3 Affine manifolds and Lagrangian fibrations [04HW]

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3 Affine manifolds and Lagrangian fibrations

Let us denote by $\aff(\mathbb{R})=\mathbb{R}^{n}\rtimes\Gl(n,\mathbb{R})$ the group of affine linear transformations, i.e. elements in $\aff(\mathbb{R})$ are maps $A:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ , $A(x)=L(x)+v$ , where $L\in\Gl(n,\mathbb{R})$ and $v\in\mathbb{R}^{n}$ . The subgroup of $\aff(\mathbb{R})$ consisting of affine linear transformations with integral linear part will be denoted by:

\aff_{\mathbb{R}}(\mathbb{Z})=\mathbb{R}^{n}\rtimes\Gl(n,\mathbb{Z}).

Let us denote by

\aff(\mathbb{R}^{n},\mathbb{R}^{n^{\prime}})=\mathbb{R}^{n^{\prime}}\times\Hom(\mathbb{R}^{n},\mathbb{R}^{n^{\prime}})

and by

\aff_{\mathbb{R}}(\mathbb{Z}^{n},\mathbb{Z}^{n^{\prime}})=\mathbb{R}^{n^{\prime}}\times\Hom(\mathbb{Z}^{n},\mathbb{Z}^{n^{\prime}}).

Definition 3.1.

Let $B$ be a topological $n$ -dimensional manifold.

(i)

An affine manifold is a pair $(B,\mathscr{A})$ where $B$ is an $n$ -dimensional manifold and $\mathscr{A}$ is a maximal atlas on $B$ whose transition maps are $\aff(\mathbb{R})$ transformations. We call $\mathscr{A}$ an affine structure on $B$ .
(ii)

An affine manifold $(B,\mathscr{A})$ is integral if the transition maps of the affine structure $\mathscr{A}$ are $\aff_{\mathbb{R}}(\mathbb{Z})$ transformations. We call $\mathscr{A}$ an integral affine structure on $B$ .
(iii)

A continuous map $\alpha:B\rightarrow B^{\prime}$ is (integral) affine if on each local coordinate chart, $\alpha$ is an element of ( $\aff_{\mathbb{R}}(\mathbb{Z}^{n},\mathbb{Z}^{n^{\prime}})$ ) $\aff(\mathbb{R}^{n},\mathbb{R}^{n^{\prime}})$ . Two (integral) affine manifolds $B$ and $B^{\prime}$ are said to be (integral) affine isomorphic if there is an (integral) affine homeomorphism between them.

It is becoming standard to call an affine manifold as in (ii) tropical manifold [9]. Though convenient for various good reasons, this is not a well established terminology at the time this paper is being written, so we prefer to stick to definition (ii) instead. Our convention coincides with that in [23] and [16] and differs from [13]. Affine manifolds whose structure group is $\aff(\mathbb{Z})=\mathbb{Z}^{n}\rtimes\Gl(n,\mathbb{Z})$ will be denoted $\aff(\mathbb{Z})$ -manifolds (these are called integral affine in [13]).

Given an affine manifold $(B,\mathscr{A})$ , consider a chart $(U,\phi)\in\mathscr{A}$ with affine coordinates $\phi=(u_{1},\ldots,u_{n})$ . The cotangent bundle $T^{\ast}_{B}$ admits a flat connection $\nabla$ defined by

\nabla du_{j}=0,

for all $j=1,\ldots,n$ and all charts $(U,\phi)\in\mathscr{A}$ . When $(B,\mathscr{A})$ is integral affine we can also define a maximal integral lattice $\Lambda\subset T^{\ast}_{B}$ by

\Lambda|_{U}=\spn_{\mathbb{Z}}\langle{du_{1},\ldots},{du_{n}}\rangle

for all $(U,\phi)\in\mathscr{A}$ . Therefore to every integral affine manifold $(B,\mathscr{A})$ we can associate the $2n$ -dimensional manifold

X(B,\mathscr{A})=T^{\ast}_{B}/\Lambda,

which together with the projection $f:X(B,\mathscr{A})\rightarrow B$ forms a $T^{n}$ fibre bundle. Also notice that the standard symplectic form $\omega$ on $T^{\ast}_{B}$ descends to $X(B,\mathscr{A})$ and the fibres of $f$ are Lagrangian.

The flat connection $\nabla$ on $T^{\ast}_{B}$ of an integral affine manifold $(B,\mathscr{A})$ has a holonomy representation $\rho^{\ast}:\pi_{1}(B,b)\rightarrow\Gl(n,\mathbb{Z})$ obtained by parallel transport along closed paths. A choice of basis of $\Lambda_{b}$ is identified naturally with a choice of basis of $H_{1}(f^{-1}(b),\mathbb{Z})$ . Under this identification, the holonomy representation $\rho^{\ast}$ coincides with the monodromy representation of the bundle $X(B,\mathscr{A})\rightarrow B$ . More precisely, if $g\in\pi_{1}(B,b)$ and $\mathcal{M}_{b}(g)$ is the corresponding monodromy matrix, then $\mathcal{M}_{b}(g)=\rho^{\ast}(g)$ . The integral affine manifold $(B,\mathscr{A})$ also induces a flat connection on $T_{B}$ whose holonomy representation, $\rho$ , is dual to $\rho^{\ast}$ , i.e. the matrix $\rho(g)$ is the inverse transpose of $\rho^{\ast}(g)$ . In what follows, unless otherwise stated, “holonomy representation” should be understood as the holonomy representation of the aforementioned flat connection on the cotangent bundle $T^{\ast}_{B}$ .

It is well known that affine manifolds arise naturally from Lagrangian fibrations. This is the classical theory of action-angle coordinates in Hamiltonian mechanics.

Action-angle coordinates.

We review here some standard facts about Lagrangian fibrations which we will use in the next Sections. For details we refer to Duistermaat [4]. Assume we are given a $2n$ -dimensional symplectic manifold $X$ with symplectic form $\omega$ , a smooth $n$ -dimensional manifold $B$ and a proper smooth submersion $f:X\rightarrow B$ whose fibres are connected Lagrangian submanifolds. For every $b\in B$ , denote by $F_{b}$ the fibre of $f$ at $b$ .

Proposition 3.2 (Arnold-Liouville).

In the above situation, for every $b\in B$ , $T^{\ast}_{b}B$ acts transitively on $F_{b}$ . In particular there exists a maximal sub-lattice $\Lambda_{b}$ of $T^{\ast}_{b}B$ such that $F_{b}$ is naturally diffeomorphic to $T^{\ast}_{b}B/\Lambda_{b}$ , therefore $F_{b}$ is an $n$ -torus.

Proof.

To every $\alpha\in T^{\ast}_{b}B$ we can associate a vector field $v_{\alpha}$ on $F_{b}$ by

\iota_{v_{\alpha}}\omega=f^{\ast}\alpha.

Let $\phi_{\alpha}^{t}$ be the flow of $v_{\alpha}$ with time $t\in\mathbb{R}$ . Then we define the action $\theta_{\alpha}$ of $\alpha$ on $F_{b}$ by

\theta_{\alpha}(p)=\phi_{\alpha}^{1}(p),

where $p\in F_{b}$ . One can check that such an action is well defined and transitive. Then, $\Lambda_{b}$ defined as

\Lambda_{b}=\{\lambda\in T^{\ast}_{b}B\ |\ \theta_{\lambda}(p)=p,\ \text{for all}\ p\in F_{b}\}

is a closed discrete subgroup of $T^{\ast}_{b}B$ , i.e. a lattice. From the properness of $F_{b}$ it follows that $\Lambda_{b}$ is maximal (in particular homomorphic to $\mathbb{Z}^{n}$ ) and that $F_{b}$ is diffeomorphic to $T^{\ast}_{b}B/\Lambda_{b}$ . ∎

We denote $\Lambda=\cup_{b\in B}\Lambda_{b}$ . Given the presheaf on $B$ defined by $U\mapsto H_{1}(f^{-1}(U),\mathbb{Z})$ , the associated sheaf is a locally constant sheaf. We can identify it with $\Lambda$ as follows. Let $U\subseteq B$ be a contractible open set. For every $b\in U$ , $H_{1}(F_{b},\mathbb{Z})$ can be naturally identified with $H_{1}(f^{-1}(U),\mathbb{Z})$ . To every $\gamma\in H_{1}(f^{-1}(U),\mathbb{Z})$ , we can associate a $1$ -form $\lambda$ on $U$ as follows. For every vector field $v$ on $U$ , if we denote by $\tilde{v}$ a lift, define

\lambda(v)=-\int_{\gamma}\iota_{\tilde{v}}\omega.

(4)

It turns out that this identifies the above sheaf with $\Lambda\subset T^{\ast}_{B}$ . If $\gamma_{1},\ldots,\gamma_{n}$ are a basis for $H_{1}(F_{b},\mathbb{Z})$ , then (4) gives us a $\mathbb{Z}$ -basis $\lambda_{1},\ldots,\lambda_{n}$ of $\Lambda$ over a contractible open neighborhood $U$ of $b$ .

In particular, one can read the monodromy of $f:X\rightarrow B$ from the monodromy of $\Lambda$ . We state now the fundamental theorem of smooth proper Lagrangian submersions:

Theorem 3.3 (Duistermaat).

Given a basis $\{\gamma_{1},\ldots,\gamma_{n}\}$ of $H_{1}(F_{b},\mathbb{Z})$ , then the corresponding 1-forms $\lambda_{1},\ldots,\lambda_{n}$ defined on a contractible open neighborhood $U$ of $b$ are closed and locally generate $\Lambda$ . In particular, $\Lambda$ is Lagrangian with respect to the standard symplectic structure in $T^{\ast}_{B}$ . A choice of functions $a_{j}$ such that $\lambda_{j}=da_{j}$ defines coordinates $a=(a_{1},\ldots,a_{n})$ called action coordinates. A covering $\{U_{i}\}$ of $B$ by contractible open sets and a choice of action coordinates on each $U_{i}$ defines an integral affine structure $\mathscr{A}$ on $B$ . Moreover, if $f$ has a Lagrangian section $\sigma:U\rightarrow X$ over an open set $U\subseteq B$ , then there is a natural symplectomorphism

\Theta:T^{\ast}_{U}/\Lambda\rightarrow f^{-1}(U).

(5)

If $\sigma$ is a global section then $X(B,\mathscr{A})$ is symplectically conjugate to $X$ . If in addition the monodromy of $\Lambda$ is trivial $X$ is symplectically conjugate to $B\times T^{n}$ . The map $\Theta$ is called the period map or action-angle coordinates map.

Proof.

We just give a sketch of the proof. Using the Weinstein neighborhood theorem one can show that in a sufficiently small tubular neighborhood of a fibre $F_{b}$ , the symplectic form is exact, i.e $\omega=-d\eta$ for some 1-form $\eta$ . Notice that $\eta|_{F_{b}}$ is a closed 1-form. Define functions $a_{j}$ on $U$ by

a_{j}=\int_{\gamma_{j}}\eta.

One can show that

\lambda_{j}=da_{j}

and therefore $\lambda_{j}$ is closed. It is clear that the coordinates $a=(a_{1},...,a_{n})$ are well defined up to an integral affine transformation and therefore they define an integral affine structure on $B$ inducing the lattice $\Lambda$ in $T^{\ast}_{B}$ . Finally, notice that given a section $\sigma:U\rightarrow X$ we have a covering map

\begin{array}[]{rcl}T^{\ast}_{U}&\rightarrow&f^{-1}(U)\\ \alpha&\mapsto&\theta_{\alpha}(\sigma(\pi(\alpha)))\end{array}

This map induces a diffeomorphism between $T^{\ast}_{U}/\Lambda$ and $f^{-1}(U)$ . One can check that in the case $\sigma$ is Lagrangian this map is a symplectomorphism. For the proof of the last statement we refer the reader to [4]. ∎

Corollary 3.4.

Let $\mathcal{F}=(X,f,B)$ and $\mathcal{F}^{\prime}=(X^{\prime},f^{\prime},B^{\prime})$ be smooth proper Lagrangian fibrations inducing integral affine structures $\mathscr{A}$ and $\mathscr{A}^{\prime}$ on $B$ and $B^{\prime}$ respectively. Assume there exist Lagrangian sections $\sigma$ and $\sigma^{\prime}$ of $f$ and $f^{\prime}$ respectively. Then an integral affine isomorphism $\phi$ between $B$ and $B^{\prime}$ induces a symplectic $(\psi,\phi)$ -conjugation between $\mathcal{F}$ and $\mathcal{F}^{\prime}$ such that $\psi\circ\sigma=\sigma^{\prime}\circ\phi$ .

Proof.

Let $\Lambda\subset T^{\ast}_{B}$ and $\Lambda^{\prime}\subset T^{\ast}_{B^{\prime}}$ be the lattices induced from the integral affine structures on $B$ and $B^{\prime}$ , respectively. From Theorem 3.3 it follows that $X$ and $X^{\prime}$ are symplectomorphic to $T^{\ast}_{B}/\Lambda$ and $T^{\ast}_{B^{\prime}}/\Lambda^{\prime}$ , respectively. Given an integral affine isomorphism $\phi$ between $B$ and $B^{\prime}$ , clearly $\phi^{\ast}$ is a symplectomorphism between $T^{\ast}_{B^{\prime}}$ and $T^{\ast}_{B}$ inducing an isomorphism between $\Lambda^{\prime}$ and $\Lambda$ . Therefore $\phi^{\ast}$ descends to a symplectomorphism $\tilde{\psi}$ between $T^{\ast}_{B^{\prime}}/\Lambda^{\prime}$ and $T^{\ast}_{B}/\Lambda$ . Defining $\psi=\Theta^{\prime}\circ(\tilde{\psi})^{-1}\circ\Theta^{-1}$ the claim follows. ∎

The following is an easy but important consequence of Arnold-Liouville-Duistermaat theorem:

Corollary 3.5.

Proper Lagrangian submersions do not have semi-global symplectic invariants. In other words, all such fibrations are symplectically conjugate to $U\times T^{n}$ when restricted to a small enough neighborhood $U$ of a base point.

It is clear that smoothness of the fibration map plays a crucial role in the above result. Semi-global invariants do arise for certain piecewise $C^{\infty}$ Lagrangian fibrations [2]. We say more about this in §6.

Affine manifolds with singularities.

When a Lagrangian fibration has singular fibres, its base is no longer an affine manifold but an affine manifold with singularities. These singularities can be a priori rather complicated. The topological properties described in §2 motivate the following:

Definition 3.6.

An (integral) affine manifold with singularities is a triple $(B,\Delta,\mathscr{A})$ , where $B$ is a topological $n$ -dimensional manifold, $\Delta\subset B$ a set which is locally a finite union of locally closed submanifolds of codimension at least $2$ and $\mathscr{A}$ is an (integral) affine structure on $B_{0}=B-\Delta$ . A continuous map between (integral) affine manifolds with singularities

\alpha:B\rightarrow B^{\prime}

is (integral) affine if $\alpha^{-1}(B_{0}^{\prime})\cap B_{0}$ is dense in $B$ and the restriction $\alpha_{0}=\alpha|_{\alpha^{-1}(B_{0}^{\prime})\cap B_{0}}$ :

\alpha_{0}:\alpha^{-1}(B_{0}^{\prime})\cap B_{0}\rightarrow B_{0}^{\prime}

is an (integral) affine map. We say that $\alpha$ is an (integral) affine isomorphism if $\alpha$ is an homeomorphism and $\alpha_{0}$ is an (integral) affine isomorphism of (integral) affine manifolds.

From now on we restrict to dimension $n=2$ or $3$ . Let $(B,\Delta,\mathscr{A})$ be an affine manifold with singularities and let $(B_{0},\mathscr{A})$ be the corresponding affine manifold. Let $X(B_{0},\mathscr{A})$ be the Lagrangian $T^{n}$ bundle over $B_{0}$ as introduced at the beginning of this section. We shall start imposing conditions on the singularities of the affine structure which, in particular, will imply that $X(B_{0},\mathscr{A})$ is of the topological type described in §2, e.g. such that $X(B_{0},\mathscr{A})$ will have semi-stable monodromy as in Theorem 2.11.

We start defining local models of integral affine manifolds with singularities. In dimension 2, the allowed behavior is described in the following:

Example 3.7 (The node).

We define an affine structure with singularities on $B=\mathbb{R}^{2}$ . Let $\Delta=\{0\}$ and let $(x_{1},x_{2})$ be the standard coordinates on $B$ . As the covering $\{U_{i}\}$ of $B_{0}=\mathbb{R}^{2}-\Delta$ we take the following two sets

U_{1}=\mathbb{R}^{2}-\{x_{2}=0\ \text{and}\ x_{1}\geq 0\},

U_{2}=\mathbb{R}^{2}-\{x_{2}=0\ \text{and}\ x_{1}\leq 0\}.

Denote by $H^{+}$ the set $\{x_{2}>0\}$ and by $H^{-}$ the set $\{x_{2}<0\}$ . Let $T$ be the matrix

T=\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right).

(6)

The coordinate maps $\phi_{1}$ and $\phi_{2}$ on $U_{1}$ and $U_{2}$ are defined as follows

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\mathrm{Id}$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\mathrm{Id}&\text{on}\ \bar{H}^{+}\cap U_{2},\\ (T^{-1})^{t}&\text{on}\ H^{-}\end{array}\right.$

The atlas $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ is clearly an affine structure on $B_{0}$ . It is easy to check that given a point $b\in B_{0}$ , we can chose a basis of $T^{\ast}_{b}B_{0}$ with respect to which the holonomy representation $\rho^{\ast}$ sends the anti-clockwise oriented generator of $\pi_{1}(B_{0})$ to the matrix $T$ .

In dimension $3$ we have the following models.

Example 3.8 (The edge).

Let $I\subseteq\mathbb{R}$ be an open interval. Consider $B=\mathbb{R}^{2}\times I$ and $\Delta=\{0\}\times I$ . On $B_{0}=(\mathbb{R}^{2}-0)\times I$ we take the product affine structure between the affine structure on $\mathbb{R}^{2}-0$ described in the previous example and the standard affine structure on $I$ .

Example 3.9 (A variation).

In the previous example the discriminant locus $\Delta$ was a straight line. We can slightly perturb $\Delta$ so that it becomes a smooth curve. More precisely, let $B=\mathbb{R}^{2}\times I$ as before and consider a smooth function $\tau:I\rightarrow\mathbb{R}$ . Let

\Delta_{\tau}=\{(\tau(s),0,s),s\in I\}\subset B

and define a covering $\{U_{i}\}$ of $B_{0}=B-\Delta_{\tau}$ to be

U_{1}=(\mathbb{R}^{2}\times I)-\{(x_{1},0,s)\ |\ x_{1}\geq\tau(s)\},

U_{2}=(\mathbb{R}^{2}\times I)-\{(x_{1},0,s)\ |\ x_{1}\leq\tau(s)\}.

Now let $H^{+}=\{x_{2}>0\}$ and $H^{-}=\{x_{2}<0\}$ . Take the following matrix

T=\left(\begin{array}[]{ccc}1&0&0\\ 1&1&0\\ 0&0&1\end{array}\right)

and define maps $\phi_{j}$ on $U_{j}$ to be

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\mathrm{Id}$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\mathrm{Id}&\text{on}\ \bar{H}^{+}\cap U_{2},\\ (T^{-1})^{t}&\text{on}\ H^{-}.\end{array}\right.$

Clearly $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ defines an affine structure on $B_{0}=B-\Delta_{\tau}$ . When $\tau=0$ , this example coincides with the previous one. Notice that the curve $(\tau(s),0,s)$ is contained inside the $2$ -plane $\{x_{2}=0\}$ , which can be viewed as an integral surface of the distribution spanned by the vectors in $TB_{0}$ which are invariant with respect to the holonomy representation $\rho$ on $TB_{0}$ . Two different curves give non-isomorphic singular affine structures, unless the curves can be taken one into the other via an integral affine transformation.

Example 3.10 (Positive vertex).

Take $B=\mathbb{R}\times\mathbb{R}^{2}$ , with coordinates $(x_{1},x_{2},x_{3})$ and identify $\mathbb{R}^{2}$ with $\{0\}\times\mathbb{R}^{2}$ . Inside $\mathbb{R}^{2}$ consider the cone over three points:

\Delta=\{x_{2}=0,\,x_{3}\leq 0\}\cup\{x_{3}=0,\,x_{2}\leq 0\}\cup\{x_{2}=x_{3},\,x_{3}\geq 0\}.

Now define closed sets in $B$

$\displaystyle R$	$\displaystyle=$	$\displaystyle\mathbb{R}\times\Delta,$
$\displaystyle R^{+}$	$\displaystyle=$	$\displaystyle\mathbb{R}_{\geq 0}\times\Delta,$
$\displaystyle R^{-}$	$\displaystyle=$	$\displaystyle\mathbb{R}_{\leq 0}\times\Delta,$

and consider the following cover $\{U_{i}\}$ of $\mathbb{R}^{3}-\Delta$ :

	$\displaystyle U_{1}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-R^{+},$
	$\displaystyle U_{2}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-R^{-}.$

It is clear that $U_{1}\cap U_{2}$ has the following three connected components

$\displaystyle V_{1}$	$\displaystyle=$	$\displaystyle\{x_{2}<0,\ x_{3}<0\},$
$\displaystyle V_{2}$	$\displaystyle=$	$\displaystyle\{x_{2}>0,\ x_{2}>x_{3}\},$
$\displaystyle V_{3}$	$\displaystyle=$	$\displaystyle\{x_{3}>0,\ x_{3}>x_{2}\}.$

Take two matrices

T_{1}=\left(\begin{array}[]{ccc}1&1&0\\ 0&1&0\\ 0&0&1\end{array}\right),\ \ \ T_{2}=\left(\begin{array}[]{ccc}1&0&-1\\ 0&1&0\\ 0&0&1\end{array}\right).

(9)

Now on $U_{1},U_{2}$ we define coordinate maps $\phi_{1}$ , $\phi_{2}$ as follows

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\I,$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\I&\text{on}\ \bar{V}_{1}\cap U_{2},\\ T_{1}^{-1}&\text{on}\ \bar{V}_{2}\cap U_{2}\\ T_{2}&\text{on}\ \bar{V}_{3}\cap U_{2}\end{array}\right.$

Again we see that $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ gives an affine structure on $B_{0}=\mathbb{R}^{3}-\Delta$ . One can compute that given a point $b\in B_{0}$ and closed paths $g_{1}$ , $g_{2}$ and $g_{3}$ as in Figure 3, we can choose a basis of $T^{\ast}_{b}B_{0}$ with respect to which the holonomy matrices satisfy $\rho^{\ast}(g_{j})=(T_{j}^{-1})^{t}$ for $j=1,2,3$ .

Example 3.11 (A variation).

In the previous example, $\Delta$ was a graph with three edges meeting in one vertex. All three edges were straight lines. In the spirit of Example 3.9 we can perturb each edge of $\Delta$ to a smooth curve starting at the vertex. Each straight edge of the previous example is contained in a $2$ -plane which is an integral plane of the distribution spanned by the vectors which are invariant with respect to the holonomy around that edge. For example, consider the edge $E_{1}=\{x_{1}=x_{2}=0,x_{3}\leq 0\}$ of $\Delta$ . Then $E_{1}$ is contained inside the half plane, $P_{1}=\{x_{2}=0,x_{3}\leq 0\}$ , whose tangent vectors are $T_{1}$ invariant, where $T_{1}=\rho(g_{1})$ is the holonomy of $T_{B_{0}}$ with respect to $E_{1}$ . An analogous thing happens with the other two edges. The union of all three half planes gives $R$ . The new perturbed edges, $E_{j}^{\prime}$ , must be curves inside the half planes $P_{j}$ . More precisely, let $\tau$ be a function on $\Delta$ which is the restriction of a smooth function defined on an open neighborhood of $\Delta$ , such that $\tau(0)=0$ . If we let $R$ be as in the previous example, define

$\displaystyle\Delta_{\tau}$	$\displaystyle=$	$\displaystyle\{(\tau(q),q)\in\mathbb{R}\times\Delta\}$
$\displaystyle R^{+}$	$\displaystyle=$	$\displaystyle\{(x_{1},q)\in\mathbb{R}\times\Delta\ \|x_{1}\geq\tau(q)\}$
$\displaystyle R^{-}$	$\displaystyle=$	$\displaystyle\{(x_{1},q)\in\mathbb{R}\times\Delta\ \|x_{1}\leq\tau(q)\}$

Now charts $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ on $B-\Delta_{\tau}$ can be defined like in the previous example, but with these new definitions of $R^{+}$ and $R^{-}$ . It is clear that $(B,\Delta_{\tau},\mathscr{A})$ defines an affine manifold with singularities. Two different choices of functions $\tau$ define non-isomorphic integral affine manifolds with singularities, unless their graphs inside $R$ can be mapped one to the other via an integral affine map.

Example 3.12 (Negative vertex).

Let $B$ and $\Delta$ be as in Example 3.10. Clearly, $\mathbb{R}^{2}-\Delta$ has three connected components, which we denote $C_{1},C_{2}$ and $C_{3}$ . Let $\bar{C}_{j}=C_{j}\cup\partial C_{j}$ . Viewing $\mathbb{R}^{2}$ embedded in $B$ as $\{0\}\times\mathbb{R}^{2}$ , consider the following three open subsets of $B_{0}$ :

$\displaystyle U_{1}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-(\bar{C}_{2}\cup\bar{C}_{3}),$
$\displaystyle U_{2}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-(\bar{C}_{1}\cup\bar{C}_{3}),$
$\displaystyle U_{3}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-(\bar{C}_{1}\cup\bar{C}_{2}).$

Let

	$\displaystyle V^{+}$	$\displaystyle=$	$\displaystyle\{x_{1}>0\},$
	$\displaystyle V^{-}$	$\displaystyle=$	$\displaystyle\{x_{1}<0\}.$

Clearly $U_{i}\cap U_{j}=V^{+}\cup V^{-}$ when $i\neq j$ . If $T_{1}$ and $T_{2}$ are as in (9), define the following coordinate charts on $U_{1}$ , $U_{2}$ , $U_{3}$ respectively:

$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\I,$
$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}(T_{1}^{-1})^{t}&\text{on}\ \bar{V}^{+}\cap U_{2}\\ \I&\text{on}\ \bar{V}^{-}\cap U_{2}\end{array}\right.$
$\displaystyle\phi_{3}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\I&\text{on}\ \bar{V}^{+}\cap U_{3}\\ (T_{2}^{-1})^{t}&\text{on}\ \bar{V}^{-}\cap U_{3}\end{array}\right.$

We can check that the affine structure defined by these charts is such that, for fixed $b\in B_{0}$ , there exists a basis of $T^{\ast}_{b}B_{0}$ with respect to which the holonomy representation is such that $\rho^{\ast}(g_{j})=T_{j}$ , where $g_{j}$ are as in Figure 3. In particular, the holonomy is given by the inverse transpose matrices of the holonomy in the previous example.

Example 3.13 (A variation).

Again, we can perturb the above example by replacing the straight edges of $\Delta$ with smooth curves starting at the origin. This time these curves have to be contained inside $\{0\}\times\mathbb{R}^{2}$ , which is the integral surface (containing $\Delta$ ) of the distribution spanned by the $\rho$ -holonomy invariant vectors in $TB_{0}$ . The perturbed $\Delta$ , which we could denote $\Delta_{\tau}$ , still separates $\mathbb{R}^{2}$ in three connected components $C_{1},C_{2}$ and $C_{3}$ . Then the definition of the affine structure carries through just like in the previous example and we denote it by $\mathscr{A}_{\tau}$ .

We are now ready to give a definition of the specific affine structures with singularities which we will consider.

Definition 3.14.

A $2$ -dimensional affine manifold with singularities $(B,\Delta,\mathscr{A})$ is said to be simple if $\Delta$ consists of a finite union of isolated points and a neighborhood of each $p\in\Delta$ is affine isomorphic to a neighborhood of $0\in\mathbb{R}^{2}$ as in Example 3.7. We call $p\in\Delta$ a node. A $3$ -dimensional affine manifold with singularities $(B,\Delta,\mathscr{A})$ is simple if it satisfies:

(i)

$\Delta$ is a trivalent graph;
(ii)

a neighborhood of each vertex of $\Delta$ is affine isomorphic to a neighborhood of $0\in\mathbb{R}^{3}$ in either Examples 3.10 or 3.11, in which case we call it a positive vertex; or to a neighborhood of $0\in\mathbb{R}^{3}$ in either Examples 3.12 or 3.13, in which case we call it a negative vertex;
(iii)

a neighborhood of each edge of the graph is affine isomorphic to a neighborhood of $\Delta$ in Example 3.8; or a neighborhood of $\Delta_{\tau}$ in Example 3.9 for a suitable $\tau$ .

The following is direct consequence of the above definition and Theorem 2.11:

Corollary 3.15.

Let $(B,\Delta,\mathscr{A})$ be a simple affine manifold with singularities and let $(B_{0},\mathscr{A})$ be the underlying integral affine manifold. Then

f_{0}:X(B_{0},\mathscr{A})\rightarrow B_{0}

is a $T^{n}$ bundle with semi-stable monodromy as in Theorem 2.11. In particular, there is an $2n$ -manifold $X$ and a topological semi-stable compactification $X(B_{0},\mathscr{A})\hookrightarrow X$ . Furthermore, the topological fibration $f:X\rightarrow B$ obtained is topologically simple.

Examples

Here we give some examples of affine manifolds with singularities and then we prove the $2$ -dimensional version of the main theorem of this article.

Example 3.16.

In $\mathbb{R}^{3}$ consider the $3$ -dimensional simplex $\Xi$ spanned by the points

P_{0}=(-1,-1,-1),\ \ P_{1}=(3,-1,-1),\ \ P_{2}=(-1,3,-1),\ \ P_{3}=(-1,-1,3).

Let $B=\partial\Xi$ . We explain how to construct a simple affine structure with singularities on $B$ . Each edge $\ell_{j}$ of $\Xi$ has $5$ integral points (i.e. belonging to $\mathbb{Z}^{n}$ ), which divide $\ell_{j}$ into $4$ segments. For each $j=1,\ldots,6$ denote by $\Delta^{j}_{k}$ , $k=1,\ldots,4$ the four barycenters of these four segments. We let

\Delta=\{\Delta^{j}_{k};j=1\ldots 6\ \text{and}\ k=1,\ldots,4\}.

A covering of $B_{0}=B-\Delta$ can be defined as follows. The first four open sets consist of the four open faces $\Sigma_{i}$ , $i=1\ldots,4$ with the affine coordinate maps $\phi_{i}$ induced by their affine embeddings in $\mathbb{R}^{3}$ . Denote by $I$ the set of integral points of $B$ which lie on an edge. For every $Q\in I$ we can choose a small open set $U_{Q}$ in $B_{0}$ such that $\{\Sigma_{i}\}_{i=1,\ldots,4}\cup\{U_{Q}\}_{Q\in I}$ is a covering of $B_{0}$ . Let $R_{Q}$ denote the $1$ -dimensional subspace of $\mathbb{R}^{3}$ generated by $Q\in I$ . One can verify that if $U_{Q}$ is small enough, the projection $\phi_{Q}:U_{Q}\rightarrow\mathbb{R}^{3}/R_{Q}$ is an homeomorphism. A computation shows that the atlas $\mathscr{A}=\{\Sigma_{i},\phi_{i}\}_{i=1,\ldots,4}\cup\{U_{Q},\phi_{Q}\}_{Q\in I}$ defines an affine structure on $B_{0}$ making $(B,\Delta,\mathscr{A})$ simple.

Example 3.17.

This three dimensional example is taken from [10] §19.3. Let $\Xi$ be the $4$ -simplex in $\mathbb{R}^{3}$ spanned by

	$\displaystyle P_{0}=(-1,-1,-1,-1),\ P_{1}=(4,-1,-1,-1),\ P_{2}=(-1,4,-1,-1),$
	$\displaystyle P_{3}=(-1,-1,4,-1),\ \ P_{4}=(-1,-1,-1,4).$

Let $B=\partial\Xi$ . Denote by $\Sigma_{j}$ the open $3$ -face of $B$ opposite to the point $P_{j}$ and by $F_{ij}$ the closed $2$ -face separating $\Sigma_{i}$ and $\Sigma_{j}$ . Each $F_{ij}$ contains $21$ integral points (including those on its boundary). These form the vertices of a triangulation of $F_{ij}$ as in Figure 7. By joining the barycenter of each triangle with the barycenters of its sides we form a trivalent graph as in Figure 7. Define the set $\Delta$ to be the union of all such graphs in each $2$ -face. Denote by $I$ the set of integral points of $B$ . Just as in the previous example, we can form a covering of $B_{0}=B-\Delta$ by taking the open $3$ -faces $\Sigma_{j}$ and small open neighborhoods $U_{Q}$ inside $B_{0}$ of $Q\in I$ . A coordinate chart $\phi_{i}$ on $\Sigma_{i}$ can be obtained from its affine embedding in $\mathbb{R}^{4}$ . If we denote again by $R_{Q}$ the linear space spanned by $Q\in I$ , as a chart on $U_{Q}$ we take the projection $\phi_{Q}:U_{Q}\rightarrow\mathbb{R}^{4}/R_{Q}$ . A computation shows that this affine structure is simple. In fact the vertices of $\Delta$ which are contained in the interior of each $2$ -face are of negative type and those which are contained in the $1$ -faces are of positive type.

Figure 7: Affine

S^{3}

with singularities.

Example 3.18 (A variation).

In the previous example, all edges of $\Delta$ were straight lines, but one can perturb them in the sense of Examples 3.9, 3.11 and 3.13. In fact we can form a new $\Delta$ by keeping the vertices fixed and connecting them through smooth curves, which are small perturbations of the straight edges of the previous example. If these curves stay inside the $2$ -faces of $B$ , then the affine structure on $B-\Delta$ can be defined just like above.

In some cases, such as in Examples 3.16 and 3.17 given an affine manifold with singularities, one can define a second affine structure $\check{\mathscr{A}}$ on $B$ , via a discrete Legendre transform of $\mathscr{A}$ (cf. Gross and Siebert [12, 13]). Here we shall not give details about how this process works. Though it is important to mention that this method produces a second integral affine manifold with singularities, $(B,\check{\Delta},\check{\mathscr{A}})$ which coincides topologically with $(B,\Delta,\mathscr{A})$ but with holonomy representation $\check{\rho}$ dual to $\rho$ . In dimension 3 this means, in particular, that the positive vertices of $\Delta$ become negative vertices of $\check{\Delta}$ and vice-versa.

These examples of singular affine manifolds are very important. The bundles associated to them satisfy the hypothesis of Theorem 2.11 so they can be used to produce topological semi-stable compactifications which are homeomorphic to well known examples of Calabi-Yau manifolds:

Theorem 3.19 (Gross [7]).

Let $(B,\Delta,\mathscr{A})$ be the integral affine manifold with singularities described in Example 3.17 and let

(B,\Delta,\mathscr{A})\rightarrow(B,\check{\Delta},\check{\mathscr{A}})

be its Legendre transform. Let $X(B_{0},\mathscr{A})\hookrightarrow X$ and $X(B_{0},\check{\mathscr{A}})\hookrightarrow\check{X}$ be the corresponding topological semi-stable compactifications. Then $X$ is homeomorphic to the quintic hypersurface and $\check{X}$ is homeomorphic to its mirror.

Later in this article we show that there are symplectic semi-stable compactifications recovering the quintic and its mirror. These compactifications rely deeply on the existence of suitable local models of Lagrangian fibrations with singular fibres. The construction of such models is a highly delicate issue.

The focus-focus fibration

In dimension 2 it is much easier to produce symplectic semi-stable compactifications. Now we will show how Example 3.16 gives rise to a symplectic semi-stable compactification diffeomorphic to a K3 surface. This will require a local model of Lagrangian $T^{2}$ fibration with a semi-stable singular fibre, such as the one in the following:

Example 3.20.

Let $X=\mathbb{C}^{2}-\{z_{1}z_{2}+1=0\}$ and let $\omega$ be the restriction to $X$ of the standard symplectic form on $\mathbb{C}^{2}$ . One can easily check that the following map $f:X\rightarrow\mathbb{R}^{2}$ is a Lagrangian fibration:

f(z_{1},z_{2})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|z_{1}z_{2}+1|\right).

(13)

The only singular fibre is $f^{-1}(0)$ , which has the topology of a $I_{1}$ fibre. It follows that this fibration is conjugate to the topological fibration in Example 2.6.

Lagrangian fibrations with semi-stable singular fibres, e.g., conjugate to the fibration in Example 2.6, are called focus-focus fibrations. They have been studied extensively in Hamiltonian Mechanics [4], [34] –where they got their name– and more recently in symplectic topology [24], [33] and Mirror Symmetry [14].

Let $\arg:\mathbb{C}^{\ast}\rightarrow\mathbb{R}$ be the multi-valued function $\rho e^{i\theta}\mapsto\theta$ . Denote by $D\subseteq\mathbb{C}$ the unit open disk and let $D^{\ast}=D-\{0\}$ . Let $\mathcal{F}=(X,\omega,f,D)$ be a focus-focus fibration. It has been shown [33] that there are coordinates $b=(b_{1},b_{2})$ on $\mathbb{R}^{2}$ , with values in $D$ , a smooth function $q:D\rightarrow\mathbb{R}$ such that $q(0)=0$ and a choice of generators of $H_{1}(f^{-1}(b),\mathbb{Z})$ with respect to which the periods $\lambda_{1}$ and $\lambda_{2}$ of $\mathcal{F}$ can be written as

	$\displaystyle\lambda_{1}$	$\displaystyle=$	$\displaystyle-\log\|b\|\ db_{1}+\arg b\ db_{2}+dq$
	$\displaystyle\lambda_{2}$	$\displaystyle=$	$\displaystyle 2\pi\,db_{2}.$

Clearly $\lambda_{1}$ is multi-valued and blows up as $b\rightarrow 0$ . The lattice

\Lambda=\spn_{\mathbb{Z}}\langle{\lambda_{1}},{\lambda_{2}}\rangle

has monodromy given by $T$ as in (6). We now describe the affine structure induced on $D^{\ast}$ . Consider the two open subsets

	$\displaystyle U_{1}$	$\displaystyle=$	$\displaystyle D-\{\im b=0\ \text{and}\re b\geq 0\},$
	$\displaystyle U_{2}$	$\displaystyle=$	$\displaystyle D-\{\im b=0\ \text{and}\re b\leq 0\}.$

On $U_{1}$ we chose the branch of $\arg$ with values in $(0,2\pi)$ and we denote it by $\arg_{1}$ . On $U_{2}$ we chose the branch with values in $(-\pi,\pi)$ which we denote by $\arg_{2}$ . Clearly on $U_{1}\cap U_{2}$ we have $\arg_{1}=\arg_{2}+2\pi$ . A computation shows that the maps $\psi_{j}:U_{j}\rightarrow\mathbb{R}^{2}$ given by

\psi_{j}(b)=(-b_{1}\log|b|+b_{1}+q(b)+b_{2}\arg_{j}b,\,2\pi b_{2}),

with $q(0)=0$ , are a choice of affine coordinates associated to $\lambda_{1}$ and $\lambda_{2}$ .

It is easy to check that the map $\psi_{1}$ (or $\psi_{2}$ ) extends continuously to $D$ . Call $\alpha:D\rightarrow\mathbb{R}^{2}$ the extended map. On a sufficiently small neighborhood $V\subseteq D$ of $0$ , the map $\alpha$ is a homeomorphism of $V$ onto $\alpha(V)$ . The reader may verify that $0\in V$ is a node with respect to the affine structure given by $\{U_{j},\psi_{j}\}$ . In other words, the map $\alpha$ restricted to $V^{\ast}=V-\{0\}$ is an affine isomorphism between $V^{\ast}$ and the affine manifold $\alpha(V^{\ast})$ whose affine structure is the restriction of the one in Example 3.7. The affine structure with singularities on $D$ induced by a focus-focus fibration is therefore simple. In particular, the affine structure induced by Example 3.20 is simple.

Remark 3.21.

Germs of focus-focus fibrations –with respect to symplectic conjugation– are classified by formal power series in two variables $\mathbb{R}[\![x,y]\!]$ with vanishing constant term [33]. Such series correspond to the Taylor coefficients of functions $q\in C^{\infty}(D)$ as above evaluated at $0\in\mathbb{R}^{2}$ . This means that there is an infinite number of different germs of focus-focus fibrations, all inducing simple affine manifolds with singularities, i.e. inducing the same singular affine structure on the base. In §4 we will see that a similar phenomenon happens in higher dimensions.

The K3 surface.

Theorem 3.22.

Let $(B,\Delta,\mathscr{A})$ be the affine manifold with singularities in Example 3.16 and let $X(B_{0},\mathscr{A})$ be the associated $T^{2}$ bundle with symplectic structure $\omega_{0}$ and projection $f_{0}$ induced by the standard ones in $T_{B_{0}}^{\ast}$ . There exists a compact symplectic manifold $(X,\omega)$ , a Lagrangian fibration $f:X\rightarrow B$ and an embedding $\iota:X(B_{0},\mathscr{A})\rightarrow X$ such that $\iota^{\ast}\omega=\omega_{0}$ and $f\circ\iota=f_{0}$ . Moreover $X$ is diffeomorphic to a smooth $K3$ surface.

Proof.

Let $f_{V}:X_{V}\rightarrow V$ be a focus-focus fibration over a small open neighborhood $V$ of its node $0\in V$ . Let $V^{\ast}=V-\{0\}$ and denote by $(V^{\ast},\mathscr{A}_{V})$ the integral affine manifold induced by $f_{V}$ . Let $X(V^{\ast},\mathscr{A}_{V})$ be the associated Lagrangian $T^{2}$ bundle over $V^{\ast}$ . It can be shown that $f_{V}$ has a Lagrangian section $s:V\rightarrow X_{V}$ such that $s(V)\cap\Crit(f_{V})=\varnothing$ . Then from Theorem 3.3 it follows that $f^{-1}_{V}(V^{\ast})\subset X_{V}$ is symplectically conjugate to $X(V^{\ast},\mathscr{A}_{V})$ .

Now let $P\in\Delta$ and let $U\subset B$ be a small neighborhood of $P$ . Denote by $U^{\ast}=U-P$ and by $X(U^{\ast},\mathscr{A})$ the Lagrangian $T^{2}$ bundle over $U^{\ast}$ given by the restriction of $X(B_{0},\mathscr{A})$ to $U^{\ast}$ . Recall that both $U$ and $V$ are simple affine manifold with singularities. Then, after taking $U$ and $V$ small enough, there exists an integral affine isomorphism $V^{\ast}\cong U^{\ast}$ . From Corollary 3.4, the latter isomorphism induces is a symplectic conjugation,

f^{-1}_{V}(V^{\ast})\cong X(V^{\ast},\mathscr{A}_{V})\cong X(U^{\ast},\mathscr{A}),

which can be used to symplectically glue $X_{V}$ to $X(B_{0})$ . Define $(X,\omega)$ to be the symplectic manifold obtained after applying this gluing over all points $P\in\Delta$ and $f:X\rightarrow B$ the resulting fibration. It is clear that $(X,\omega)$ is a semi-stable compactification of $(X(B_{0},\mathscr{A}),\omega_{0})$ such that $\iota^{\ast}\omega=\omega_{0}$ . It is easy to check that $(X,\omega,f,B)$ is topologically conjugate to a simply connected elliptic fibration with 24 singular fibres of type $I_{1}$ . It follows that $X$ is diffeomorphic to a K3 surface. ∎

Corollary 3.23.

In view of Remark 3.21, given $(B,\Delta,\mathscr{A})$ as in Example 3.16, a compactification $X(B_{0},\mathscr{A})\hookrightarrow(X,\omega)$ as above is uniquely determined up to symplectic conjugation by a choice of 24 formal power series in two variables:

\mathfrak{q}_{1},\ldots,\mathfrak{q}_{24}\in\mathbb{R}[\![x,y]\!]

corresponding to germs of focus-focus fibrations $\mathcal{F}_{1},\ldots\mathcal{F}_{24}$ . In particular, there are infinitely many Lagrangian fibrations of a symplectic K3 surface, fibering over $(B,\Delta,\mathscr{A})$ , which are all topologically conjugate but not symplectically conjugate.

The space $\mathbb{R}[[x,y]]$ being contractible, implies that every two focus-focus fibrations can be connected with a path in $\mathbb{R}[[x,y]]$ . The standard Moser’s argument implies that the corresponding total spaces are symplectomorphic. Similarly, any two symplectic structures obtained using Theorem 3.22 can be connected with a path in $\mathbb{R}[[x,y]]^{24}$ . Moser’s argument implies that all such manifolds are symplectomorphic.

Following an alternative approach, Zung obtained a Lagrangian fibration of a symplectic 4-manifold which is also diffeomorphic to a K3 surface (cf. [35]Example 4.19). Leung and Symington [24] use affine geometry as starting point to construct and classify –up to diffeomorphism– the so-called almost toric symplectic 4-manifolds. The fibration we obtained in Theorem 3.22 coincides with one of the list in [24].

Other ways of constructing affine manifolds with singularities have been proposed by Gross and Siebert [12, 13], Hasse and Zharkov [16, 17, 18]. In [8], Gross finds a combinatorial method to obtain simple affine manifolds with singularities out of the geometry of the polytopes which Batyrev and Borisov use to construct pairs of Calabi-Yau varieties as complete intersections inside Fano toric varieties. From Theorem 0.1 of [8] (proved by Gross and Siebert in [11]) it follows that these affine manifolds give rise to topological semi-stable compactifications homeomorphic to the two Batyrev-Borisov’s Calabi-Yau varieties. We shall see in this paper that similar compactifications can be carried out in the symplectic category.