The main theorem [04LY]

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The main theorem

Finally, having completed the construction of the negative fibration, in this last section we prove the main result of the article. In order to give a correct statement of the theorem, we need first to make a few observations.

We start with a compact simple integral affine $3$ -manifold with singularities $(B,\Delta,\mathscr{A})$ . The goal is to symplectically compactify the torus bundle $X(B_{0},\mathscr{A})$ by gluing to it singular fibres. We have already seen in Section 4, Proposition 4.17 how the gluing of positive or generic singular fibres is quite straightforward. In the case of negative vertices we have seen that our construction gives a fibration whose discriminant locus contains components of type $\Delta_{a}$ , i.e. of codimension $1$ . For this reason around negative points one needs to replace $\Delta$ with a slightly perturbed discriminant locus containing components of type $\Delta_{a}$ .

Let us consider the fibration of Example 5.8. The periods of this fibration were computed in [3] and they are given by formulas (78). Let us consider the corresponding primitives (action coordinates) restricted to the plane $\{b_{1}=0\}$ , which is the plane where the discriminant locus lies. We can easily see that the action coordinates map $\alpha$ transforms the amoeba with thin legs into a slightly different shape, depicted in in Figure 16. This shape does not change much after we have done the smoothings of Lemmas 7.4, 7.6 and 7.12, what may happen is that the codimension $2$ part –i.e. the legs– may become slightly curved. Nevertheless, it is not difficult to see that we can prolong the legs of a negative fibration so that they become straight toward their ends. This can be done by gluing suitable generic-singular Lagrangian fibrations using the methods of Proposition 4.18.

Refer to caption — Figure 16: The affine image of the amoeba with thin legs

Definition 8.1.

Given a simple integral affine $3$ -manifold with singularities $(B,\Delta,\mathscr{A})$ , all of whose negative vertices are straight (i.e. locally affine isomorphic to Example 3.12), 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. This replacement takes place in the plane corresponding to $\{x_{1}=0\}$ of the local model, Example 3.12.
(ii)

$\mathcal{N}$ is the set of negative vertices and for each $p^{-}\in\mathcal{N}$ , $D_{p^{-}}$ is a submanifold of $B$ , homeomorphic to a disk and containing the codimension $1$ component of $\Delta^{\blacklozenge}$ around the negative vertex $p^{-}$ . Moreover, $D_{p^{-}}$ is contained in the plane $\{x_{1}=0\}$ . We depict $D_{p^{-}}$ as the gray area in Figure 17.

The requirement that all negative vertices are straight is only to avoid unnecessary complications. Given a localized thickening of $\Delta$ , define

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

Clearly, the integral affine structure $\mathscr{A}$ on $B-\Delta$ restricts to an integral affine structure on $B_{\blacklozenge}$ which we denote by $\mathscr{A}_{\blacklozenge}$ , therefore we can form the torus bundle $X(B_{\blacklozenge},\mathscr{A}_{\blacklozenge})$ .

Now we can state and prove the theorem:

Theorem 8.2.

Given a compact simple integral affine $3$ -manifold with singularities $(B,\Delta,\mathscr{A})$ , all of whose negative vertices are straight (i.e. locally isomorphic to Example 3.12), there is a localized thickening $(\Delta_{\blacklozenge},\{D_{p^{-}}\}_{p^{-}\in\mathcal{N}})$ of $\Delta$ and a smooth, compact symplectic $6$ -manifold $(X,\omega)$ together with a piecewise smooth Lagrangian fibration $f:X\rightarrow B$ such that

(i)

$f$ is smooth except along $\bigcup_{p^{-}\in\mathcal{N}}\,f^{-1}(D_{p^{-}})$ ;
(ii)

the discriminant locus of $f$ is $\Delta_{\blacklozenge}$ ;
(iii)

there is a commuting diagram

$\begin{CD}X(B_{\blacklozenge},\mathscr{A}_{\blacklozenge})@>{\Psi}>{}>X\\ @V{f_{0}}V{}V@V{}V{f}V\\ B_{\blacklozenge}@>{\iota}>{}>B\end{CD}$

where $\psi$ is a symplectomorphism and $\iota$ the inclusion;
(iv)

over a neighborhood of a positive vertex of $\Delta_{\blacklozenge}$ the fibration is positive, over a neighborhood of a point on an edge the fibration is generic-singular, over a neighborhood of $D_{p^{-}}$ the fibration is Lagrangian negative.

Proof.

The proof is quite simple. First we glue positive fibrations over sufficiently small neighborhoods of positive vertices of $\Delta$ using Proposition 4.17. Now given a negative vertex $p^{-}\in\mathcal{N}$ , we have that a neighborhood of $p^{-}$ is affine isomorphic to a neighborhood $U$ of zero in the local model Example 3.12. Consider a negative Lagrangian fibration $\mathcal{F}^{-}=(X^{-},\omega^{-},f^{-},B^{-})$ (cf. Definition 7.1), which we have constructed in Theorem 7.3. The discriminant locus $\Delta^{-}$ of $f^{-}$ has the shape of an amoeba with thin legs and there is a disc $D$ containing the codimension $1$ part of $\Delta^{-}$ such that $f^{-}$ is smooth except at points of $(f^{-})^{-1}(D)$ (cf. part (i) and (ii) of Definition 7.1). Moreover we may assume that $B^{-}-(\Delta^{-}\cup D)$ is affine isomorphic to $(U^{\prime}-(D^{\prime}\cup\Delta_{\tau}),\mathscr{A}_{\tau})$ , where $U^{\prime}$ is a neighborhood of $0$ in the affine manifold with singularities of Example 3.13 and $D^{\prime}\subset\{x_{1}=0\}$ contains $0$ and is homeomorphic to a disc (cf. point (iii) of Definition 7.1). It may happen that $U^{\prime}$ is too big for us to glue the Lagrangian negative fibration as it is. However, if we replace $\omega^{-}$ with $\epsilon\,\omega^{-}$ for a sufficiently small $\epsilon>0$ , this has the effect of scaling the affine coordinates on the base by a factor of $\epsilon$ (i.e. of making the amoeba as small as we please). Therefore we may assume that $U^{\prime}\subset U$ . Moreover, we may also assume that the legs of $\Delta^{-}$ (in affine coordinates) are straight towards their ends, i.e. they coincide with the legs of $\Delta$ outside an open subset $U^{\prime\prime}$ such that $D^{\prime}\subset\bar{U}^{\prime\prime}\subset U^{\prime}$ . The localized thickening $\Delta_{\blacklozenge}$ of $\Delta$ around $p^{-}$ consists in replacing $U^{\prime}\cup\Delta$ with $\Delta^{-}$ and defining $D_{p^{-}}=D^{\prime}$ . The affine structure $\mathscr{A}_{\blacklozenge}$ is inherited from $\mathscr{A}$ . This can be done at every negative vertex $p^{-}$ . Tautologically, we have that $X(U^{\prime}-(D_{p^{-}}\cup\Delta_{\blacklozenge}),\mathscr{A}_{\blacklozenge})$ is symplectically conjugate to $(f^{-})^{-1}(B^{-}-(\Delta^{-}\cup D))$ and therefore we can glue $X^{-}$ to $X(B_{\blacklozenge},\mathscr{A}_{\blacklozenge})$ .

Finally, now that singular fibres have been glued on top of all vertices, it only remains to glue generic-singular fibres along the edges. This can be easily done by applying directly Proposition 4.18, notice in fact that Lagrangian negative fibrations are smooth and generic-singular towards the ends of the legs. ∎

We remark that the manifolds we obtain with this theorem are diffeomorphic to Gross’ semi-stable compactifications of Theorem 2.11. Also, as a corollary of this construction we have

Corollary 8.3.

A smooth quintic $X$ in $\mathbb{P}^{4}$ has a symplectic form $\omega$ with a piecewise smooth Lagrangian fibration $f:X\rightarrow S^{3}$ .

Proof.

If we apply Theorem 8.2 to Example 3.17 we obtain a symplectic manifold $X$ with a piecewise smooth Lagrangian fibration $f:X\rightarrow S^{3}$ . By Gross’ Theorem 3.19, $X$ is homeomorphic to a non-singular quintic. ∎

We do not know whether the symplectic manifold $(X,\omega)$ obtained in this corollary is actually symplectomorphic to a quintic with a Kähler form, although we conjecture it is.