A map $f:X\rightarrow B$ from a smooth symplectic manifold onto a smooth manifold is a Lagrangian fibration if the regular locus of fibres has half the dimension of $X$ and the symplectic form restricts to zero there. The fibration is allowed to have singular fibres. In fact, interesting examples should in general include singular fibres. If the fibration map is smooth and proper, it is a well-known fact that the non-singular fibres are tori. Furthermore, away from the discriminant locus parametrizing the singular fibres, the base has the structure of an integral affine manifold. In other words, $B$ has an atlas whose change of coordinates are integral affine linear transformations.

Lagrangian fibrations lie at the crossroads of integrable systems, toric symplectic geometry and more recently, Mirror Symmetry. For all three subjects, important issues are: the global topology of the fibration, the singularities of the fibres, the regularity of the fibration map and the affine structures induced on the base. In the recent years, integral affine geometry started to play a remarkably important role in Mirror Symmetry. The first evidence of this is given by Hitchin [20], who observed that the SYZ duality [32] can be interpreted as a Legendre transform between integral affine manifolds. Later, Kontsevich and Soibelman [22] and Gross and Wilson [14] proposed a landmark conjecture which, roughly speaking, says:

• (1)

  Degenerating families of Calabi-Yau manifolds approaching large complex structure limits should collapse down to a singular integral affine $S^{n}$.

• (2)

  Mirror families should be (re)constructed starting from the affine manifolds in (1).

The first part of this conjecture is referred to as the Gromov-Hausdorff collapse, while the second part is usually called the reconstruction problem [9]. We know that the Gromov-Hausdorff collapse does happen in dimension two [14]. More recently, Gross and Siebert [12, 13] develop a program to reconstruct the “complex side” of the mirror using Logarithmic geometry. Kontsevich and Soibelman [23] approach the complex reconstruction problem using non-Archimedean analytic spaces. The final explanation of Mirror Symmetry is likely to emerge from the work deriving from these two main streams.

On the “symplectic side” of the mirror, there is an analogous reconstruction problem. This paper is motivated by the following question. Can we construct symplectic manifolds starting from integral affine manifolds with singularities and obtain total spaces homeomorphic to mirror pairs of Calabi-Yau manifolds?

To answer this question we take Gross’ Topological Mirror Symmetry [7] as a starting point. Gross developed a method to construct topological $T^{3}$ fibrations of 6-manifolds. This method consists, roughly, on the compactification of certain $T^{3}$ bundles by means of gluing suitable singular fibres. The discriminant locus in this case is a 3-valent graph with vertices labeled positive or negative. There are three types of singular fibres: generic fibres, positive fibres and negative fibres, mapping to either points on the edges, or positive or negative vertices of the graph, respectively. The names are given according to the Euler characteristic of the fibres which can be 0, $+1$ or $-1$ respectively¹¹ 1 Gross uses a different convention: $(2,2)$, $(1,2)$ and $(2,1)$, for generic, positive and negative fibres, respectively. Gross’ compactification produces a class of fibrations that can be dualized. As an example of this construction, Gross obtained a pair of smooth manifolds with dual topological $T^{3}$ fibrations, the first one being homeomorphic to the quintic 3-fold and the second one homeomorphic to a mirror of the quintic.

The main result of this paper is the proof that a compactification similar to that of Gross can be carried out in the symplectic category. The basic idea is the following. We start with an integral affine manifold with singularities $(B,\Delta,\mathscr{A})$ with 3-valent graph singular locus $\Delta$. The affine structure on $B_{0}=B-\Delta$ induces a family of maximal lattices $\Lambda\subseteq T^{\ast}B_{0}$, together with a symplectic manifold $X(B_{0})$ and an exact sequence

$$0\rightarrow\Lambda\rightarrow T^{\ast}B_{0}\rightarrow X(B_{0})\rightarrow 0.$$

This gives us a Lagrangian $T^{n}$ bundle $f_{0}:X(B_{0})\rightarrow B_{0}$. When $\mathscr{A}$ is simple (cf. Definition 3.14), $X(B_{0})$ can be compactified to a topological 6-manifold $X(B)$ using Gross method. To define a symplectic structure on $X(B)$, in other words, to achieve a symplectic compactification of $X(B_{0})$, one needs Lagrangian models of generic, positive and negative singular fibres. The first two models have already been studied by the first author [1]. The construction of a Lagrangian negative model is much more delicate. An important part of this article is devoted to the construction of Lagrangian fibrations of negative type.

While the generic and positive models are given by smooth maps and have codimension two discriminant loci, our model for the negative fibration is piecewise smooth and has mixed codimension one and two discriminant: it is an “amoeba” whose three legs are pinched down to codimension 2 (cf. Figure 5). In fact it can be described as a perturbation of Gross’ negative fibration, localized in a small neighborhood of the ‘figure eight’ (i.e. the singular locus of the negative fibre), which forces the singularities of the fibres to become isolated points and the discriminant locus to jump to codimension one near the vertex. The topology of the total space is unchanged by this perturbation. Joyce [21] had already conjectured that special Lagrangian fibrations should be in general piecewise smooth and should have codimension 1 discriminant locus. Over the codimension 1 part of the discriminant locus, our model has exactly the topology which Joyce proposed as the special Lagrangian version of Gross’ negative fibre.

Our first attempt to construct a model of a Lagrangian negative fibration produces a fibration which fails to be smooth along a large codimension one subset, a whole plane containing the discriminant locus (cf. Example 5.8). This model is not suitable for the symplectic compactification. This is essentially due to the fact that piecewise smooth fibrations in general do not induce integral affine structures on the base. The affine structure induced by fibrations of this sort consists of two pieces separated by the codimension one wall. Piecewise smooth fibrations of this type are called stitched and have been studied in great detail by the authors [2, 3]. It turns out that the information on the lack of regularity of these fibrations can be encoded into certain invariants. This allows us to have good control on the regularity of stitched fibrations. In particular, we are able to modify Example 5.8 to a Lagrangian fibration which induces an integral affine structure on the complement of a closed 2-disc containing the codimension one component of the discriminant. Moreover, away from this ‘bad disc’, where the fibration fails to be smooth, the induced integral affine structure is simple.

Given a simple integral affine $3$-manifold with singularities $(B,\Delta,\mathscr{A})$ a localized thickening of $\Delta$ is given by the data $(\Delta^{\blacklozenge},\{D_{p^{-}}\}_{p^{-}\in\mathcal{N}})$ where:

• (i)

  $\Delta^{\blacklozenge}$ is the closed subset obtained from $\Delta$ after replacing a neighborhood of each negative vertex with a shape of the type depicted in Figure 17 (an “amoeba” with thin legs).

• (ii)

  $\mathcal{N}$ is the set of negative vertices and for each $p^{-}\in\mathcal{N}$, $D_{p^{-}}$ is a disk containing the codimension $1$ component of $\Delta^{\blacklozenge}$ around $p^{-}$ (depicted as the gray area in Figure 17).

Given a localized thickening define

$$B_{\blacklozenge}=B-\left(\Delta\cup\bigcup_{p^{-}\in\mathcal{N}}D_{p^{-}}\right).$$

and denote by $\mathscr{A}_{\blacklozenge}$ the restriction of the affine structure on $B_{\blacklozenge}$

The main result of this paper is the following (cf. Theorem 8.2):

As a corollary of Theorem 8.2 and Gross’ topological compactification [7], when $(B,\Delta,\mathscr{A})$ is as in Example 3.17, the symplectic manifold obtained is homeomorphic to the quintic Calabi-Yau 3-fold. Applying the Legendre transform to Example 3.17 produces a compact simple integral affine manifold with singularities $(\check{B},\check{\Delta},\check{\mathscr{A}})$ [12]. The latter induces a bundle $X(\check{B}_{0})$, dual to $X(B_{0})$. By applying the Theorem we obtain a compact symplectic manifold $(\check{X},\check{\omega})$ homeomorphic to Gross’ topological compactification $X(\check{B}_{0})$, therefore homeomorphic to a mirror of the quintic.

The affine structures we consider here satisfy a property called simplicity. Essentially, our notion of simplicity coincides with Gross and Siebert’s simplicity in dimensions $n=2$ and $3$. Theorem 8.2 should produce pairs of compact symplectic manifolds fibering over Gross and Siebert’s integral affine manifolds, therefore producing a vast number of examples of dual Lagrangian $T^{3}$ fibrations. For example, in [8], Gross shows that to the pairs of Calabi-Yau’s constructed with the method of Batyrev and Borisov as complete intersections in dual Fano toric varieties, one can associate a pair of simple affine manifolds with singularities which, when compactified, give back a pair of manifolds homeomorphic to the two Calabi-Yau’s. The latter statement is the content of [8]Theorem 0.1, which is proved in [11] by Gross and Siebert. Combining this with our result, we obtain a construction of symplectic manifolds fibred by Lagrangian tori, which are homeomorphic to the Batyrev and Borisov mirror pairs of Calabi-Yau manifolds. Also, another source of examples may come from the structures constructed in [16, 17, 18], provided they are simple.

We should mention at this point that Lagrangian $T^{3}$ fibrations of Calabi-Yau manifolds have been constructed before by Ruan [27, 29, 30]. Ruan’s construction does not use integral affine geometry, rather, it depends on a gradient flow argument. In particular Ruan’s construction depends on the embedding inside an ambient manifold. We suspect that Ruan’s fibrations share many similarities with our symplectic compactifications but we haven’t been able to verify this. It is not clear what kind of regularity Ruan’s fibrations have, therefore whether they induce integral affine structures on the base. One interesting aspect of our method is that it makes explicit connection with the formulation of Mirror Symmetry in [22] and [13], where affine geometry is essential.

The main motivation of this paper is Mirror Symmetry but we expect interesting applications in symplectic topology to emerge from the results we present here. Our construction of Lagrangian fibrations has a flavor similar to the work on almost toric symplectic geometry of Leung and Symington [24]. A theory on almost toric 6-folds could emerge from the methods applied in this article. On the other hand, being our construction so explicitly connected to affine geometry, it is possible that the construction in Theorem 8.2 will eventually shed light onto the new methods in symplectic enumerative problems arising from tropical geometry.

The material of this paper is organized as follows. We start giving in §2 the description of Gross’ compactification of topological $T^{n}$ bundles with semi-stable monodromy. Here we explain how to modify Gross’ negative fibration to a fibration with a localized thickening near the negative vertex. In §3 we introduce the integral affine manifolds we use in the rest of the paper. We formalize our notion of simplicity by means of standard models of affine manifolds with singularities with prescribed holonomy. Our notion of simplicity coincides with the one in [12] in dimension $n=2$ and $3$. Simplicity is, essentially, a condition which guarantees that the induced Lagrangian $T^{n}$ bundles have semi-stable monodromy that can be compactified. We describe some examples of non-compact and compact simple integral affine manifolds with singularities. As an illustration of some of the methods we use, we show in Theorem 3.22 how, in dimension $n=2$, one can produce symplectic manifolds diffeomorphic to K3 surfaces. In §3 we describe Lagrangian models of positive and generic fibrations and prove that they induce integral affine structures which are simple. These models can be used to produce semi-stable symplectic compactifications over simple affine manifolds without negative vertices (cf. Theorem 4.19). This is not enough, in general, to construct symplectic manifolds homeomorphic to Calabi-Yaus –such as a quintic and its mirror– as one should normally include negative vertices. In any event, given the existence of simple affine bases with positive vertices only (or without any vertices at all) Theorem 4.19 tells us how to construct a symplectic manifold together with a Lagrangian fibration over it. In this case, the Lagrangian fibrations obtained are everywhere smooth and the thickening of the discriminant is not necessary. There are explicit Examples of integral affine manifolds structures with no vertices [8] and Theorem 4.19 can be used to produce symplectic compactifications. In §5 we move on to piecewise smooth fibrations. We give concrete examples of piecewise smooth Lagrangian $T^{3}$ fibrations. In particular, in Example 5.8 we explicitly construct a Lagrangian version of the topological negative fibration with fat discriminant given in §2. This model is piecewise smooth over a large region. In §6 we review some of the techniques we developed in [2], which allow us to make certain non-smooth Lagrangian fibrations into smoother ones, such as the one in Example 5.8. The material of this section is rather technical and the reader may skip it in a first reading. In §7 we construct Lagrangian fibrations of negative type. These are local models whose discriminant is a localized thickening of a 3-valent negative vertex $p-$. The fibration is smooth away from a 2-disc $D_{p-}$ containing the codimension 1 component of the discriminant. Away from $D_{p-}$, the affine structure is integral and simple. Finally, in §8 we prove Theorem 8.2.

Aknowledgments: The second author was partially funded by an EPSRC Research Grant GR/R44041/01, UK. Both authors would like to thank Mark Gross and Richard Thomas for useful discussions. Moreover they thank the following institutions for hosting them while working on this project: the Abdus Salam International Centre for Theoretical Physics in Trieste, the Department of Mathematics at Imperial College in London, the Max Plank Institute in Leipzig and in Bonn, the Dipartimento di Scienze e Tecnologie Avanzate of the University of Piemonte Orientale in Alessandria (Italy), the Department of Mathematics of the University of Pavia (Italy), the IHES in Paris.