that this will be useful for constructing Lagrangian torus fibrations on surfaces of general type. Further, we describe a Lagrangian torus fibration for the n-dimensional pair-of-pants in Section 8.3.
• We give a Lagrangian torus fibration on the affine variety W = {(x, y, u 1 , u 2 ) ∈ C 2 × (C * ) 2 : xy = u 1 + u 2 + 1} (1) such that the base is a 3-ball and the discriminant locus (set of singular fibres) is a Y-graph. The existence of such a Lagrangian torus fibration was conjectured by Gross [17], who gave a topological torus fibration on this space with the same discriminant locus. The singular fibre over the vertex of the graph appears to be the same as the one Gross conjectured (a space with Euler characteristic −1 which we will call the Gross fibre).
This second example is of particular significance. In the early days of mirror symmetry, there was an expectation that the discriminant locus for a Lagrangian torus fibration on a Calabi-Yau 3-fold should be a trivalent graph. The only fibres with nontrivial Euler characteristic should live over the vertices of this graph, so there should be two local models where the discriminant is a Y-graph: one where the fibre over the vertex has Euler characteristic 1, and one where it has Euler characteristic −1 (as there are Calabi-Yau 3-folds with both positive and negative Euler characteristic). The local model with Euler characteristic 1 is easy to construct (see [16,Example 1.2]).
An explicit Lagrangian fibration on the variety given by Eq. (1) was found by Castaño-Bernard and Matessi [5,6], but the discriminant locus was a codimension 1 thickening of the Y-graph. Indeed, Joyce [22] had earlier argued that the property of having codimension 1 discriminant locus should be generic amongst singularities of special Lagrangian torus fibrations. For this reason, attention in recent years has focused on understanding the case where the discriminant locus has codimension 1. For example, the powerful method introduced by W.-D.
In light of our construction, it seems reasonable to expect Lagrangian torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus, as originally expected.
An alternative approach to the SYZ conjecture, explained in [33] and dating back to the seminal works [28, §3.3] and [29], aims to replace the quest for special Lagrangian fibrations with the more accessible construction of nonarchimedean avatars. According to [29,Conjecture 1], the smooth part of the Gromov-Hausdorff limit of maximally degenerating families of Calabi-Yau varieties should carry an integral affine structure, of which it should be possible to give a purely algebraic description; see [29,Conjecture 3] and [33,Theorem 6.1]. In Section 8, we show how the integral affine structure constructed in [33,Theorem 6.1] arises indeed from a Lagrangian torus fibration on the total space of the degenerating family, and we work out the example of a Dwork pencil of quartic surfaces.

Outline
In Sections 2.1 to 2.3, we give some basic definitions and setup; in particular, we define precisely what we mean by a Lagrangian torus fibration and a coisotropic fibration. In Section 3, we discuss the dual complex of a simple normal crossing divisor and prove some results about the topology of the dual complex under the assumption that the complement of the simple normal crossing divisor is affine.
Next, we explain the construction of a Lagrangian torus fibration on X \ Y in several steps: • In Section 4, we construct a coisotropic fibration on the contact boundary of X \ Y such that the base of the fibration is the dual complex of the compactifying divisor Y .
• In Section 5, we explain how to refine the coisotropic fibration from Section 4 to a Lagrangian torus fibration given the additional data of a suitable (dovetailed ) Lagrangian torus fibration on the simple normal crossing divisor. We give conditions under which the base of this torus fibration is homeomorphic to the dual complex and, in Section 6, we establish that all the fibrations we construct are homotopic.
• In Section 7, we explain how to construct a Lagrangian torus fibration on W and we apply this to find Lagrangian torus fibrations on the 4dimensional pair-of-pants and on the negative vertex (the variety defined by Eq. (1)).
• In Section 8, we explain an inductive construction of Lagrangian torus fibrations with codimension 2 discriminant locus which is a symplectic analogue of the non-archimedean SYZ fibration constructed in [33], and we apply it to a special degeneration of K3 surfaces.

Acknowledgements
where each B d is a closed subset such that: We say that B is finite-dimensional if the d-stratum is empty for sufficiently large d, and we say that B is n-dimensional if B is finite-dimensional and n is the maximal number for which S n (B) is nonempty, in which case we call S n (B) the top stratum, S top (B).

Lagrangian torus fibrations
We are interested in proper continuous maps F : X → B with connected fibres where B is a stratified space and, either: • (X, ω) is a symplectic manifold of dimension 2n, or • (X, α) is a contact manifold of dimension 2n − 1 with a chosen contact 1-form α. In this case, we write ω = dα. • the fibres over other strata are themselves stratified spaces with isotropic strata.
We call the complement of the top stratum the discriminant locus Discr(F ) = B \ S top (B).
Remark 2.6. The Arnold-Liouville theorem tells us that the Lagrangian fibres of F are tori and, if X is symplectic, that the top stratum of B inherits an integral affine structure.
Remark 2.7. In the definition of coisotropic fibration, the fibres can be very large. For example, the fibres over the 0-strata of B are codimension zero submanifolds. One example of a coisotropic fibration would be taking a Lagrangian torus fibration X → B and postcomposing with a map which collapses an open set in B containing the discriminant locus. This motivates the following definition: Definition 2.8. We will say that a Lagrangian torus fibration F 1 : X → B 1 is a refinement of a coisotropic fibration F 2 : X → B 2 if there is a continuous surjection g : We finish this section with some remarks on Lagrangian torus fibrations on contact manifolds.
Remark 2.9. Recall that the Reeb flow for a contact manifold (M, α) is always tangent to a dα-Lagrangian submanifold of M .
Remark 2. 10. Suppose that f : X → B is a Lagrangian torus fibration on a contact (2n − 1)-manifold (X, α). The action integrals C1 α, . . . , Cn α of the contact 1-form around a basis for the homology of a general fibre define a map from the universal cover of the top stratum S top (B) to R n . This map necessarily avoids the origin. Otherwise the pullback of α to the fibre is a nullhomologous closed 1-form, which therefore vanishes somewhere on the fibre. However it must always evaluate positively on the Reeb vector, which is tangent to the fibre by Remark 2.9.
Remark 2.11. In the context of the previous remark, the action integrals e r C1 α, . . ., e r Cn α on the symplectisation then give action coordinates for the Lagrangian torus fibration (r, f ) : R × X → R × B on the symplectisation (R × X, d(e r α)).
Remark 2.12. The following rigidity result for Lagrangian torus fibrations on contact manifolds with particular conditions on the Reeb dynamics will apply to some examples later (see Remark 4.12).
Lemma 2.13. Let (M, α) be a contact manifold with contact 1-form α and let F 1 : M → B 1 , F 2 : M → B 2 be Lagrangian torus fibrations. Suppose that for an open set U ⊂ (B 1 ) top of regular fibres there is a dense set V ⊂ U such that, for any v ∈ V , the fibre F −1 1 (v) contains a dense Reeb orbit. Then F 2 is constant along the fibres of F 1 . In other words, the fibration F 2 coincides with F 1 over Proof. The fibre of F 2 through a point x ∈ F −1 1 (v), v ∈ V , necessarily contains the closure of the Reeb orbit through x, so it contains the whole fibre F −1 1 (v) (recall that we have a standing assumption that fibres are connected). Since this is true for a dense set V ⊂ U , and since F 1 and F 2 are continuous, this means F 2 | F −1 1 (u) is constant for all u ∈ U , as required.

Affine varieties
Let X be a smooth complex projective variety of complex dimension n. Let Y ⊂ X be a simple normal crossing divisor with components Y i , i ∈ I, for some indexing set I. Suppose that there is an ample divisor i∈I m i Y i , m i ≥ 1, fully supported on Y . Let L i be the holomorphic line bundle with c 1 (L i ) = Y i , so that Y i is the zero locus of a transversely vanishing section s i ∈ H 0 (X, L i ).
Let | · | i be a Hermitian metric on L i and let ∇ i be the corresponding Chern connection (uniquely determined by the condition of being a metric connection for |·| i ) with curvature F ∇i . By [14, p.148], we can choose the Hermitian metrics in such a way that ω := −2πi i∈I m i F ∇i is a Kähler form on X with Kähler potential In other words, ψ is a plurisubharmonic exhausting function on W .
We now have a standard package of geometric objects associated to ψ: • its gradient (with respect to the Kähler metric on W ) is a Liouville vector field Z := ∇ψ; in other words L Z ω = ω. The 1-form λ := ι Z ω is a primitive for ω called the Liouville form. The flow φ t for time t along Z is called the Liouville flow : it dilates the symplectic form in the sense that φ * t ω = e t ω. • the critical locus of ψ is compact [41].
• if ψ is Morse or Morse-Bott, the union of all downward manifolds of the Liouville flow is isotropic (called the skeleton, Σ). Since the critical locus of ψ is compact, this can be achieved by perturbing the Hermitian metrics on the bundles L i (and hence the symplectic form) over a compact subset of W .
• if M ⊂ W is a (2n−1)-dimensional submanifold transverse to the Liouville flow then M inherits a contact form α pulled back from the Liouville form λ. Moreover, if M is disjoint from the skeleton Σ then the complement W \ Σ is symplectomorphic to a subset of the symplectisation SM of the form 3 Dual complexes

Definition
Let Y be a pure-dimensional simple normal crossing variety of dimension n − 1 (see Definition 1.8 of [25]), for example, a simple normal crossing divisor as in Section 2.3. Note that Y is stratified by the intersections of its irreducible Definition 3.1 (Dual complex). The dual complex D(Y ) of Y is a regular ∆-complex [21, Section 2.1] whose vertices are in correspondence with the irreducible components Y i and whose d-cells correspond to connected components of S d (Y ). Like any ∆-complex, the dual complex is stratified (the d-stratum is the union of its open d-cells).
Definition 3.2 (Maximal intersection). We say that Y has maximal intersection if it admits a stratum of dimension zero. Equivalently, the corresponding cells of D(Y ) have real dimension dim C (Y ), and we say that D(Y ) has maximal dimensional.
Remark 3.3. Suppose that Y is a simple normal crossing divisor in X. In non-archimedean geometry, the cone over the dual complex coincides with the Berkovich skeleton of the formal completion X of X along Y (see [44]). The retraction of the analytic generic fibre X η to the Berkovich skeleton is a nonarchimedean analogue of the Lagrangian torus fibration defined in Theorem 7.1. See Section 8 for more discussion.

Dual boundary complexes of affine varieties
Let Y ⊂ X be a simple normal crossing divisor of dimension n − 1. In this section, we study the homotopy type of the dual complex D(Y ), under the assumption that X \ Y is affine.
First, recall the following result due to Danilov [8,Proposition 3]. The hypothesis of Proposition 3.4 is too restrictive for our purposes (for instance, it does not include all toric boundaries). Moreover, the statement does not provide any control on the number of spheres in the bouquet.
The following propositions can be regarded as generalizations and refinements of Proposition 3.4, and they are inspired by [34] and [27].
Proposition 3.5 (Rational cohomology). Let Y ⊂ X be a simple normal crossing divisor of dimension n − 1 and W := X \ Y be an affine variety. Then, If X has Hodge coniveau ≥ 1, i.e. h 0,i (X) = 0 for all i > 0, and Y J := j∈J Y j does not admit global holomorphic canonical sections for any J ⊆ I (e.g. if X and Y J are rationally connected), then Proof. The vanishing (Eq. (2)) of the rational cohomology is noted in [34,Section 6].
In order to prove Eq. (3), we identify the (n − 1)th cohomology group of D(Y ) with the (n − 1)th cohomology group of the structure sheaf O Y . Note indeed that the cohomology of O Y is computed by a spectral sequence whose page E 1 is given by and which degenerates at the page E 2 ; see [10, Proof of Proposition 1.5.3]. Since H dim YJ (Y J , O YJ ) = 0 for any J ⊆ I, and dim Y J > 0, we have that since the complex (E 0, * 1 , d 1 ) computes the cellular cohomology of D(Y ). Further, the short exact sequence induces the following isomorphism in cohomology Proposition 3.6 (Fundamental group). Let Y ⊂ X be a simple normal crossing divisor of dimension ≥ 2 and W := X \ Y be an affine variety. If X is simply connected (e.g. if X is rationally connected), then D(Y ) is so as well.
Proof. By the Lefschetz hyperplane theorem, π 1 (X) ≃ π 1 (Y ). Note also that there is a natural surjective map π 1 (Y ) ։ π 1 (D(Y )), induced for instance by the evaluation map in Definition 4.1; see also [27,Lemma 26]. We conclude that π 1 (D(Y )) is a quotient of π 1 (X). Proposition 3.7 (Homotopy type). Suppose that the hypotheses of Proposition 3.5 hold, that n := dim(X) ≤ 3, and that X is simply connected. Then, D(Y ) has the homotopy type of a bouquet of h 0 (X, K X +Y ) spheres of dimension dim X − 1, unless it is contractible.
Proof. Since X is simply-connected, Proposition 3.6 implies that D(Y ) is simplyconnected. Since n ≤ 3, dim(D(Y )) ≤ 2, so being simply-connected means that D(Y ) has torsion-free integral homology. Therefore Proposition 3.5 says that D(Y ) has the integral homology of a bouquet of h 0 (X, K X +Y ) spheres of dimension n − 1, or of a point. Hurewicz's theorem then gives a continuous map from such a bouquet to D(Y ) inducing an isomorphism on homology. Since D(Y ) is simply-connected, Whitehead's theorem implies that this map is a homotopy equivalence.

The evaluation map
In this section, we use the following standard notation: if S, T are sets then S T denotes the space of maps T → S.

Evaluation map
Let Y = i∈I Y i ⊂ X be a simple normal crossing divisor. Here and in the following, we can assume that Y J = j∈J Y j is connected for any J ⊂ I (the assumption is not essential but it makes the notation lighter; further, it can always be achieved via a sequence of blowups along connected components of the strata of Y ).
By definition of partition of unity, the image of N • J ∩ Link(Y ) via ev is the convex hull of e j with j ∈ J, where {e i } i∈I is the standard basis of R |I| . In particular, it is a standard simplex of dimension |J| − 1, and it corresponds to the (|J| − 1)-cell of D(Y ) associated to Y J .
We will see that if (X, ω) is a symplectic manifold then the partition of unity can be chosen to make ev into a generically Lagrangian coisotropic fibration. For this, we need to recall the symplectic neighbourhood theorem for simple normal crossing divisors. The key point that we address in this paper is that ev can be adapted to a symplectic form ω, meaning that it can be turned into a generically Lagrangian fibration, unique up to homotopy.

Symplectic plumbing neighbourhoods
Suppose we are given a simple normal crossing divisor Y = i∈I Y i ⊂ X. By a theorem of Ruan [36, Theorem 7.1] we can modify the Hermitian metrics | · | i on the line bundles L i , thereby changing the Kähler form, to ensure that these components meet pairwise ω-orthogonally. Guadagni [20] has proved a symplectic neighbourhood theorem for such divisors, which we now describe.
The bundles ν J inherit Hermitian metrics from the ambient Kähler metric; let U J → Y J be the underlying principal U (1) J -bundle, so that ν J is the associated bundle U J × U(1) J C J . Let µ j : ν j → R be the Hamiltonian function which generates the circle action rotating the fibres of π j and let µ J : ν J → R J be the function µ J (x)(j) = µ j (x), which generates the torus action rotating the summands of ν J independently.
Given a unitary connection α J = j∈J α j on U J we get a 2-form on U J × C J (where θ j are angular coordinates on the summands of C J ), which descends to give a symplectic form ω J on ν J [20, Lemma 2].
The symplectic neighbourhood theorem tells us that for each i ∈ I there is a neighbourhood N i ⊂ ν i of Y i and a symplectic embedding There is a natural compatibility condition on the choices of connections α i and symplectic embeddings ι i [20, Definitions 7 and 8] which can be achieved if the components Y i intersect pairwise symplectically orthogonally [20,Theorem 6.2]. In what follows, we assume that such choices have been made, we will blur the distinction between N i and ι i (N i ). We will call the union N := i∈I N i of the neighbourhoods a symplectic plumbing neighbourhood of Y and, if J ⊂ I, we will write N J := j∈J N j .
Remark 4.4. There is also a symplectic neighbourhood theorem due to Tehrani, McLean and Zinger [43] which tells us that we can find a symplectic plumbing neighbourhood after deforming the symplectic form. However, we want to ensure that our symplectic form remains Kähler for a fixed complex structure, so we prefer to use Ruan's manifestly Kähler deformation to achieve orthogonality followed by Guadagni's version of the neighbourhood theorem.
Definition 4.5. If K ⊂ J then we will write rest J,K : R J → R K for the natural projection map given by restriction from J to K. We will write µ K :

Evaluation map as a coisotropic fibration
We now show that the evaluation map can be made into a coisotropic fibration on the contact hypersurface Link(X \ Y ). We first need to write an explicit equation for Link(X \ Y ).
Proof. This argument is due to Seidel [41, Section 4(b)] but he only explains it in the case where Y is a curve, so we include it (and elaborate on it) for the reader's convenience 2 .
Let g = g i so that the link is g −1 (ǫ ′ ). We will show that for sufficiently small ǫ ′ , dg(Z) < 0 where Z = ∇ψ is the Liouville field (this implies the lemma). If this condition fails then there is a sequence ǫ 1 , ǫ 2 , . . . tending to zero and a sequence of points P α ∈ g −1 (ǫ α ) such that dg(∇Z) ≥ 0 at P α . Because X is compact, and because g −1 (0) = Y , after passing to a subsequence we can assume that P α converges to some point y ∈ Y . Suppose that y lives in the interior of the stratum Y J for some J and, without loss of generality, suppose that J = {1, . . . , j}.
Pick a trivialisation of the bundles {L i } i∈J in such a way that the holomorphic section s i ∈ H 0 (L i ) which vanishes along Y i is given by z i . With respect to this trivialisation, the Hermitian metric on L i can be written as e hi |·|, where | · | is the usual Euclidean norm. Define h = m i h i so that the Kähler potential for ω on this chart is . Note that because the Y i intersect symplectically orthogonally, we can assume that ω is standard at y (i.e. at the origin) after making a complex linear change of coordinates .
Our goal is to show that dg(∇ψ) < 0 at P α for sufficiently large α (which will give a contradiction). Since dg(∇ψ) = dψ(∇g) and since g > 0 away from Y , it is sufficient to show that dψ(∇g/g) < 0 at P α for sufficiently large α.
For points sufficiently close to y, µ i < ǫ/3 for i = 1, . . . , j, and so g( where C is the maximum of |dh| over our neighbourhood. Notice that µ i has Y i as a Morse-Bott critical manifold, so ∇µ i vanishes cleanly along z i = 0. Moreover, the linearisation of the circle action generated by µ i at the fixed point z = (z 1 , . . . , z i−1 , 0, z i+1 , . . . , z n ) is a circle action on T z X which fixes the subspace z i = 0; it is generated by the Hamiltonian vector field associated to (half) the Hessian of µ i at z. This Hessian is a positive-definite quadratic form involving only the real and imaginary parts of z i . We will write Q i for the Hessian of µ i at y.
By shrinking our neighbourhood, we can assume that the Kähler form is approximately constant, equal to the standard form. Moreover, we can assume that µ i is approximately equal to 1 2 Q i . In particular, we get that More precisely: The term |dh||∇µ i | goes to zero as we approach y because all the µ i have a critical point at y, so lim α→∞ C|∇µ i (P α )| = 0. Putting these observations together, we see that, for sufficiently large α, and hence, by Eq. (8), which gives a contradiction. Proof. Choose the following partition of unity on N subordinate to {N i } i∈I : .
These functions Poisson commute with one another and with i∈I g i , so the restrictions of these functions to the link define a coisotropic fibration over the dual complex. More precisely, if N • J := N J \ k ∈J N k is the subset where precisely the functions {χ j } j∈J are non-zero, then the evaluation map projects In particular, ev is generically Lagrangian if Y has maximal intersection.
is an R-action inside that torus, and for a dense set of points in the corresponding n-cell of the dual complex, this R-action has dense image in the n-torus. In particular, Lemma 2.13 applies to the evaluation map over these regions, and hence to any Lagrangian torus fibration which is a refinement of it.

Dovetailing
Given a suitable system of Lagrangian torus fibrations on the irreducible components Y i of Y , we will construct a Lagrangian torus fibration f : Link(X \ Y ) → B which is a refinement of the evaluation map. The idea is clearer when Y is a smooth divisor (and the evaluation map is constant). In this special case, let N be a symplectic neighbourhood of Y as in Section 4.
is a Lagrangian torus fibration on N , which restricts to a Lagrangian torus fibration on the link Link(X \ Y ) ⊂ N . More generally, we need to impose compatibility conditions between the Lagrangian torus fibrations on the components Y i . • If K ⊂ J then there is an embedding e J,K :

Dovetailed systems of Lagrangian torus fibrations
and such that the following diagram commutes.
Let (X, ω) be a symplectic toric manifold and Y be its toric boundary. Then, the restrictions of the toric moment map to Y J provide a dovetailed system of Lagrangian torus fibrations.
Remark 5.3. Note that given a dovetailed system of Lagrangian torus fibrations we can define the pushout colim J B J of the system of bases B J to be the quotient of J B J by the equivalence relation identifying b ∈ B J with e J,K (b) ∈ B K whenever K ⊂ J. Likewise, we can define a thickening B of colim J B J by taking the quotient of J B J × [0, ǫ) J by the equivalence relation identifying Proof. On the subset N J we have the Lagrangian torus fibration Since N = i∈I N i , it suffices to show that these torus fibrations agree on plumbing regions, in other words that if K ⊂ J then the following diagram commutes where the top horizontal map is inclusion and the bottom horizontal map is In other words, we need to show that Since µ J\K is constant along fibres of π K , we have µ J\K (x) = µ J\K (π K (x)).

f is a refinement of the evaluation map
where f is the cutoff function defined in Eq. (5). By construction, the map 3 glue to give a global continuous function g :

B and the dual complex
In this section, we will give conditions which imply that B is homeomorphic to D(Y ). For our purposes, it is convenient to describe the base of f as a complex of spaces (see Section 4.G of [21]). The realisation comes naturally equipped with two projections, since iterated mapping cylinders do as well. Proof. All three spaces embed into B: Proof. Consider the complex of spaces whose underlying graph is the 1-skeleton of the barycentric subdivision of D(Y ), and such that its vertices v i are decorated with the faces, containing v i , of the stars of the vertices of D(Y ) in its barycentric subdivision. The pushout of this complex of spaces is homeomorphic to D(Y ) (but it is endowed with a dual tessellation; see for instance [45,Chapter 6]). By hypothesis, this complex of spaces agrees with B • , so the pushouts colim J B J and D(Y ) are homeomorphic. The result now follows from Lemma 5.8.  Proof. We construct a collection of homotopies

The complex of spaces
where σ J is a cell in D(Y ) associated to Y J . As in Proposition 4.11, we can sup- The linear path (1 − t)r(b) + t id(b), 0 ≤ t ≤ 1, in a cell of D(Y ) containing b, gives a homotopy between r and the identity id. Hence, without loss of generality, we can suppose that H n−1 (−, 1) = ev.
The following proposition shows that the homotopy class of a Lagrangian torus fibration Link(X \ Y ) → D(Y ) (with connected fibres, but not necessarily the map constructed in Corollary 5.6) is unique, provided that D(Y ) has the homotopy type of a bouquet of spheres.
Triangulating S n−1 i as a finite simplicial complex, we can suppose that h i is simplicial by simplicial approximation theorem (see [   Remark 7.3. The Lagrangian skeleton of W is not easy to find in general. Ruddat, Sibilla, Treumann and Zaslow [38] have given a conjectural description of the skeleton when W is a hypersurface in (C * ) m × C n , and all the varieties we consider in this paper fall into this class. When W is hypersurface in (C * ) m , P. Zhou [46] has confirmed that there is a Liouville structure whose skeleton is precisely the RSTZ skeleton.
Remark 7.4. In the situation of the SYZ conjecture, where W is a local model for a Calabi-Yau variety, Cone(B) should be homeomorphic to a codimension zero submanifold of the n-sphere. One might wonder in that case if the map F is smooth across the different strata for some chosen smooth structure on Cone(B). That is not a question we will consider in this paper, as it does not appear natural from our perspective: in our more general setting, Cone(B) will not always be a topological manifold.

Example: pair-of-pants
Let Y be a union of four lines Y 1 , . . . , Y 4 in general position in CP 2 . The complement CP 2 \ Y is called the 4-dimensional pair-of-pants. On each line Y j , you can pick a function ϕ j : Y j → B j where B j is a Y -shaped graph: and, because each pair of lines intersects at one of the points living over an external vertex of B j , the base B looks like this graph: The cone on this graph can be visualised in R 3 as the cone on the 1-skeleton of a tetrahedron, and the result of applying Theorem 7.1 in this case is a Lagrangian torus fibration of CP 2 \ Y over this cone. The fibres over the cones on the blue edges are tori; the fibres over the cones on the red vertices are S 1 × 8 (where 8 denotes the wedge of two circles). The fibre over the cone point is any Lagrangian skeleton for the pair-of-pants. In particular, since the pair-of-pants is a hypersurface in (C * ) 3 , Zhou's result [46] implies that we can take the RSTZ skeleton [38]. In this particular case, the RSTZ skeleton is obtained from three disjoint 2-tori by attaching three cylinders and a triangular 2-cell as indicated in the figure below. This is homotopy equivalent to the 2-skeleton of a 3-torus.
Remark 7.5. In this case, Mikhalkin gave a purely topological torus fibration of the pair-of-pants over the same base, namely the tropicalisation map [31]. See also the paper of Golla and Martelli [13] for a description of the Mikhalkin fibration. The fact that the 4-dimensional pair-of-pants is homotopy equivalent to the 2-skeleton of a 3-torus was proved much earlier by Salvetti [39].

Example: negative vertex
Take the affine variety We call this variety the negative vertex. It can be compactified inside CP 3 as follows. Rewrite the equation for W as u 2 = xy − u 1 − 1; W is the complement of u 2 = 0 in C 2 xy × C * u1 , in other words it is the complement in CP 3 [x:y:u1:w] of (xy − u 1 w − w 2 )u 1 w = 0. Taking However, this subvariety is not normal crossing: the components Y 1 and Y 3 intersect non-transversely at [0 : 0 : 1 : 0] (Y 3 is the tangent plane to the quadric Y 1 at that point).
If we blow up along the line [x : 0 : u 1 : 0] then the total transform of Y becomes simple normal crossing. We continue to write Y j for the proper transform of Y j , j = 1, 2, 3, and we write Y 4 for the exceptional divisor (a copy of CP 1 × CP 1 ). We will consider the ample divisor giving all components multiplicity one). Writing H for the cohomology class Poincaré dual to the proper transform of a generic plane and E for the class of the exceptional divisor, this gives us a symplectic form in the cohomology class 4H − E.
In Fig. 1, we draw almost toric base diagrams for a dovetailing system of Lagrangian torus fibrations ϕ j : Y j → B j . Recall that crosses indicate focus-focus singularities and dotted lines indicate branch cuts. Some of the edges are broken by a branch cut: they are nonetheless straight lines in the affine structure and the break point is not to be considered a vertex. The colours on edges indicate how these edges are to be identified in the pushout. Broken edges are decorated with the same colour on each segment, which is taken to mean that the edge continues beyond the break point, not that some kind of self-identification should be made. Dots on the edges are there to indicate interior integral points of the edges (for the integral affine structure). You can read the affine lengths of edges off from the integrals of [ω] = 4H − E over the corresponding spheres.
We obtain a Lagrangian torus fibration on the link. The base B is a topological 2-sphere and the discriminant locus consists of three points (the focus-focus singularities of the almost toric fibrations). We further obtain a Lagrangian torus fibration on W ; the base is a 3-ball and the discriminant locus is the cone over three points, i.e. a Y-graph.
Theorem 7.6. The negative vertex admits a Lagrangian torus fibration over the 3-ball such that the discriminant locus is a Y-graph. The fibre over the vertex of the Y-graph is a topological space obtained by attaching a solid torus to a wedge of two circles via an attaching map which is freely homotopic to a map φ : Remark 7.7. Note that since S 1 ∨S 1 is an Eilenberg-MacLane space, the induced map on fundamental groups determines the attaching map up to homotopy.
Remark 7.8. Note that the Gross fibre is also obtained by attaching a solid torus to a wedge of two circles by an attaching map in this homotopy class.
Remark 7.9. It seems harder to see the relationship between the RSTZ skeleton in this case and the Gross fibre, though they are necessarily homotopy equivalent.
Proof of Theorem 7.6. By Theorem 7.1, and the previous discussion, we get a Lagrangian torus fibration over the 3-ball with discriminant locus a Y-graph (the cone on three points). It remains to identify the fibre over the vertex, that is the Lagrangian skeleton of the negative vertex. The theorem then follows from Lemma 7.10 and Lemma 7.12 below: • Lemma 7.10 identifies the critical locus of a suitably-chosen plurisubharmonic function ψ. It shows that there are three isolated critical points inside the locus xy = 0 (whose downward manifolds trace out a copy of S 1 ∨ S 1 ) and a circle of critical points with xy = 0 (whose downward manifold is a solid torus).
• Lemma 7.12 identifies the attaching map for the solid torus to S 1 ∨S 1 .
The downward manifolds of the critical points P 2 , P 3 trace out a figure 8 with vertex at P 1 . The downward manifold of the circle of critical points is a solid torus.
Proof. We need to find the critical points of the constrained functional where λ and µ are Lagrange multipliers imposing the constraint xy = 1 + u j . Differentiating, the critical point equations are λ sin θ j = µ cos θ j .
Case XY = 0. If X and Y are nonzero then Eq. (11) implies (λ, µ) = k(cos Φ, sin Φ) for some k = 0. Equation (13) implies that, for all j, (λ, µ) = k j (cos θ j , sin θ j ) for some k j . Overall, this means that the angles θ 1 , . . . , θ n and Φ agree modulo π and that k j e iθj = ke iΦ , so k/k j = ±1. The constraint equation becomes XY − (ke Ri /k j ) e iΦ = 1, which implies that Φ is either 0 or π. Equations (9) and (10) tell us that X = kY and Y = kX, so X = k 2 X and k = ±1. Indeed, since both X and Y are positive, this implies k = 1 and X = Y . Since |k j | = |k| = 1, this implies that k j = ±1. Case XY = 0: If one of X or Y vanishes, then so does the other (using Eq. (9) and Eq. (10)). One could find the critical points from the statement of the lemma by a detailed computation as in the other case, but there is a nice "picture-proof" in this case. The subset of points in W with x = y = 0 is the curve C = {(u 1 , u 2 ) ∈ (C * ) 2 : u 1 + u 2 + 1 = 0} in (C * ) 2 . If we draw the image of this curve under the map (R 1 , R 2 ) : W → R 2 then we get the amoeba shown below. We also show the level sets of ψ| C as dotted circles, and it is easy to see these critical points (blue dots). In red, we have given an idea of how the gradient flowlines from these critical points look: the downward manifold from each of P 2 and P 3 is an interval, whose boundary points must tend to the index 0 critical point P 1 in the limit. Note that it is not true in general that one can find the critical points and flowlines by restricting to a submanifold in this way, but in this case, the Hessian is positive definite on the normal directions to C.
Note that U is a 4-dimensional pair-of-pants; U therefore deformation retracts onto the 2-skeleton of a 3-torus, so π 1 (U ) = Z 3 . Let γ be a circle in a smooth fibre of π onto which the fibre deformation retracts. The fundamental group of V is a central extension of π 1 (U ) by Z γ . We have i * (γ) = 0 because the loop γ is a vanishing cycle for the conic fibration. Therefore π 1 (W ) is a quotient of π 1 (U ) = Z 3 , hence abelian.
Lemma 7.12. Let φ : ∂(D 2 × S 1 ) → S 1 ∨ S 1 be the attaching map for the solid torus to the 1-skeleton. Then, after possibly precomposing with a diffeomorphism of the solid torus and postcomposing with a conjugation, φ * : This determines φ completely up to free homotopy (and precomposition by a diffeomorphism of the solid torus).
Proof. The fact that φ is determined by φ * follows from the fact that S 1 ∨ S 1 is an Eilenberg-MacLane space.
The image of φ * is a subgroup of a free group and hence free, however it is also abelian, so it is either trivial or has rank 1. In other words, φ * (1, 0) = c m and φ * (0, 1) = c n for some c ∈ a, b and m, n ∈ Z. Suppose that (1, 0) is the loop in T 2 which bounds a disc in the solid torus.
The Dehn twist around the loop (0, 1) in T 2 extends to a diffeomorphism of the solid torus, and precomposing with a power of this diffeomorphism allow us to change m by a multiple of n. Since n = 1, we can achieve m = 0. This proves the lemma.

Analogue of non-archimedean SYZ fibration
Let (X, ω) be a smooth complex projective variety of complex dimension n.
Suppose that Y = i∈I Y i ⊂ X is a simple normal crossing divisor with maximal intersection and log Calabi-Yau along 1-dimensional components Y K (|K| = n− 1), i.e. each Y K is a copy of CP 1 which hits precisely two points Y L (|L| = n) in the 0-stratum of Y . For ℓ = 0, 1 and for each L ⊂ I of size n − ℓ with Y L = ∅, let B L be an ℓ-simplex. Attach B K to B L if L ⊂ K and hence form the colimit B 1 : this will be the 1-skeleton of our base. Let B 1 be a thickening of this in the sense of Remark 5.3.
By the symplectic neighbourhood theorem (Section 4.2), a neighbourhood of Y K , denoted (N K , ω| NK ), is symplectomorphic to a neighbourhood of the zero section of a toric vector bundle ν K of rank n − 1 on CP 1 , endowed with the symplectic form Eq. (4). As Guadagni's result has an equivariant strengthening, we can assume that the induced torus action on N K restricts respectively to the T n -and T n−1 -actions rotating the fibres of π L : N L → Y L and π K : N K → Y K . Hence the toric moment map φ K : N K → R n restricts to µ L : N L → [0, ǫ) L ⊆ R n . This property is a dovetailing condition which yields a Lagrangian torus fibration with no singularities Φ : The construction induces a Lagrangian torus fibration on Link(Y ), as shown in the following theorem. Proof. The former statement is obtained by restricting Φ to Link(Y ).
For the latter, start with the standard Lagrangian torus fibration on CP 1 (given by a height function with maximum and minimum at the two points in the 0-stratum and no other critical points). We can use Theorem 7.1 inductively to construct a dovetailed system of Lagrangian torus fibrations ϕ J : Y J → B J where B J is the cone over the dual complex of J K Y K . In other words, B J is the dual block of the cells in D(Y ) corresponding to Y J . Due to the dovetailing condition satisfied by ϕ K , the Lagrangian torus fibration Φ restricts to ϕ J on Y J ∩ K:|K|=n−1 N K .
We can identify the discriminant locus.
• If |J| = n − 2 then Y • J is complex 2-dimensional, and ϕ J : Y • J → B • J has at most one singular fibre (the Lagrangian skeleton of the affine surface Y • J , if that skeleton is not a smooth torus). In other words, the discriminant locus comprises at most a point for each 2-dimensional stratum.
J is a complex 3-fold and ϕ J : Y • J → B • J has discriminant locus equal to the cone on the union of the discriminant loci for the surfaces Y K with J ⊂ K. In other words, this is a graph. It is the graph whose vertices are in bijection with the 3-dimensional components of Y and whose edges correspond to the intersections between these components.
• Continuing in this manner, we see that the discriminant locus is homeomorphic to the codimension 2 subcomplex of the dual complex of Y .
This should be thought of as a symplectic analogue (or dual) of the affinoid torus fibration constructed in [33,Theorem 6.1]. The base is again the (cone over the) dual complex and the discriminant locus is its codimension 2 subcomplex.
In addition, the integral affine structure induced by Φ is dual to that induced by the non-archimedean affinoid torus fibration ([33, §2.6]), in the sense of Proposition 8.3. This is the content of the following section.

Monodromy of the integral affine structure
The non-archimedean SYZ fibration, defined in [33], is a retraction of the analytic generic fibre X η of X along Y over the triviallyvalued 8 field C to the cone over D(Y ). Let Z be the union of the strata of codimension ≥ 2 in Cone(D(Y )). One of the main results of [33] is that ρ X is an affinoid torus fibration over Cone(D(Y )) \ Z, which induces an integral affine structure on it.
). An integral affine structure on a manifold B of dimension n is a maximal atlas such that the transition functions belong to Gl(n, Z) ⋊ Z n . Equivalently, it is a sheaf Aff B of continuous functions on B such that (B, Aff B ) is locally isomorphic to R n endowed with the sheaf of degree one polynomials with integral coefficients.
The cone Cone(D(Y )) \ Z is covered by two types of open sets, whose integral affine structure is described in [33]. . An algebraic version of the tubular neighbourhood theorem for the corresponding curve Y K holds: the formal completion of X along Y K is isomorphic to the formal completion of the zero section of a toric vector bundle ν K of rank n − 1 along its zero section ≃ CP 1 . The integral affine structure on Star K is that of the toric fan of ν K .
In [33,Proposition 5.4], the algebraic tubular neighbourhood theorem is proved under the assumption that the conormal bundle of Y K is ample. This condition can be always achieved by a sequence of stratum blow-ups at Y L ⊂ Y K (or rather at the corresponding 0-strata in the strict transform of Y K ) Note that the operation does not alter the integral affine structure on Cone(σ L ).
The positivity assumption can be actually removed in the following way. The case of positive conormal bundle implies that the formal completion X m Y m K of X m along the strict transform Y m K of Y K is isomorphic to the formal completion ν m K of the toric vector bundle ν m K along its zero section. The isomorphism, we observe that f m−1 is an isomorphism, as required.
By duality, the inverse transpose (β −1 K1K2 ) t is the linear part of the affine transformation which identifies open subsets of the associated moment polytopes. Equivalently, these are the images of the maps φ K . Therefore, (β −1 K1K2 ) t gives the monodromy representation ρ Lagr : π 1 (D(Y ) \ Z) → GL(n, Z) of the integral affine structure induced on Cone(D(Y )) \ Z ≃ B 1 \ B 1 by the Lagrangian torus fibration Φ; see for instance [9, §2].
Lagr ) t . Remark 8.4 (Singularities). In [33], the pairs (X, Y ) of interest are more singular than simple normal crossing. Usually the divisors are special fibres of semistable good minimal dlt degenerations of Calabi-Yau varieties. The singularities of these pairs do not intersect the one-dimensional strata of Y ; see [33,Corollary 4.6]. Therefore, the singularities do not interfere in the construction of Φ, and if φ extends to Link(Y ), they are mapped into the discriminant locus Z.
In high dimension, the singularities of the Lagrangian torus fibration φ will usually be too awful to contemplate. We limit ourself to describe a pencil of quartic surfaces.

Example: Dwork pencil of quartics
Consider the variety The projection π : X → C is a Dwork pencil of quartic K3 surfaces. Its special fibre Y := π −1 (0) = 3 i=0 Y i is the toric boundary of CP 3 , i.e. the union of four planes Y i = {x i = 0} ≃ CP 2 . Along each 1-dimensional component Y ij ≃ CP 1 , X has four ordinary double points, for a total of 24 singularities.
A birational modification X of X is in minus-one-form [32], if • X is a smooth complex (not necessarily projective) manifold with trivial canonical bundle, in a neighbourhood of the special fibre Y over 0; i=0 Y i has simple normal crossing; • any curve Y ij has self-intersection −1 in the surfaces Y i and Y j .
In our case, we can achieve this configuration in the following way. By blowingup the 24 singularities, we replace the singular points with 24 exceptional divisors E k ≃ CP 1 × CP 1 , k = 1, . . . , 24. Then, we contract a ruling on each E k to a rational curve c k . We can choose the rulings in a symmetric fashion, namely in such a way that half of the curves c k intersecting Y ij are contained in Y i , and the other half in Y j , as illustrated below. The model X obtained in this way is isomorphic to X away from the special fibre, and it is the union of four cubic surfaces (i.e. blow-up of CP 2 in six general points) glued along triangles of lines. We draw the special fibre Y as a tetrahedron and mark the 12 exceptional curves c k contained in Y 0 and Y 1 , in the style of [11].
By the Fujino-Nakano criterion [12], the contraction yields a complex manifold (actually an algebraic space, by [ is obtained from the cubic surface Y i by excising a generic tritangent hyperplane section (i.e. an anticanonical divisor comprising a cycle of three lines). The dovetailing system has four singular fibres, which are precisely the Lagrangian skeleta of these affine cubics. These singular fibres have the homotopy type of a wedge of five spheres.
In fact, following Ruan [35], one can use symplectic parallel transport to construct Lagrangian torus fibrations on the nearby smooth fibres of the Dwork pencil. Because X is a semistable degeneration (in particular, Y is snc), the model of [35,Section 3.3] applies, so there are no further singular fibres introduced, and we obtain a Lagrangian torus fibration on π −1 (c), c = 0, with four singular fibres, each having the homotopy type of a wedge of five spheres.
Remark 8.5. The cohomology class of the symplectic form on π −1 (c) is no longer the standard polarisation for the quartic surface. Indeed, the Lagrangian spheres in the quartic surface which are vanishing cycles of the Dwork pencil have nonzero symplectic area because they are homologous to the exceptional curves of X → X.
Remark 8.6. Note that an elliptic fibration on a K3 surface with four singular fibres which have the homotopy type of the wedge of five spheres can be obtained in the following way. Consider a Kummer surface associated to the product of two elliptic curves. A projection of the product descends to the required fibration. This would yield a Lagrangian torus fibration after hyperKähler rotation.
Lemma 8.7. In the notation above, there exists a relatively ample line bundle L on X (or at least in a neighbourhood of Y ) satisfying the following property ( * ) the first Chern class c 1 (L| Yi ) ∈ H 2 ( Y i , C) is a multiple of the anticanonical polarisation of Y i for any i = 0, . . . , 3.
Proof. This will follow from a sequence of lemmas below. In Lemma 8.8, we construct an ample line bundle L Y on Y with Property ( * ). In Lemma 8.9, we show that L Y extends to a line bundle on X, i.e. the completion of X along Y . Finally, by Artin's approximation theorem [2, Theorem 1.12], L Y is approximated by a line bundle L over a Euclidean neighbourhood of Y in X.
More precisely, this means that the restrictions of L Y and L to an infinitesimal thickening of Y coincide. Note that, in particular, L satisfies Property ( * ), and it is ample by the openness of amplitude for algebraic spaces (apply [30, Proposition 1.2.17] to a scheme which is a finite cover of X [24, Lemma 2.8 and Theorem 3.11]).
The proofs below use the Mayer-Vietoris complex [15], whose definition we now recall, following [ We now prove the lemmas used in Lemma 8.7. Proof. The spectral sequence Eq. (14) yields the exact sequence By definition, After possibly flopping some of the curves c k in X, we can suppose that ι sends the centres of the blow-down Y 0 → Y 0 to those of Y 1 → Y 1 , while it permutes those of Y 2 → Y 2 , equivalently of Y 3 → Y 3 . In this configuration, for any of the 24 singular points p k ∈ Y of X, we can find Euclidean neighbourhoods U p k ⊂ X with the following properties: • ι(U p k ) = U ι(p k ) ; • if V p k is the preimage of U p k in X obtained by blowing up the Weil (but non-Cartier) divisor Y i ∩ U p k , then V ι(p k ) is isomorphic to the blow-up of U ι(p k ) along the divisor ι(Y i ) ∩ U ι(p k ) .
The local models show that ι lifts to a regular involution of X.
The exact sequence Eq. (15) is equivariant with respect to the Z/2Z-action induced by ι, and E 0,2 2 is a non-trivial Z/2Z-module. Indeed, the involution ι lifts to a reflection of the tetrahedron D( Y ), and acts on E 0,2 2 ≃ H 2 (D( Y ), C * ) ≃ C * as z → 1/z. Moreover, (K Yi ) i is fixed by ι.
where the isomorphisms at the edges follow from the identification H i (O Y ) = H i (D( Y ), C); see for instance [25,Lemma 3.63]. We may argue as in the proof of Lemma 8.8. Indeed, since L Y is fixed by ι but H 2 (O Y ) has no non-trivial fixed subspace, δ(L Y ) must be trivial, so L Y lifts to a line bundle on any infinitesimal thickening of Y , thus on X by Grothendieck

Example: Higher-dimensional pairs-of-pants
Let X = CP n and Y be a union of n + 2 hyperplanes in general position. The variety W = X \ Y is the 2n-dimensional pair-of-pants. Each component Y i of Y is a copy of CP n−1 and the intersection of Y i with the other components is again a union of (n − 1) + 2 hyperplanes in general position, so the top strata Y • i are 2(n − 1)-dimensional pairs-of-pants. By applying Theorem 7.1 repeatedly (as in the proof of Theorem 8.1), we can construct a Lagrangian torus fibration inductively on the 2n-dimensional pair of pants; the example worked out in Section 7.2 was a special case of this.