1.1.1 On torsion

We now discuss the assumption concerning torsion in Theorem 1.1. We repeat that we fully expect that this assumption can be lifted from the statement. Proving this would have required working a lot more at the chain level, making the paper more technical than it already is, but we do not expect any serious difficulties.

In the text we prove a statement that is more refined than Theorem 1.1, which applies without assumptions on the torsion. Namely, we prove that the truncated relativeFootnote ² symplectic cohomologies $SH^*_{Y,\lambda }(K)$ and $SH^*_{M,\lambda }(\iota (K))$ (see § 2.2) satisfy locality in the sense of Theorem 1.1 (with the same assumptions on $M,Y$ and $\iota$) for all $\lambda \geqslant 0$ and all compact $K\subset Y$. See Theorem 3.5 for the precise statement.

The torsion assumption comes in when we try to recover relative symplectic cohomology from truncated symplectic cohomologies. We need some definitions to explain this. The maximal torsion of a module $V$ over the Novikov ring $\Lambda _{\geqslant 0}$ is the supremum of the torsions of all of its torsion elements (Definition 6.15), which is possibly equal to $\pm \infty$. If it is not $+\infty$, then we say that $V$ has finite torsion.

Let us state the specific result that we use to relate the relative symplectic cohomology to truncated symplectic cohomologies as we think it is instructive in its own right. We define

\[ SH_{M,{\rm red}}^*(K):=\varprojlim_{\lambda}SH_{M,\lambda}^*(K). \]

Proposition 1.2 Assume that $SH^{i}_{M}(K)$ has finite torsion for some $i\in \mathbb {Z}$, then the canonical map

\[ SH_M^i(K)\to SH_{M,{\rm red}}^i(K) \]

is an isomorphism.

The proof is an exercise in homological algebra and it is provided in § 6.3. Let us stress that in Theorem 1.1 (or even better in its slightly sharper version Theorem 6.21) the assumption on finiteness of torsion is made only with respect to $Y$ (and not $M$), making the result much more useful.

We now comment on when the assumption on homological finiteness of torsion is known or expected to hold and when it does not hold. Let us define the homological torsion in degree $i$ of $K\subset M$ as the maximal torsion of $SH_M^i(K)$.

• – By the computations in [Reference Floer, Hofer and WysockiFHW94] ellipsoids and polydisks in ${\mathbb {C}}^n$ have homologically finite torsion in each degree.

• – Let $Y=T^*{\mathbb {T}}^n$ with its projection $\pi :T^*{\mathbb {T}}^n\to {\mathbb {R}}^n$. Let $K=\pi ^{-1}(P)\subset Y$ for $P\subset {\mathbb {R}}^n$ a compact convex domain. Then $SH_Y^*(K)$ has no torsion, so finiteness of torsion in any degree holds trivially. This result can be derived by a direct computation using Viterbo-type acceleration data along with careful perturbations to deal with the Morse–Bott $T^n$-families of orbits that appear (similar to § 6.4).

• – For $Y$ as in the previous item and $K=\pi ^{-1}(P)\subset Y$ such that $P\subset {\mathbb {R}}^n$ is star-shaped but not convex, it can be shown that homological finiteness of torsion does not hold in all degrees. We give the argument for a particular such domain in § 6.4.

• – Let $Y$ be a symplectic cluster manifold with nodal almost toric fibration $\pi :Y\to B$ of cluster type (see § 1.2.1 for quick definitions). Let $P\subset B$ be a compact domain (possibly with corners) whose boundary does not contain any nodes. Let $K=\pi ^{-1}(P)$. Then $K$ has homologically finite torsion in degree $0$. If we assume further that $P$ is a convexFootnote ³ polygon with rational slope sides and the restriction of the symplectic form to $K$ is exact, then $K$ has homologically finite torsion in degree $1$ as well. This can be shown by combining Theorems 1.9 and 1.1 and the argument given in [Reference PascaleffPas19, Theorem 6.19 and Remark 6.20].

• – For $Y$ as in the previous item, it can again be shown that if $P$ is star shaped but not convex, homological finiteness of torsion does not hold in degree 1; cf. [Reference PascaleffPas19, Remark 6.17].

• – If $K$ is an index bounded Liouville domain (as in, e.g., Definition 2.2 of [Reference SunSun21]) inside a symplectic manifold $Y$ with some (homotopy class) of a trivialization of its canonical bundle, then $SH^{i}_{Y}(K)$ has finite torsion for all $i$. Compare with Proposition 4.6 of [Reference SunSun21].

1.1.2 Idea of the proof

We conclude this part of the introduction with a brief discussion of the method of proof. Consider the setup of Theorem 1.1. We start with a comment on why one should generally expect any type of locality result. The first condition for such result is that the contribution of the $1$-periodic orbits which are, roughly speaking, present in $M$ but not in $Y$ be negligible. This is expected to always hold as a result of the truncation and completion operations involved in the definition of our invariants, but it is not always easy to prove. If this holds, we can consider the underlying $\Lambda _{\geqslant 0}$-modules for $SH^*_Y(K)$ and $SH^*_M(\iota (K))$ at the chain level as being identical. The locality question then involves understanding the contribution, or lack thereof, of Floer trajectories that connect orbits in $\iota (Y)$ but are not contained in $\iota (Y)$.

Let us assume that $\iota$ is the inclusion of an open subset $Y\subset M$ equipped with the symplectic form $\omega |_Y$.Footnote ⁴ The geometrically finite-type hypothesis made on $Y$ implies that we can find a geometrically bounded almost complex structure $J_Y$, an admissible function $f$ as defined in the beginning of § 3, and a constant $r$ such that all the $1$-periodic orbits of $f$ are contained in the sub-level $V_r:=f^{-1}(-\infty,r)$. For any $R>0$ the annulus $A_R:=f^{-1}([r,R+r])$ forms a separating region in the following sense.

Proposition 1.5 Fix $J_Y$ as above. There are constants $\epsilon >0,\ C>0,\ D,$ depending on the bounds on the geometry of $J_Y$ such that the following holds. Let $f$ be a function on $Y$ as above and assume the Lipschitz constant of $f$ is $\leqslant \epsilon$. Let $R>0$ and let $H:M\to {\mathbb {R}}$ be a Hamiltonian such that $H|_{A_R}= f|_{A_R}$. Let $J$ be a compatible almost complex structure on $M$ such that $J|_{A_R}=J_Y|_{A_R}$. Then any Floer trajectory $u$ for the datum $(H,J)$ which meets both $V_{r}$ and $M\setminus V_{R+r}$ satisfies

(1)\begin{equation} E(u)\geqslant CR-D. \end{equation}

The proofs of more detailed forms of this statement are given in §§ 4.1 and 4.2.

Given a compact set $K$ we can choose $r$ so that, in addition, $K\subset V_r$. The proposition implies that at each energy truncation level we can Floer theoretically separate $K$ from $M\setminus Y$ by making $R$ large enough. The upshot is that at each truncation level we can make the underlying complexes for $K\subset Y$ and ${\iota (K)}\subset M$ coincide after justifying that the orbits that are not common to both do not contribute.

1.2.1 Symplectic cluster manifolds

Let $X$ be a symplectic manifold and $\pi : X\to B$ be a nodal Lagrangian torus fibration. That is, an almost toric fibration as in [Reference SymingtonSym03] with a finite number of singularities all of which are of focus–focus type. The torus fibration $\pi$ induces on $B$ the structure of an integral affine manifold with singularities. That is, denoting by $N\subset B$ the critical values of $\pi$, the open and dense subset $B^{\rm reg}:=B\setminus N$ carries an integral affine structure. Around each $n\in N$ the integral affine structure has monodromy of shear type. A detailed description is given in § 7.

A Lagrangian tail $L$ emanating from a focus–focus point $p$ is a properly embedded Lagrangian plane in $X$ which surjects under $\pi$ onto a smooth ray $l$ emanating from $m=\pi (p)$ with fiber over $m$ being $p$ and with all the other fibers diffeomorphic to circles disjoint from the critical points of $\pi$. Here, by a smooth ray we mean the image of a smooth embedding $[0,\infty )\to B$. We show in Lemma 7.13 that $l$ is, in fact, an eigenray of $p$ in the sense of Definition 7.4. See Figure 1. Note that $X$ might admit different nodal Lagrangian torus fibrations and each of these give rise to Lagrangian tails in their own right.

Definition 1.7 A four-dimensional symplectic cluster manifold is a symplectic 4-manifold $X$ for which there exists a nodal Lagrangian torus fibration $\pi :X\to B$ with $B$ simply connected, and a choice of pairwise disjoint Lagrangian tails $L_p$ for each critical point $p$ of $\pi$ such that the following condition holds.

• – The eigenrays $l_p:=\pi (L_p)$ are proper in the sense of Definition 7.4. Moreover, an affine geodesic starting at any point $b\in B^{\rm reg}$ in any direction can be extended indefinitely unless it converges in finite time to a point on $\bigcup _{p\in {\rm Crit} (\pi )}l_p$.

Let us refer to this condition as weak geodesic completeness.

Figure 1. A depiction of a Lagrangian tail.

For the remainder of the paper we will refer to four-dimensional symplectic cluster manifolds simply as symplectic cluster manifolds. In Proposition 7.16 we show that symplectic cluster manifolds are of geometrically finite type.

We refer to the data $P=\{\pi,\{L_p\}_{p\in {\rm Crit}(\pi )}\}$ as a cluster presentation of $X$. We say that a nodal Lagrangian torus fibration $\pi$ on $X$ is of cluster type if it is part of a cluster presentation on $X$. The multiset $\{\pi (L_p)\}_{p\in {\rm Crit}(\pi )}$ of rays in $B$ associated to a given cluster presentation, together with the integral affine structure induced by $\pi$ give rise to a combinatorial structure called an eigenray diagram in the beginning of § 7.2.Footnote ⁵ We refer to it as an eigenray presentation of $X$ and denote it by $\mathcal {R}_P$. It is shown in Proposition 7.14 that an eigenray diagram $\mathcal {R}$ gives rise to a symplectic cluster manifold $M_{\mathcal {R}}$ which is unique up to symplectomorphism (see § 7.1 for the uniqueness part). In particular, for the case where $\pi$ in a cluster representation has no singularities we have $M_0=T^*T^2$.

A given cluster manifold may possess many different eigenray presentations. These may be related by nodal slides which change the fibration $\pi$ and by $\textit {branch moves}$ which roughly amount to replacing an eigenray by its opposite (see § 7.2 for precise definition). Eigenray diagrams are the symplectic counterpart of the seeds familiar from the cluster variety literature, and one could think of a cluster representation as being analogous to that of toric models from [Reference Gross, Hacking and KeelGHK15].

The defining property of cluster varieties in algebraic geometry is the existence of a family $T_{\alpha }$ of embeddings of the algebraic torus parametrized by the set of seeds $\alpha$. The following theorem gives the symplectic counterpart of this story.

Theorem 1.9 Let $X$ be a symplectic cluster manifold, and let

\[ P=\{\pi,\{L_p\}_{p\in {\rm Crit}(\pi)}\} \]

be a cluster presentation of $X$. For any subset $S\subset {\rm Crit}(\pi )$ let $\mathcal {R}_{P\setminus S}$ be the eigenray diagram obtained from $\mathcal {R}_P$ by deleting (in the multiset description used above) the eigenrays $\pi (L_p)$ for $p\in S$. Then $X\setminus \bigcup _{p\in S} L_p$ is symplectomorphic to $M_{\mathcal {R}_{P\setminus S}}$. In particular,

(2)\begin{equation} X\setminus \bigcup_{p\in {\rm Crit}(\pi)}L_p\simeq T^*T^2. \end{equation}

The proof of Theorem 1.9 is given in § 7.5. There, we also show that we have good control over how the nodal Lagrangian torus fibrations on the two symplectic cluster manifolds relate to each other. The basic idea is explained in § 1.2.3.

Let us illustrate how Theorem 1.9 is used in an example.

Example 1.11 Consider the eigenray diagram $\mathcal {R}$ with two nodes of multiplicity one at the points $(1,0)$ and $(0,1)$ in ${\mathbb {R}}^2$ with rays going along the positive $x$- and $y$-axes, respectively. See the left-hand side of Figure 2. Let $\pi : M\to B$ be a compatible nodal Lagrangian torus fibration, i.e. in the notation of § 7.2, one of the form $\pi _\mathcal {R}: M_\mathcal {R}\to B_\mathcal {R}$. We describe $5$ symplectic embeddings of $M_0=T^*T^2$ into $M$ using the statement of Theorem 1.9. Let us denote the critical values in $B$ corresponding to $(1,0)$ and $(0,1)$ by $n_x$ and $n_y$. We then denote the eigenrays of $n_x$ and $n_y$ corresponding to the defining rays by $l_x$ and $l_y$, and the other eigenrays by $l_x^-$ and $l_y^-$. There are $4$ different ways of choosing an eigenray for each node. Three out of these four choices, namely $(l_x,l_y)$, $(l_x^-,l_y),$ and $(l_x,l_y^-)$ lead to non-intersecting eigenrays. The corresponding Lagrangian tails are clearly disjoint, and hence for each of these three cases we obtain a cluster representation of $M$ (along with $\pi$). By Theorem 1.9, we therefore obtain $3$ embeddings $M_0\to M$. For the choice $(l_x^-,l_y^-)$, however, one immediately sees that, no matter how the Lagrangian tails are chosen, they cannot be disjoint as they intersect the torus over the intersection point along cycles representing mutually independent homology classes (well defined up to sign). Indeed, under the identification of the tangent space to the base with the cohomology of the fiber, these non-zero homology classes annihilate the tangent line to the respective tail.

Figure 2. Two eigenray diagrams related by a nodal slide.

Using nodal slides we can produce two additional embeddings. As mentioned above, here we are thinking of a nodal slide operation as changing the initial nodal Lagrangian torus fibration on the underlying symplectic manifold $M$ to a new fibration $\pi ^{\rm new}:M\to B^{\rm new}$. Here $B^{\rm new}=B$ as smooth manifolds, but with a different induced nodal integral affine structure. Moreover, $B^{\rm new}$ as a nodal integral affine manifold can be described by an eigenray diagram $\mathcal {R}^{\rm new}$ obtained by applying a nodal slide to $\mathcal {R}$, which we will use in what follows. The reader might want to consult the end of § 7.3 for how this works in an example.

The first option is to slide $n_x$ in the direction of $l_x^{-}$ until in the eigenray diagram the node $(1,0)$ arrives at $(-1,0)$. See the right-hand side of Figure 2. This results in a new nodal Lagrangian torus fibration $\pi ':M\to B'$ as in the previous paragraph. Let us denote the nodes in $B'$ by $n_x'$ and $n_y'$ with eigenrays $l'_x$ and $l'_y$ corresponding to the rays of the eigenray diagram, and $(l'_x)^{-}$ and $(l'_y)^{-}$ for the others. Lagrangian tails corresponding to $l'_x, l'_y, (l'_x)^{-}$ are Hamiltonian isotopic to those of $l_x, l_y, l_x^{-}$ respectively, but crucially, that for $(l'_y)^-$ has changed drastically and it is not Hamiltonian isotopic to that for $l_y^-$.Footnote ⁷ More important for our point here, $(l'_y)^-$ does not intersect $(l'_x)^-$ and, hence, their Lagrangian tails give rise to a cluster presentation (along with $\pi '$) of $M$. This gives our fourth embedding via Theorem 1.9. We later refer to the Lagrangian tail of $(l'_y)^-$ as a scattered Lagrangian for the diagram $\mathcal {R}$.

Finally, we can apply the same procedure but this time sliding $n_y$. This produces the fifth embedding. The fact that these five embeddings are not Hamiltonian isotopic is not trivial but we give a convincing sketch of an argument in the following.

Interestingly, the two scattered Lagrangians are Hamiltonian isotopic to each other. In fact, it can be shown that they are both Hamiltonian isotopic to the tropical Lagrangian from Figure 3. Working slightly harder one can also show that iterations of the nodal slide and mutate procedure produces Lagrangian tails, which are either Hamiltonian isotopic to the initial Lagrangian tails (those of $l_x,l_y,l_x^-$ and $l_y^-$) or to the scattered Lagrangian.

Figure 3. The tropical curve drawn above in the base of the original fibration $\pi :M\to B$ gives rise to a tropical Lagrangian inside $M$. The scattered Lagrangians in Example 1.11 are both Hamiltonian isotopic to this tropical Lagrangian.

We now formulate the following conjecture inspired by the combination of mirror symmetry, Theorems 1.9 and 1.1, and the theory of the local Fukaya category being developed by Abouzaid jointly with the present authors. It is also a special case of the more general expectation (see [Reference HicksHic20] for first steps) that the support a tropical Lagrangian should be equal to the defining tropical variety.

Without loss of generality fix $i=1$. We will assume for simplicity that $\pi (L_1)$ does not contain any $\pi (L_i)$ in its interior and that the fiber in question is a smooth one. By nodal slides supported away from $\pi (L_1)$, the ray $\pi (L_1)$ can be taken to be in mutable position. Taking $L'_1$ to be an opposite tail (i.e. one that lies over the opposite ray and emanates from the same critical point), we get a new cluster presentation $\{\pi ',L'_1,L_2,\dots, L_n\}$. Taking $Y=X\setminus L_1'\cup _{i\geqslant 2}L_i$ we have that $L_1\cap Y$ is a properly embedded Lagrangian cylinder. Under the symplectomorphism (2) $Y\cap L_1$ maps to the conormal $N$ of a rational affine line $l$ in the base of the standard fibration $\pi _0:T^*{\mathbb {T}}^2\to {\mathbb {R}}^2$. It is straightforward to show that for any rational convex polygon $P\subset {\mathbb {R}}^2$ which meets $l=\pi _0(N)$ we have

(3)\begin{align} HF^*_{T^*{\mathbb{T}}^2}(N,N; \pi^{-1}(P))\neq 0. \end{align}

The left-hand side is the relative Lagrangian Floer cohomology, which was introduced in § 2.3 of [Reference Tonkonog and VarolgunesTV20]. In this case $HF^*_{T^*{\mathbb {T}}^2}(N,N; \pi ^{-1}(P))$ can be shown to be a certain completion of the wrapped Floer cohomology of $N$ which is straightforward to compute and so prove (3). It should not be hard to generalize the locality Theorem 1.1 to give an isomorphism

(4)\begin{equation} HF^*_Y(N_1\cap Y,N_2\cap Y;K)\simeq HF^*_M(N_1,N_2;K) \end{equation}

whenever $Y\subset M$ is a complete embedding, $N_1,N_2\subset M$ are properly embedded, and for a compact subset $K\subset Y$ the Lagrangians $N_1\cap Y\setminus K, N_2\cap Y\setminus K$ can be displaced a distance $\epsilon$ from each other by a Lipschitz Hamiltonian with respect to a metric determined by an almost complex structure for which $Y$ is geometrically bounded. We deduce from this that

(5)\begin{equation} HF^*_{X}(L_1,L_1; \pi^{-1}(P))\neq 0 \end{equation}

for any polygon $P\subset B$ meeting $\pi (L_1)$. Strictly speaking, this requires discussing potential obstructions, but we ignore this at the level of rigor of the current discussion. Since (5) holds for arbitrary $P\subset B$ meeting $\pi (L_1),$ Conjecture (1) follows.

We can use Conjecture 1 to produce the complete embeddings version of [Reference Shende, Treumann and WilliamsSTW16] on distinguishing exact Lagrangian tori in exact symplectic cluster manifolds. Given two complete embeddings $i_1,i_2$ of, say, $T^*T^2$ inside $X$ each obtained as in Theorem 1.9 by choosing a cluster presentation and removing Lagrangian tails we may ask whether the two complete embeddings are related by Hamiltonian isotopy. Equivalently we may ask the same question for their complements. This leads us to the question of distinguishing Lagrangian tails associated with possibly different fibrations $\pi$ up to Hamiltonian isotopy.

Conjecture 1 has the following immediate corollary.

This corollary can be applied to distinguish complete embeddings. We expand on how this works in Example 1.11. In this case the corollary shows that the four initial Lagrangian tails corresponding to $l_x^\pm,l_y^\pm$ together with the scattered Lagrangian are pairwise Hamiltonian non-isotopic. Indeed, for the initial rays this is an immediate consequence of Conjecture 1. We now consider a pair consisting of an initial tail and the scattered one. According to Proposition 7.8 we can take the nodal slide involved in producing the scattered ray to be supported on arbitrarily small neighborhood of the segment containing the intersection point and one of the nodes. Thus, $V$ in the corollary can be taken to be the complement of the closure of this neighborhood and the result follows.

1.2.2 Some three-dimensional examples

The ideas presented so far are not limited to dimension $4$. There are open symplectic manifolds in any even dimension for which analogues to Theorem 1.9 hold. Here we content ourselves by analyzing one of the two classes of six-dimensional analogues to a symplectic cluster manifold with global $S^1$ symmetry (e.g. one with a single node). Specifically, we consider symplectic $3$-manifolds carrying Lagrangian torus fibrations $\mathcal {L}:M\to {\mathbb {R}}^3$ which:

• – have global ${\mathbb {T}}^2$ symmetry generated by the last two coordinate functions $x_1$ and $x_2$;

• – satisfy an appropriate completeness condition; and

• – has singular values along a one-dimensional graph $\Delta \subset {\mathbb {R}}^3$ which lies in the plane $x_0=0$.

For a complete definition of the class we consider see Definitions 8.1 and 8.4. Important examples of this geometric setup is given by Gross fibrations from [Reference GrossGro01, Theorem 2.4]. We describe in detail two special cases of Gross’ construction in Examples 8.2 and 8.3.

Let $P^{\pm }= {\mathbb {R}}_{\pm }\times \Delta \subset {\mathbb {R}}^3$. Let $C^{\pm }\subset M$ be a ${\mathbb {T}}^2$ invariant lift of $P^{\pm }$. Note that $C^{\pm }$ is a stratified coisotropic subspace. The pair of spaces $C^{\pm }$ are the analogues of the pair of Lagrangian tails emanating from the singular point in the symplectic cluster manifold with a single node. We prove the following theorem in § 8.

We briefly comment on the significance of this result in relation to the locality theorem. The Gross fibrations have been studied from the point of view of Strominger–Yau–Zaslow (SYZ) mirror symmetry in [Reference GrossGro01, Reference Abouzaid, Auroux and KatzarkovAAK16, Reference Chan, Lau and LeungCLL12]. It has been shown that, up to real codimension 4, the SYZ mirror is a conic bundle which is glued together from a pair of algebraic tori $(\Lambda ^*)^3$ over the Novikov field according to some wall crossing formula. Theorem 1.14 together with our locality theorem gives a reinterpretation of this SYZ construction using relative symplectic cohomology. Explaining this further is outside the scope of the present paper.

1.2.3 Basic idea

The method we use to prove the embedding theorems above is an anti-surgery operation on symplectic manifolds equipped with nodal Lagrangian submersions. We can also define a surgery operation, which we develop in § 7.4 and give applications, but our focus is on the anti-surgery operation in the introduction.

As a toy version of the anti-surgery operation we describe in the following, the reader may consider the case of removing the ray $\gamma ={\mathbb {R}}_{\geqslant 0}\times \{1\}$ from the cylinder ${\mathbb {R}}\times S^1$. The operation would start with the standard Lagrangian circle fibration ${\mathbb {R}}\times S^1\to {\mathbb {R}}$, produce a Lagrangian circle fibration ${\mathbb {R}}\times S^1\setminus \gamma \to {\mathbb {R}}$ so that the integral affine structure induced on ${\mathbb {R}}$ is the standard one, and hence, in particular, prove that ${\mathbb {R}}\times S^1\setminus \gamma$ is symplectomorphic to ${\mathbb {R}}\times S^1$.

To explain the basic idea in the context of symplectic cluster manifolds, we focus on the embedding described in (2) of Theorem 1.9 in the $S={\rm Crit}(\pi )$ case. The procedure starts with a nodal Lagrangian torus fibration $\pi :X\to B$ that is part of a cluster presentation and modifies it in a neighborhood $U_p$ of each Lagrangian tail $L_p$. The end result is a new Lagrangian torus fibration

\[ \pi^{\rm new}:X\setminus \bigcup_{p\in {\rm Crit}(\pi)}L_p\to B \]

which is:

• – proper;

• – has no singularities; and, crucially,

• – induces on $B$ an integral affine structure which is isomorphic to the standard one on ${\mathbb {R}}^2$.

We describe this now in more detail. We show in Proposition 7.14 that we may identify the base $B$ with ${\mathbb {R}}^2$ so that with respect to the induced integral affine structure, the identity map is a piecewise linear (PL) homeomorphism which is an affine isomorphism on the complement of the projections $l_p=\pi (L_p)$ of the tails. For simplicity, we assume that the rays of the form $l_p$ for $p\in {\rm Crit}(\pi )$ are pairwise disjoint. We focus on one Lagrangian tail at a time. A local model for the fibration in a neighborhood of such a tail is as follows. Consider $\mathbb {C}^2$ with its standard Kahler structure and denote the complex coordinates by $z_1$ and $z_2$. The Hamiltonian function $\mu (z_1,z_2):=\pi (|z_1|^2-|z_2|^2)$ generates a $S:=\mathbb {R}/\mathbb {Z}$ action on $\mathbb {C}^2$ by $\theta \cdot (z_1,z_2)=(e^{2\pi i\theta }z_1, e^{-2\pi i\theta }z_2)$. We have a smooth map ${\it Hopf}: \mathbb {C}^2\to \mathbb {C} \times \mathbb {R}$ defined by

(6)\begin{equation} (z_1,z_2)\mapsto \big(2\pi z_1z_2, \mu(z_1,z_2)\big). \end{equation}

The fibers of ${\it Hopf}$ are precisely the orbits of the $S$-action. Let

(7)\begin{equation} \pi^{\rm st}=({\rm Re}(z_1z_2),\mu(z_1,z_2)), \end{equation}

and let

(8)\begin{equation} L^{\rm st}:=\{{\rm Re}(z_1z_2)\geqslant 0\}\cap \{{\rm Im}(z_1z_2)= 0\}\cap \{|z_1|^2-|z_2|^2=0\}. \end{equation}

Then there is an open neighborhood $W_p$ of $L_p$, an $S$-equivariant neighborhood $W^{\rm st}$ of $L^{\rm st}$, and a symplectomorphism $\psi :W_p\to W^{\rm st}$ mapping $L_p$ to $L^{\rm st}$ and intertwining $\pi$ with $\pi ^{\rm st}$.

We now make the following crucial observation: if $f:{\mathbb {C}}\times {\mathbb {R}}\setminus Q\to {\mathbb {R}}$ is a smooth submersion with $Q$ a closed subset containing $(0,0)$, then the fibers of the map $(f\circ {\it Hopf},\mu ):{\mathbb {C}}^2\setminus {\it Hopf}^{-1}(Q)\to {\mathbb {R}}^2$ are Lagrangian submanifolds.

We apply this as follows. Let $l:={\it Hopf}(L^{\rm st})$ and let $U={\it Hopf}(W^{\rm st})$. By being a little careful in the choice of $W^{\rm st}$ we can ensure that there is a diffeomorphism $\Phi : U-l\to U$ which preserves the projection to $\mathbb {R}$ and is the identity near the boundary of $U$. Let $h:{\mathbb {C}}\times {\mathbb {R}}\to {\mathbb {R}}$ be the map $(z,x)\mapsto {\rm Re} (z)$. Then the map

\[ \pi':=(h\circ \Phi\circ {\it Hopf},\mu):W^{\rm st}\setminus L^{\rm st}\to U \]

is a Lagrangian submersion with no singularities. Moreover, $\pi '=\pi ^{\rm st}$ near the boundary of $W_p$.

We may thus define $\pi ^{\rm new}|_{W_p}=\pi '\circ \psi$ for each of the critical points of $\pi$, and $\pi ^{\rm new}=\pi$ on $X\setminus \bigcup _{p\in {\rm Crit}(\pi )}W_p$. The resulting Lagrangian submersion is readily seen to be proper. A graphic illustration of the modification of the foliation near one of the Lagrangian tails is given in Figure 4. The result is a proper Lagrangian submersion $\pi ^{\rm new}:X^{\rm new}:=X\setminus \bigcup _{p\in {\rm Crit}(\pi )} L_p\to B$.

Figure 4. The foliation before (top) and after (bottom) an anti-surgery (along a Lagrangian tail) on the reduced space containing the $S$ quotient of the tail (depicted as the smooth ray going to infinity with an arrow on it). Note that at the bottom image, this ray has been removed.

To conclude, we need to show that the integral affine structure induced on $B$ by $\pi ^{\rm new}$ via the Arnold–Liouville theorem makes it isomorphic to ${\mathbb {R}}^2$ with its standard integral affine structure. To proceed, recall that $L_p$ projects under $\pi$ to an eigenray $l_p$, i.e. an affine ray whose direction is fixed under the integral affine monodromy around $\pi (p)$. In fact, there is an integral affine coordinate $i:\pi (W_p)\to {\mathbb {R}}$, such that $i$ generates the local $S$-action and such that $\pi (L_p)$ is contained in a level set of $i$. The function $i$ is determined by the equation $i\circ \pi =\mu \circ \psi _p$. Without loss of generality we choose $W_p\subset B$ so that the non-empty level sets of $i$ are affine isomorphic to half infinite intervals.

The modifications do not affect the integral affine structure outside of the neighborhoods $\pi (W_p)$, so we focus on analyzing what happens in the neighborhoods $\pi (W_p)$. Note that $i\circ \pi ^{\rm new}|_{W_p\setminus L_p}=i\circ \pi _{W_p\setminus L_p}$ generates a $1$-periodic flow and so is still an integral affine coordinate with respect to $\pi ^{\rm new}$. Now let us show that, with respect to $\pi ^{\rm new}$, the non-empty level sets of $i$ are integral affine isomorphic to half infinite intervals. Equivalently, we need to show that if $\gamma$ is any simple loop in a fiber of $\pi ^{\rm new}$ which is transverse to the $S$-orbits, then the cylinder $C$ traced out by transporting $\gamma$ along any non-empty level set $i^{-1}(c)$ has infinite area. This is readily seen by comparing the area of such a cylinder $C$ with that of a cylinder $C'$ traced by transporting $\gamma$ along fibers of the original fibration $\pi$ over the same level set $i^{-1}(c)$. Figure 4 extended to a larger portion of the reduced space so as to include some of the unchanged fibers could help visualizing this. The effect of the modification in the neighborhood $W_p$ is thus to make it affine isomorphic to the corresponding neighborhood in ${\mathbb {R}}^2$. Doing this for all the tails, the integral affine structure induced by $\pi ^{\rm new}$ on $B$ becomes isomorphic to ${\mathbb {R}}^2$.

Let us also discuss the application of the surgery operation that we have in mind. Recall that a Looijenga interior $U$ is a log Calabi–Yau surface with maximal boundary. By [Reference Gross, Hacking and KeelGHK15], $U$ admits a toric model. Fixing a toric model, we obtain a decomposition of $U$ into an open dense $(\mathbb {C}^*)^2$ and some disjoint union of exceptional curves coming from the Kaliman modifications. Assume that the holomorphic volume form $\Omega$ of $U$ is normalized so that it is equal to $d\log (x)\wedge d\log (y)$ on $(\mathbb {C}^*)^2$. Using integrable surgery one can prove the following result, which could be of independent interest.

The sketch of the proof is given in Remark 7.32 in the first non-trivial example. Note that with this symplectic form the exceptional curves become Lagrangians. In the case exceptional curves are all complex planes the theorem follows immediately from doing integrable surgery to the standard Lagrangian torus fibration on

\[ ((\mathbb{C}^*)^2, {\rm Im}(\Omega))=(T^*T^2, d\lambda_{\rm taut}) \]

along the exceptional curves. When some of the exceptional curves are a chain of ${\mathbb {C}} P^1$ followed by a complex plane, we can modify them to pairwise disjoint Lagrangian tails emanating from each of the double points in the exceptional curves or use a straightforward generalization of the integrable surgery.

In [Reference Bardwell-Evans, Cheung, Hong and LinBECHL21, Lemma 4.3], which came out days after the first version of our preprint appeared, the authors construct special Lagrangian fibrations $\pi '_\epsilon :U\to B$ on $(U,\Omega )$ for a family of Kahler forms $\omega _\epsilon$. In particular, the $\pi '_\epsilon$ are all nodal Lagrangian torus fibrations with respect to ${\rm Im}(\Omega )$. This appears to be an alternative construction to that from Theorem 1.15. It will be interesting to compare the two approaches more thoroughly in the future.