Monodromy [04KK]

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Monodromy

We now study stitched fibrations defined over non simply connected bases. In this case, the underlying topological $T^{n}$ bundle may have monodromy. When $\mathcal{F}$ is smooth, monodromy can be read from the holonomy of the affine structure on the base. This is no longer true for stitched fibrations in general. This is the case, for instance, of Example 5.5; in fact, in [3]Proposition 7 (cf. also Remark 5) we gave explicit evidence of this. We show now that monodromy can alternatively be detected from the behavior of the first order invariant $\ell_{1}$ . We restrict to some specific examples with unipotent monodromy.

Example 6.14.

Let $U\subset\mathbb{R}^{2}$ be an open annulus in $\mathbb{R}^{2}$ centered at the origin. As usual denote $U^{+}=U\cap\{b_{1}\geq 0\}$ , $U^{-}=U\cap\{b_{1}\leq 0\}$ and $\Gamma=U^{+}\cap U^{-}$ . This time $\Gamma$ is disconnected. We let $\Gamma_{u}=\Gamma\cap\{b_{2}\geq 0\}$ and $\Gamma_{d}=\Gamma\cap\{b_{2}\leq 0\}$ be the upper and lower parts of $\Gamma$ respectively. Now let $f:X\rightarrow\mathbb{R}^{2}$ be a stitched Lagrangian fibration such that $f(X)=U$ . Observe that the seam $Z$ has two connected components: $Z_{u}=f^{-1}(\Gamma_{u})$ and $Z_{d}=f^{-1}(\Gamma_{d})$ . Denote by $\bar{Z}_{u}$ and $\bar{Z}_{d}$ the respective $S^{1}$ quotients, i.e. the connected components of $\bar{Z}$ . Let $b\in\Gamma_{u}$ and choose as generator of $\pi_{1}(U,b)$ an anti-clock-wise oriented curve starting at $b$ and going once around $0$ . Suppose that with respect to a basis $\{\gamma_{1},\gamma_{2}\}$ of $H_{1}(F_{b},\mathbb{Z})$ the monodromy is

\left(\begin{array}[]{cc}1&-m\\ 0&1\end{array}\right),

(57)

for some integer $m\neq 0$ . In this case we must have that $\gamma_{1}$ is represented by the orbits of the $S^{1}$ action. As usual let $X^{\pm}=f^{-1}(U^{\pm})$ . Since $U-\Gamma_{d}$ is contractible we can think of $\{\gamma_{1},\gamma_{2}\}$ as a basis of $H_{1}(f^{-1}(U-\Gamma_{d}),\mathbb{Z})$ . Consider the diagrams:

induced by inclusions and restrictions. The map $j_{+}$ identifies $\{\gamma_{1},\gamma_{2}\}$ with a basis $\{\gamma_{1},\gamma_{2}^{+}\}$ of $H_{1}(f^{-1}(U-\Gamma_{u}),\mathbb{Z})$ , whereas $j_{-}$ with a basis $\{\gamma_{1},\gamma_{2}^{-}\}$ . Notice that monodromy is given by $j_{+}^{-1}\circ j_{-}$ . Therefore we must have $\gamma_{2}^{+}=m\gamma_{1}+\gamma_{2}^{-}$ . Hence $\{\gamma_{1},\gamma_{2}^{+}\}$ and $\{\gamma_{1},\gamma_{2}^{-}\}$ satisfy conditions (a) and (b) in the previous section. Applying Proposition 6.5 to $f$ restricted to $f^{-1}(U-\Gamma_{u})$ we can consider the action coordinates map $\alpha$ constructed by taking action coordinates with respect to $\{\gamma_{1},\gamma_{2}^{+}\}$ on $U^{+}$ and with respect to $\{\gamma_{1},\gamma_{2}^{-}\}$ on $U^{-}$ . Denote by $(b_{1}^{d},b_{2}^{d})$ such coordinates. Similarly on $U-\Gamma_{d}$ we can consider action angle coordinates with respect to the basis $\{\gamma_{1},\gamma_{2}\}$ . Denote by $(b_{1}^{u},b_{2}^{u})$ these coordinates. In particular we have the identifications

\bar{Z}_{d}=T^{\ast}\Gamma_{d}\,/\,\langle db_{2}^{d}\rangle_{\mathbb{Z}}

and

\bar{Z}_{u}=T^{\ast}\Gamma_{u}\,/\,\langle db_{2}^{u}\rangle_{\mathbb{Z}}.

With respect to this choice of coordinates we can compute the first order invariants of $f$ , $\ell_{1}^{u}$ and $\ell_{1}^{d}$ on $\bar{Z}_{u}$ and $\bar{Z}_{d}$ , respectively. Then (49) should hold, therefore we obtain

\int_{[db_{2}^{u}]}\ell_{1}^{u}=0\ \ \text{and}\ \ \int_{[db_{2}^{d}]}\ell_{1}^{d}=m.

This tells us that monodromy can be read from a jump in cohomology class of the first order invariant associated to action coordinates.

Using the methods of Theorem 6.11 we can also construct stitched Lagrangian fibrations with prescribed monodromy and invariants. In fact we have

Theorem 6.15.

Let $U\subset\mathbb{R}^{2}$ be an annulus as above with coordinates $(b_{1},b_{2})$ . Let $\bar{Z}_{d}=T^{\ast}\Gamma_{d}\,/\,\langle db_{2}\rangle_{\mathbb{Z}}$ and $\bar{Z}_{u}=T^{\ast}\Gamma_{u}\,/\,\langle db_{2}\rangle_{\mathbb{Z}}$ with projections $\bar{\pi}^{d}$ and $\bar{\pi}^{u}$ and bundles $\mathfrak{L}_{d}=\ker\bar{\pi}^{d}_{\ast}$ and $\mathfrak{L}_{u}=\ker\bar{\pi}^{u}_{\ast}$ respectively. Given an integer $m$ and sequences $\ell^{d}=\{\ell_{k}^{d}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{d}}$ and $\ell^{u}=\{\ell_{k}^{u}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{u}}$ such that

\int_{[db_{2}]}\ell_{1}^{u}=0\ \ \text{and}\ \ \int_{[db_{2}]}\ell_{1}^{d}=m,

there exists a smooth symplectic manifold $(X,\omega)$ and a stitched Lagrangian fibration $f:X\rightarrow U$ having monodromy (57) with respect to some basis $\gamma=\{\gamma_{1},\gamma_{2}\}$ of $H_{1}(f^{-1}(U-\Gamma_{d}),\mathbb{Z})$ and satisfying the following properties:

(i)

the coordinates $(b_{1},b_{2})$ are action coordinates of $f$ with moment map $f^{\ast}b_{1}$ ;
(ii)

the periods $\{db_{1},db_{2}\}$ , restricted to $U^{\pm}$ correspond to the basis $\{\gamma_{1},\gamma_{2}\}$ ;
(iii)

there is a Lagrangian section $\sigma$ of $f$ , such that $(\bar{Z}_{u},\ell^{u})$ and $(\bar{Z}_{d},\,\ell^{d})$ are the invariants of $(f^{-1}(U-\Gamma_{d}),\,f,\,U-\Gamma_{d},\,\sigma,\,\gamma)$ and $(f^{-1}(U-\Gamma_{u}),\,f,\,U-\Gamma_{u},\,\sigma,\,j_{+}(\gamma))$ respectively.

The fibration $(X,f,U)$ satisfying the above properties is unique up to fibre preserving symplectomorphism.

Proof.

This is just a repetition of the arguments in Theorem 6.11 for each component of $\Gamma=\Gamma_{d}\cup\Gamma_{u}$ . We leave the details as an exercise. ∎

Remark 6.16.

Notice that the stitched fibrations discussed in Example 6.14 are more general than the ones constructed in Theorem 6.15. We illustrate this with an example. Let $U^{-}$ and $U^{+}$ be two “half annuli” of the same width but of different radii (as depicted in Figure 10). If $b^{\pm}=(b_{1}^{\pm},b_{2}^{\pm})$ denote coordinates on $U^{\pm}$ and we let $\Lambda^{\pm}=\langle{db_{1}^{\pm}},{db_{2}^{\pm}}\rangle_{\mathbb{Z}}$ , then we can glue together $X^{+}=T^{*}U^{+}/\Lambda^{+}$ and $X^{-}=T^{*}U^{-}/\Lambda^{-}$ after choosing suitable invariants and applying the usual method of Theorem 6.11. We first glue the lower boundaries of $X^{+}$ and $X^{-}$ and then the upper boundaries, (as indicated by the arrows in Figure 10). This produces a stitched fibration of the type discussed in Example 6.14, in fact we would obtain a total space $X$ which fibres over a base obtained as the result of the gluing of the two half annuli, which is clearly diffeomorphic to an annulus. The fibration is not of the type constructed in Theorem 6.15. There are two main differences between the two constructions. In the examples from Theorem 6.15 action coordinates extend continuously to the whole annulus and the symplectic form on the total space is exact. These two facts do not hold in the example just described, in fact if the symplectic form were exact then the action coordinates would extend continuously to the whole annulus (to show this one can use an argument similar to the one used in Proposition 4.11).

Refer to caption — Figure 10: Gluing half annuli with different radii.

Example 6.17.

An example of a stitched Lagrangian fibration constructed using Theorem 6.15 is the following. We can choose the elements of the sequence $\ell^{u}$ to be all zero, while the elements of the sequence $\ell^{d}$ to be all zero except $\ell_{1}^{d}$ which we define to be

\ell_{1}^{d}=m\,dy_{2}.

It is clear that the resulting fibration is only fake stitched, in fact the invariants are fibrewise constant. One can also see that, in the case $m=1$ and $U=\mathbb{R}^{2}-\{0\}$ , the fibration is symplectically conjugate to $(X,\alpha\circ f)$ , where $(X,f)$ is a smooth focus-focus fibration (where the singular fibre has been removed) and $\alpha$ is the action coordinates map (see the discussion after Example 3.20 and Example 6.13). In particular this fibration induces an affine structure on the base which is simple.

We now discuss a three dimensional example.

Example 6.18.

In $\mathbb{R}^{3}$ consider the 3-valent graph

\Delta=\{(0,0,-t),\ t\geq 0\}\cup\{(0,-t,0),\ t\geq 0\}\cup\{(0,t,t),\ t\geq 0\}

and let $D$ be a tubular neighborhood of $\Delta$ . Take $U=\mathbb{R}^{3}-D$ and assume we have a stitched Lagrangian fibration $f:X\rightarrow\mathbb{R}^{3}$ such that $U=f(X)$ and the seam is $Z=f^{-1}(\{b_{1}=0\}\cap U)$ . Again we let $U^{+}=U\cap\{b_{1}\geq 0\}$ , $U^{-}=U\cap\{b_{1}\leq 0\}$ and $\Gamma=U^{+}\cap U^{-}$ . Also let $X^{\pm}=f^{-1}(U^{\pm})$ . This time $\Gamma$ (hence $Z$ ) has three connected components

$\displaystyle\Gamma_{c}$	$\displaystyle=$	$\displaystyle\{(0,t,s),\ t,s<0\}\cap U,$
$\displaystyle\Gamma_{d}$	$\displaystyle=$	$\displaystyle\{(0,t,s),\ t>0,s<t\}\cap U,$
$\displaystyle\Gamma_{e}$	$\displaystyle=$	$\displaystyle\{(0,t,s),\ s>0,t<s\}\cap U.$

Also denote by $Z_{c}$ , $Z_{d}$ and $Z_{e}$ the corresponding connected components of $Z$ and by $\bar{Z}_{c}$ , $\bar{Z}_{d}$ and $\bar{Z}_{e}$ their $S^{1}$ quotients.

Fix $b\in\Gamma_{c}$ and suppose that there is a basis $\{\gamma_{1},\gamma_{2},\gamma_{3}\}$ of $H_{1}(F_{b},\mathbb{Z})$ and generators $g_{1},g_{2},g_{3}$ of $\pi_{1}(U,b)$ , satisfying $g_{1}g_{2}g_{3}=1$ , with respect to which the monodromy transformations are

\mathcal{M}_{b}(g_{1})=T_{1}=\left(\begin{array}[]{ccc}1&-m_{1}&0\\ 0&1&0\\ 0&0&1\end{array}\right),\ \ \ \mathcal{M}_{b}(g_{2})=T_{2}=\left(\begin{array}[]{ccc}1&0&-m_{2}\\ 0&1&0\\ 0&0&1\end{array}\right)

(58)

and $\mathcal{M}_{b}(g_{3})=T_{3}=T_{2}^{-1}T_{1}^{-1}$ , for non zero integers $m_{1}$ and $m_{2}$ . We have that $\gamma_{1}$ is represented by the orbits of the $S^{1}$ action, since it is the only monodromy invariant cycle. Now, since $U-(\Gamma_{d}\cup\Gamma_{e})$ is contractible, $\{\gamma_{1},\gamma_{2},\gamma_{3}\}$ is a basis of $H_{1}(f^{-1}(U-(\Gamma_{d}\cup\Gamma_{e})),\mathbb{Z})$ . Consider the diagrams:

induced by inclusions and restrictions. The map $j_{+}$ identifies $\{\gamma_{1},\gamma_{2},\gamma_{3}\}$ with a basis of $H_{1}(f^{-1}(U-(\Gamma_{c}\cup\Gamma_{d})),\mathbb{Z})$ , which we call $\{\gamma_{1},\gamma_{2}^{+},\gamma_{3}^{+}\}$ , while $j_{-}$ identifies it with another basis, which we call $\{\gamma_{1},\gamma_{2}^{-},\gamma_{3}^{-}\}$ . Notice that the monodromy map $\mathcal{M}_{b}(g_{2})=j_{+}^{-1}\circ j_{-}$ . We must have

\begin{cases}\gamma_{2}^{+}=\gamma_{2}^{-},\\ \gamma_{3}^{+}=m_{2}\gamma_{1}+\gamma_{3}^{-}.\end{cases}

(59)

Applying Proposition 6.5 to $f$ restricted to $f^{-1}(U-(\Gamma_{c}\cup\Gamma_{d}))$ , we can consider the action coordinates map $\alpha$ on $U-(\Gamma_{c}\cup\Gamma_{d})$ computed with respect to $\{\gamma_{1},\gamma_{2}^{+},\gamma_{3}^{+}\}$ on $U^{+}$ and with respect to $\{\gamma_{1},\gamma_{2}^{-},\gamma_{3}^{-}\}$ on $U^{-}$ . Let us denote these coordinates by $(b_{1}^{e},b_{2}^{e},b_{3}^{e})$ . Similarly we can consider action coordinates on $U-(\Gamma_{d}\cup\Gamma_{e})$ with respect to the basis $\{\gamma_{1},\gamma_{2},\gamma_{3}\}$ of $H_{1}(f^{-1}(U-(\Gamma_{d}\cup\Gamma_{e})),\mathbb{Z})$ . We denote them by $(b_{1}^{c},b_{2}^{c},b_{3}^{c})$ . We have the identifications

\bar{Z}_{e}=T^{\ast}\Gamma_{e}\,/\,\langle db_{2}^{e},db_{3}^{e}\rangle_{\mathbb{Z}}

and

\bar{Z}_{c}=T^{\ast}\Gamma_{c}\,/\,\langle db_{2}^{c},db_{3}^{c}\rangle_{\mathbb{Z}}.

With respect to these coordinates we can compute the first order invariants $\ell_{1}^{e}$ and $\ell_{1}^{c}$ on $\bar{Z}_{e}$ and $\bar{Z}_{c}$ respectively. From Proposition 6.5 and identities (59) applied to $\ell_{1}^{c}$ and $\ell_{1}^{e}$ we obtain

\int_{[db_{2}^{c}]}\ell_{1}^{c}=\int_{[db_{3}^{c}]}\ell_{1}^{c}=0

and

\int_{[db_{2}^{e}]}\ell_{1}^{e}=0\ \ \text{and}\ \ \int_{[db_{3}^{e}]}\ell_{1}^{e}=m_{2}.

Similarly we construct the first order invariant $\ell_{1}^{d}$ on $\bar{Z}_{d}$ . It will satisfy

\int_{[db_{2}^{d}]}\ell_{1}^{d}=m_{1}\ \ \text{and}\ \ \int_{[db_{3}^{d}]}\ell_{1}^{d}=0.

Again, monodromy is understood in terms of the difference in the cohomology class of the first order invariant. Example 5.5 is a special case of this situation, where $m_{1}=m_{2}=1$ .

Conversely, we can construct stitched fibrations like the one in previous example by specifying gluing data and applying Theorem 6.11. In fact we can prove

Theorem 6.19.

Let $U\subset\mathbb{R}^{3}$ , $\Gamma_{c}$ , $\Gamma_{d}$ and $\Gamma_{e}$ be as in Example 6.18 and let $(b_{1},b_{2},b_{3})$ be coordinates on $U$ . Define $\bar{Z}_{c}=T^{\ast}\Gamma_{c}\,/\,\langle db_{2},db_{3}\rangle_{\mathbb{Z}}$ , $\bar{Z}_{d}=T^{\ast}\Gamma_{d}\,/\,\langle db_{2},db_{3}\rangle_{\mathbb{Z}}$ and $\bar{Z}_{e}=T^{\ast}\Gamma_{e}\,/\,\langle db_{2},db_{3}\rangle_{\mathbb{Z}}$ with projections $\bar{\pi}^{c}$ , $\bar{\pi}^{d}$ , $\bar{\pi}^{e}$ and bundles $\mathfrak{L}_{c}=\ker\bar{\pi}^{c}_{\ast}$ , $\mathfrak{L}_{d}=\ker\bar{\pi}^{d}_{\ast}$ , $\mathfrak{L}_{e}=\ker\bar{\pi}^{e}_{\ast}$ . Suppose we are given integers $m_{1}$ , $m_{2}$ and sequences $\ell^{c}=\{\ell_{k}^{c}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{c}}$ , $\ell^{d}=\{\ell_{k}^{d}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{d}}$ and $\ell^{e}=\{\ell_{k}^{e}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{e}}$ satisfying

$\displaystyle\int_{[db_{2}]}\ell_{1}^{c}$	$\displaystyle=$	$\displaystyle\int_{[db_{3}]}\ell_{1}^{c}=0,$
$\displaystyle\int_{[db_{2}]}\ell_{1}^{e}$	$\displaystyle=$	$\displaystyle 0\quad\ \ \text{and}\quad\int_{[db_{3}]}\ell_{1}^{e}=m_{2},$	(60)
$\displaystyle\int_{[db_{2}]}\ell_{1}^{d}$	$\displaystyle=$	$\displaystyle m_{1}\quad\text{and}\quad\int_{[db_{3}]}\ell_{1}^{d}=0.$

Then there exists a smooth symplectic manifold $(X,\omega)$ and a stitched Lagrangian fibration $f:X\rightarrow U$ having the same monodromy of Example 6.18 with respect to some basis $\gamma=\{\gamma_{1},\gamma_{2},\gamma_{3}\}$ of $H_{1}(f^{-1}(U-(\Gamma_{d}\cup\Gamma_{e})),\mathbb{Z})$ and satisfying the following properties:

(i)

the coordinates $(b_{1},b_{2},b_{3})$ are action coordinates of $f$ with moment map $f^{\ast}b_{1}$ ;
(ii)

the periods $\{db_{1},db_{2},db_{3}\}$ , restricted to $U^{\pm}$ correspond to the basis $\gamma$ ;

(iii)

there is a Lagrangian section $\sigma$ of $f$ , such that $(\bar{Z}_{c},\ell^{c})$ , $(\bar{Z}_{d},\,\ell^{d})$ and $(\bar{Z}_{e},\,\ell^{e})$ are respectively the invariants of:

(f|_{U-(\Gamma_{d}\cup\Gamma_{e})},\sigma,\,\gamma),\ (f|_{U-(\Gamma_{c}\cup\Gamma_{e})},\sigma,\,j_{+}(\gamma))\ \textrm{and}\ (f|_{U-(\Gamma_{c}\cup\Gamma_{d})},\sigma,\,j_{+}(\gamma)).

The fibration $(X,f,U)$ satisfying the above properties is unique up to fibre preserving symplectomorphism.

Remark 6.20.

Also in this case (cf. Remark 6.16) we notice that fibrations of the type discussed in Example 6.18 are more general than the ones constructed using Theorem 6.19. To show this one can use higher dimensional versions of the fibration in Remark 6.16, with discontinuous action coordinates. We leave the details to the reader.

Example 6.21.

A simple example of stitched Lagrangian fibration which can be constructed using Theorem 6.19 is as follows. Define the sequence $\ell^{c}$ to be identically zero and choose the terms of $\ell^{d}$ and $\ell^{e}$ to be zero except the first order ones, which we define to be

\ell_{1}^{d}=m_{1}\,dy_{2}\ \ \text{and}\ \ \ell_{1}^{e}=m_{2}\,dy_{3}.

Clearly $\ell_{1}^{c}$ , $\ell_{1}^{d}$ and $\ell_{1}^{e}$ satisfy the integral conditions of Theorem 6.19, moreover they are fibrewise constant, therefore they define fake stitched fibrations. Since the fibration is smooth after a change of coordinates on the base, it induces an affine structure on the base. One can easily see that in the case $m_{1}=-1$ and $m_{2}=1$ and $U=\mathbb{R}^{3}-\Delta$ , this affine structure is simple and affine isomorphic to a negative vertex of Example 3.12. Notice that we could also replace $\Delta$ with $\Delta_{\tau}$ and obtain an affine structure which is isomorphic to the one in Example 3.13.