3.4. The torus fibration [03E5]

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3.4. The torus fibration

Using the foliation $\mathcal{F}$ we are going to define a decomposition of the hypersurface $H_{s}=H_{s}^{\mathrm{sm}}\sqcup H_{s}^{\mathrm{sing}}$ , construct a torus fibration $H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D)$ and show that it is isomorphic to the fibration $W^{\epsilon}\rightarrow\Sigma\backslash N(D)$ .

For any closed subset $J\subset\Sigma$ we will denote by $X_{s}(J)\subset X_{\Delta_{\nu}}$ the closure of $\mathrm{Log}_{s}^{-1}\left(\bigcup_{q\in J}\mathcal{F}_{q}\right)$ in $X_{\Delta_{\nu}}$ .

Definition.

Let $N(D)$ be a regular neighborhood of $D$ in $\Sigma$ . Then the smooth part of the hypersurface is $H_{s}^{\mathrm{sm}}:=H_{s}\cap X_{s}(\Sigma\backslash N(D))$ , and the rest $H_{s}^{\mathrm{sing}}:=H_{s}\backslash H_{s}^{\mathrm{sm}}$ is singular.

Since $D=\partial\mathcal{U}\cap\partial\mathcal{V}$ , there exist regular neighborhoods $N_{1}(\partial\mathcal{U})$ of $\partial\mathcal{U}$ and $N_{2}(\partial\mathcal{V})$ of $\partial\mathcal{V}$ in $\Sigma$ , such that $N(D)\supset N_{1}(\partial\mathcal{U})\cap N_{2}(\partial\mathcal{V})$ . This means that $\Sigma\backslash N(D)$ can be covered by the union of the closed sets:

\mathcal{U}^{\epsilon}=\{U^{\epsilon}_{v}\}=\{U_{v}\backslash{N_{1}(\partial\mathcal{U})}\}\ \text{ and }\ \mathcal{V}^{\delta}=\{V^{\delta}_{w}\}=\{V_{w}\backslash{N_{2}(\partial\mathcal{V})}\}.

The amoebas $\mathcal{A}^{\lambda}_{s}$ , for a large enough $s$ , all lie in $\mathbb{R}^{d}\backslash Q^{\lambda}_{\{0\}}(\epsilon)$ . This means that $\mathcal{F}$ defines a projection $\mathcal{A}^{\lambda}_{s}\to\Sigma$ and, by composition with $\mathrm{Log}_{s}$ , the projection $H_{s}^{\operatorname{af{}f}}\to\Sigma$ . Also $\mathcal{A}^{\lambda}_{s}$ lie in $\mathbb{R}^{d}\backslash Q^{\lambda}_{v}(\epsilon)$ , for any $v\in\operatorname{vert}(S)$ and large $s$ . Since the unbounded ends of flow lines $\mathcal{F}_{q}$ , for $q\in U^{\epsilon}_{v}$ , are in $Q^{\lambda}_{v}(\epsilon)$ their closures do not contain any extra points of the hypersurface:

H_{s}^{\operatorname{af{}f}}\cap\mathrm{Log}_{s}^{-1}\left(\bigcup_{q\in U^{\epsilon}_{v}}\mathcal{F}_{q}\right)=H_{s}^{\operatorname{af{}f}}\cap X_{s}(U^{\epsilon}_{v})=H_{s}\cap X_{s}(U^{\epsilon}_{v}).

Thus, the map $H_{s}\cap X_{s}(U^{\epsilon}_{v})\to U^{\epsilon}_{v}$ is well defined. On the other hand, for two distinct points $q_{1},q_{2}$ in $V^{\delta}_{w}$ the corresponding leaves are straight lines. Written in local coordinates (see Lemma 3.9) this implies that the sets $X_{s}(\mathcal{F}_{q_{1}})$ and $X_{s}(\mathcal{F}_{q_{2}})$ are disjoint. Hence, the map $H_{s}\cap X_{s}(V^{\delta}_{w})\to V^{\delta}_{w}$ is well defined. Combined together we have (for large enough $s$ ) the well defined projection

f_{s}\colon H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D),\qquad f_{s}(x):=q\ \Leftrightarrow\ x\in X_{s}(q).

Theorem 3.7.

There exists a real number $s_{0}$ , such that for any $s$ with $|s|\geq s_{0}$ ,

f_{s}\colon H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D)

is a torus fibration isomorphic to $W^{\epsilon}\rightarrow\Sigma\backslash N(D)$ .

Before proving the theorem we need to make a comment about smoothness. $H_{s}^{\mathrm{sm}}$ is missing all singular points (if any) of the toric variety $X_{{\Delta_{\nu}}}$ , which are all in the moment map preimage of the $(d-2)$ -skeleton of ${\Delta_{\nu}}$ (see Lemma 3.9). On the other hand, $\Sigma\backslash N(D)$ carries a canonical smooth structure induced by the affine structure on $\Sigma\backslash D$ . So given the topological fibration $H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D)$ of Theorem 3.7, standard techniques apply to make it smooth.

The strategy of proving Theorem 3.7 will be as follows. First, we analyze the map $f_{s}$ in $U^{\epsilon}_{v}$ and $V^{\delta}_{w}$ for every $v\in\operatorname{vert}(S),w\in\operatorname{vert}(T)$ . Then the proof of the theorem can be completed in three steps: we show that $f_{s}\colon H_{s}^{\mathrm{sm}}\to\Sigma\backslash N(D)$ is a torus fibration over the two kinds of covering patches, and then check that it has the monodromy of our model.

Let $v\in\operatorname{vert}(S)$ . For a fixed $s$ we consider the $(\Delta\cap(\mathbb{Z}^{d})^{*}-2)$ -parameter family of hypersurfaces $H_{s}^{v}(a)$ in $X_{s}(U^{\epsilon}_{v})$ :

s^{\lambda(0)}+s^{\lambda(v)}x^{v}+\sum\limits_{m\neq\{0\},v}a_{m}s^{\lambda(m)}x^{m}=0,\quad 0\leq a_{m}\leq 1.

Lemma 3.8.

There exists $s_{0}$ such that whenever $|s|\geq s_{0}$ , all $H_{s}^{v}(a)$ are smooth and transversal to $X_{s}(q)$ for every $q\in U^{\epsilon}_{v}$ .

Proof.

According to Proposition 3.2 we can choose $s$ big enough so that the $\mathrm{Log}_{s}$ -image of every hypersurface $H_{s}^{v}(a)$ lies in the $\epsilon$ -neighborhood of $U^{\epsilon}_{v}$ . Recall from Lemma 3.3 that a small neighborhood of $U^{\epsilon}_{v}$ lies in the domain $Q^{\lambda}_{(\{0\}\mid v)}(\epsilon)$ . Thus we can assume that all hypersurfaces $H_{s}^{v}(a)$ lie entirely in $\mathrm{Log}_{s}^{-1}(Q^{\lambda}_{(\{0\}\mid v)}(\epsilon))$ .

Whenever $\mathrm{Log}_{s}(x)\in Q^{\lambda}_{(\{0\}\mid v)}(\epsilon)$ , we have

\langle m,\frac{\log|x|}{\log|s|}\rangle+\lambda(m)\leq\lambda(0)-\epsilon,\text{ for all }m\neq v,\{0\}

or, equivalently,

|x^{m}s^{\lambda}(m)|\leq|s|^{-\epsilon}|s|^{\lambda(0)}.

This means that the values of all monomials $x^{m}s^{\lambda(m)},\ m\neq v,\{0\}$ , for $x\in\mathrm{Log}_{s}^{-1}(Q^{\lambda}_{(\{0\}\mid v)}(\epsilon))$ , are (uniformly) bounded by $|s|^{-\epsilon}|s|^{\lambda(0)}$ . Note also, that their log-derivatives are bounded by $C|s|^{-\epsilon}|s|^{\lambda(0)}$ , some constant $C\geq 0$ , since

x\frac{\partial}{\partial x}(a_{m}s^{\lambda(m)}x^{m})=m\cdot a_{m}s^{\lambda(m)}x^{m}.

For any basis $\{e_{i}\}$ of $(\mathbb{Z}^{d})^{*}$ , the functions $y_{i}=x^{e_{i}}$ give affine coordinates on $(\mathbb{C}\backslash\{0\})^{d}$ . We choose $e_{1}=-v$ , multiply the equations of the hypersurfaces in our family $H_{s}^{v}(a)$ by $y_{1}=x^{-v}$ , and look for critical points:

\frac{\partial}{\partial y_{1}}\bigl(y_{1}s^{\lambda(0)}+s^{\lambda(v)}+y_{1}\sum\limits_{m\neq\{0\},v}a_{m}s^{\lambda(m)}x^{m}\bigr)\\ =s^{\lambda(0)}+\bigl(1+y_{1}\frac{\partial}{\partial y_{1}}\bigr)\sum\limits_{m\neq\{0\},v}a_{m}s^{\lambda(m)}x^{m}=s^{\lambda(0)}(1+O(|s|^{-\epsilon}))\neq 0,

for large enough $s$ . Thus, there are no critical points, hence every member of our family $H_{s}^{v}(a)$ is smooth.

Finally, note that $\bigcup_{q\in U^{\epsilon}_{v}}\mathcal{F}_{q}$ is in $Q_{v}^{\lambda^{\epsilon}}$ , but Lemma 3.6 asserts that the vectors $\xi\in\mathfrak{X}$ in $Q_{v}^{\lambda^{\epsilon}}$ satisfy $\langle v,\xi\rangle=1$ . Thus, for any point of intersection $H_{s}^{v}(a)\cap X_{s}(q)$ the corresponding tangent vector to $X_{s}(q)$ has the form:

\bar{\xi}=y_{1}\frac{\partial}{\partial y_{1}}+\alpha_{2}y_{2}\frac{\partial}{\partial y_{2}}+\dots+\alpha_{d}y_{d}\frac{\partial}{\partial y_{d}}.

Differentiating the defining equation for $H_{s}^{v}(a)$ with respect to $\bar{\xi}$ gives:

\bar{\xi}\bigl(y_{1}s^{\lambda(0)}+s^{\lambda(v)}+y_{1}\sum\limits_{m\neq\{0\},v}a_{m}s^{\lambda(m)}x^{m}\bigr)\\ =y_{1}s^{\lambda(0)}(1+O(|s|^{-\epsilon}))+\sum\limits_{i=2}^{d}\alpha_{i}y_{i}\frac{\partial}{\partial y_{i}}\bigl(y_{1}\sum\limits_{m\neq\{0\},v}a_{m}s^{\lambda(m)}x^{m}\bigr)\\ =y_{1}s^{\lambda(0)}(1+O(|s|^{-\epsilon}))+\sum\limits_{i=2}^{d}y_{1}s^{\lambda(0)}O(|s|^{-\epsilon})=y_{1}s^{\lambda(0)}(1+O(|s|^{-\epsilon}))\neq 0.

Thus, we can conclude that $\bar{\xi}$ is transversal to the tangent planes to $H_{s}^{v}(a)$ , that is $X_{s}(q)$ is transversal to all $H_{s}^{v}(a)$ . ∎

Remark.

The estimates for the monomials in the lemma can be used to give another proof of the Hausdorff convergence in Proposition 3.2. Note that for any $\epsilon>0$ for large enough $s$ , the monomial $x^{v}s^{\lambda(v)}$ become dominant in $\mathrm{Log}_{s}^{-1}(Q^{\lambda}_{v}(\epsilon))$ . Hence the equation for $H_{s}^{\operatorname{af{}f}}$ cannot have solutions in this domain. This means that the amoebas $\mathcal{A}^{\lambda}_{s}$ are $\epsilon$ -close to their spine $\mathcal{A}^{\lambda}_{\infty}$ .

Now let $w\in\operatorname{vert}(T)$ . Recall that $w^{\perp}$ is the set of integral points in $(\operatorname{carrier}_{\Delta^{\vee}}w)^{\vee}$ . For a fixed $s$ we consider the $(\Delta\cap(\mathbb{Z}^{d})^{*}-w^{\perp}-1)$ -parameter family of hypersurfaces:

s^{\lambda(0)}+\sum\limits_{m\in G_{w}}s^{\lambda(m)}x^{m}+\sum\limits_{m\notin G_{w}\cup\{0\}}a_{m}s^{\lambda(m)}x^{m}=0,\quad 0\leq a_{m}\leq 1,

and let $H_{s}^{w}(a)$ be its closure in $X_{s}(V^{\delta}_{w})$ . Now we can repeat the arguments of Lemma 3.8 to prove the analogous statement for the family $H_{s}^{w}(a)$ .

Lemma 3.9.

There exists $s_{0}$ such that whenever $|s|\geq s_{0}$ , all $H_{s}^{w}(a)$ are smooth and transversal to $X_{s}(q)$ for every $q\in V^{\delta}_{w}$ .

Proof.

According to Proposition 3.2 we can choose $s$ big enough so that the $\mathrm{Log}_{s}$ -image of the affine part of every hypersurface $H_{s}^{w}(a)$ lies in the $\epsilon$ -neighborhood of the Minkowski sum $V^{\delta}_{w}+\operatorname{cone}(w)$ . Also, recall from Lemma 3.3 that $V^{\delta}_{w}+\operatorname{cone}(w)$ lies in the domain $Q^{\lambda}_{(\{0\}\mid w^{\perp})}(\epsilon)$ . Thus, we can assume that the affine parts of all hypersurfaces $H_{s}^{w}(a)$ lie in $\mathrm{Log}_{s}^{-1}(Q^{\lambda}_{(\{0\}\mid w^{\perp})}(\epsilon))$ .

We choose a basis $\{e_{i}\}$ of $(\mathbb{Z}^{d})^{*}$ such that

\langle e_{1},w\rangle=-1\text{ and }\langle e_{i},w\rangle=0,i=2,\dots,d.

Then the affine coordinate functions $y_{i}=x^{e_{i}}$ can be extended (by allowing zero values for $y_{1}$ ) to the open part of the toric divisor $Z_{w}$ corresponding to the facet $F_{w}\subset\partial{\Delta_{\nu}}$ . Moreover, in these coordinates the preimage of each flow line $\mathcal{F}_{q}$ in $(\mathbb{C}\backslash\{0\})^{d}$ is defined by fixing the values of $|y_{2}|,\dots,|y_{d}|$ , so that its closure $X_{s}(q)$ is defined by the same equations, but allowing the zero value for $y_{1}$ . Hence, we can use $\{y_{i}\}$ as global coordinates on $X_{s}(V^{\delta}_{w})$ .

Multiplying the affine equation of $H_{s}^{w}(a)$ by $y_{1}$ we note that the Laurent polynomial

y_{1}\sum\limits_{m\in G_{w}}s^{\lambda(m)}x^{m}+y_{1}\sum\limits_{m\notin G_{w}\cup\{0\}}a_{m}s^{\lambda(m)}x^{m}

has only positive powers of $y_{1}$ , where as its first part $P_{1}(y)=y_{1}\sum\limits_{m\in G_{w}}s^{\lambda(m)}x^{m}$ is independent of $y_{1}$ at all. Thus, we get the global equation for the family in $X_{s}(V^{\delta}_{w})$ .

Now we can repeat the estimates for the monomials and their log-derivatives. Whenever $\mathrm{Log}_{s}(x)\in Q^{\lambda}_{(\{0\}\mid w^{\perp})}(\epsilon)$ , we have

\langle m,\frac{\log|x|}{\log|s|}\rangle+\lambda(m)\leq\lambda(0)-\epsilon,\text{ for all }m\notin G_{w}\cup\{0\},

or, equivalently,

|x^{m}s^{\lambda}(m)|\leq|s|^{-\epsilon}|s|^{\lambda(0)}.

When written in the $y$ -coordinates these estimates extends by continuity from the affine part to the entire $X_{s}(V^{\delta}_{w})$ .

To see that $H_{s}^{w}(a)$ has no critical points we differentiate its defining equation with respect to $y_{1}$ :

\frac{\partial}{\partial y_{1}}\bigl(y_{1}s^{\lambda(0)}+y_{1}\sum\limits_{m\in G_{w}}s^{\lambda(m)}x^{m}+y_{1}\sum\limits_{m\notin G_{w}\cup\{0\}}a_{m}s^{\lambda(m)}x^{m}\bigr)\\ =s^{\lambda(0)}+\bigl(1+y_{1}\frac{\partial}{\partial y_{1}}\bigr)\sum\limits_{m\notin G_{w}\cup\{0\}}a_{m}s^{\lambda(m)}x^{m}=s^{\lambda(0)}(1+O(|s|^{-\epsilon}))\neq 0,

for large enough $s$ .

Finally, Lemma 3.6 asserts that the vector field $\mathfrak{X}$ in $\bigcup_{q\in V^{\delta}_{w}}\mathcal{F}_{q}$ is constant and equal to $w$ . It means that $\frac{\partial}{\partial y_{1}}$ is a tangent vector to $X_{s}(q)$ , $q\in V^{\delta}_{w}$ , and it is transversal to $H_{s}^{w}(a)$ by the above calculation. ∎

Proof of Theorem 3.7.

Note that if all $a_{i}=1$ in family $H_{s}^{v}(a)$ , then we have the original equation of $H_{s}$ . On the other hand, if all $a_{i}=0$ , then the family $H_{s}^{v}(a)$ degenerates to the hyperbola:

H_{s}^{v}(0):=\{x\in X_{s}(U^{\epsilon}_{v})\ :\ s^{\lambda(0)}+s^{\lambda(v)}x^{v}=0\}.

Because $X_{s}(\mathcal{F}_{q})$ , $q\in U^{\epsilon}_{v}$ , intersect every $H_{s}^{v}(a)$ transversally, the corresponding fibers $F_{q}:=H_{s}\cap X_{s}(\mathcal{F}_{q})$ and $F_{q}^{v}:=H_{s}^{v}(0)\cap X_{s}(\mathcal{F}_{q})$ are diffeomorphic.

If $\theta=\{\theta_{i}\}$ denote the coordinates of the torus $\mathbb{T}$ and $\theta_{s}$ is the phase of $s$ , then the fiber $F_{q}^{v}$ of $H^{v}_{s}(0)$ is the torus

F_{q}^{v}=\{\theta\in\mathbb{T}\ :\ \langle v,\theta\rangle+(\lambda(0)-\lambda(v))\theta_{s}\equiv 0\ \mathrm{mod}\ 2\pi\},

which, for a fixed $s$ , can be identified with the torus $\mathbb{T}_{v}$ (though, see the remark below about monodromy as $\theta_{s}\mapsto\theta_{s}+2\pi$ ).

Similarly, the fibers $F_{q}=H_{s}\cap X_{s}(\mathcal{F}_{q})$ and $F_{q}^{w}:=H_{s}^{w}(0)\cap X_{s}(\mathcal{F}_{q})$ for $q\in V^{\delta}_{w}$ are diffeomorphic. But $F_{q}^{w}$ can be naturally identified with the torus $\mathbb{T}/w$ , which follows from writing the equation for $H_{s}^{w}(0)$ in the local coordinates $\{y_{i}\}$ from Lemma 3.9:

s^{\lambda(0)}y_{1}+P_{1}(y_{2},\dots,y_{d})=0,

where $P_{1}(y_{2},\dots,y_{d})$ is a Laurent polynomial independent of $y_{1}$ . Restricting to the fiber $X_{s}(q)$ means fixing absolute values of $y_{i},\ i=2,\dots,d$ . A point on the torus $\mathbb{T}/w$ determines the phases of $y_{i},\ i=2,\dots,d$ . Once $y_{i},\ i=2,\dots,d$ , are fixed, there is a unique solution to the equation of $H_{s}^{w}(0)$ .

Thus, $f_{s}:H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D)$ is a torus fibration. The only thing left to check is that it has the correct monodromy.

Note that all diffeomorphisms $F_{q}\cong F^{v}_{q}$ , $q\in U^{\epsilon}_{v}$ , and $F_{q}\cong F^{w}_{q}$ , $q\in V^{\delta}_{w}$ , are deformation diffeomorphisms. Hence, the transitions maps between $\mathbb{T}_{v}$ and $\mathbb{T}/w$ , for $q\in U_{v}\cap V_{w}$ , are homotopic to the map $f_{vw}:\mathbb{T}_{v}\to\mathbb{T}/w$ . But monodromy is a homotopy invariant, hence, it has to be equal to the one given by the maps $f_{vw}$ . This completes the proof. ∎

Remark.

The same statement was proven in [Zha00] for regular hypersurfaces in smooth toric varieties using partition of unity arguments. This method can also be applied in our situation since we do not touch the singular part of $X_{\Delta_{\nu}}$ at all.

Remark.

The fiber isomorphisms $F_{p}\cong\mathbb{T}_{v}$ depend on the value of the phase of $s$ . If we go around a loop $s\mapsto se^{2\pi i}$ , we won’t come back to the original diffeomorphism $H_{s}^{\mathrm{sm}}\to W^{\epsilon}$ . Rather, it will be a composition with a generalized Dehn twist, namely, the diffeomorphism $W^{\epsilon}\to W^{\epsilon}$ which is the fiber wise shift by a section of $W^{\epsilon}\to\Sigma\backslash N(D)$ (the tori are abelian groups). Such a section was explicitly written down in [Zha00].