3.4. A localization Q n ⊂ ( ℂ ∗ ) n + 1 of the standard hyperplane [04ST]

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3.4. A localization $Q^{n}\subset(\mathbb{C}^{*})^{n+1}$ of the standard hyperplane

The toric part $H^{\circ}\subset(\mathbb{C}^{*})^{n+1}$ of a hyperplane from 3.3 is a nice embedding of $\mathcal{P}_{n}$ to $(\mathbb{C}^{*})^{n+1}$ . However for our purposes it is convenient to modify it in a neighborhood of infinity to get a different submanifold $Q^{n}$ which is better suited for gluing.

Note that the symmetric group $S_{n+2}$ acts on $(\mathbb{C}^{*})^{n+1}$ by interchanging the functions $z_{1},\dots,z_{n+1},\frac{1}{z_{1}\dots z_{n+1}}$ . This action is inherited from the action of $S_{n+2}$ on ${\mathbb{C}}{\mathbb{P}}^{n+1}$ interchanging the homogeneous coordinates since $(\mathbb{C}^{*})^{n+1}\subset{\mathbb{C}}{\mathbb{P}}^{n+1}$ is invariant. The hyperplane $H^{\circ}$ is invariant with respect to this action.

Proposition 3.6.

There exists a proper submanifold $Q^{n}\subset(\mathbb{C}^{*})^{n+1}$ such that

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$Q^{n}$ is embedded to $(\mathbb{C}^{*})^{n+1}$ symplectically, i.e. so that the restriction of the form (2) to $Q^{n}$ is a symplectic form.
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$Q^{n}$ is isotopic to $H^{\circ}$ in $(\mathbb{C}^{*})^{n+1}$ .
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The composition $\pi_{\mathcal{F}}\circ\operatorname{Log}_{t}|_{Q^{n}}$ is a stratified $T^{n}$ -fibration that satisfies to all hypotheses of Theorem 3.
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the closure $\bar{Q^{n}}$ of $Q^{n}$ in ${\mathbb{C}}{\mathbb{P}}^{n+1}\supset(\mathbb{C}^{*})^{n+1}$ is a smooth manifold isotopic to $H$ .
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$Q^{n}$ is invariant with respect to the action of the symmetric group $S_{n+2}$ on $(\mathbb{C}^{*})^{n+1}$ (see above).
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For a sufficiently large $M>0$

$Q^{n}\cap(\mathbb{C}^{*})^{n+1}_{-M}=Q^{n-1}\times\\ C^{*}_{-M},$

where $(\mathbb{C}^{*})^{n+1}_{-M}=\{(z_{1},\dots,z_{n+1})\in(\mathbb{C}^{*})^{n+1}\ |\ \log|z_{n+1}|<-M\}$ and $\mathbb{C}^{*}_{-M}=\{z\in\mathbb{C}^{*}\ |\ \log|z|<-M\}$ . In particular, the intersection $Q^{n}\cap(\mathbb{C}^{*})^{n+1}_{-M}$ is invariant under a translation $z_{n+1}\mapsto cz_{n+1}$ , $0<c<1$ .

Refer to caption — Figure 7. The amoeba of the localization $Q^{n}$ of a hyperplane.

Proof.

We construct $Q^{n}$ inductively by dimension $n$ . If $n=0$ then $H^{\circ}$ is a point and $Q^{0}=H^{\circ}$ . Assume that $Q^{k}$ , $k<n$ is already constructed. Consider the simplex

\Delta_{n}(R)=\{x\in\mathbb{R}^{n+1}\ |\ -x_{j}\leq R,\sum\limits_{j}x_{j}\leq R\}.

Each its $k$ -dimensional face is dual to a $(n+1-k)$ -cell of $\Sigma_{n}$ . Fix a sufficiently large number $R_{n}>0$ .

First we define $Q^{n}\cap\operatorname{Log}^{-1}(\partial\Delta(R_{n}))$ . Each $k$ -face of $\Delta(R_{n})$ is contained in a unique affine $k$ -space $A$ in $\mathbb{R}^{n+1}$ . Furthermore, the adjoint faces cut the polyhedron $\Delta_{k-1}(R_{n})\subset A$ . Thus we may identify $A$ with $\mathbb{R}^{k}$ and, therefore, $\operatorname{Log}^{-1}(A)$ with $(\mathbb{C}^{*})^{k}$ . By the induction assumption we already have $Q^{k-1}\subset(\mathbb{C}^{*})^{k}\to\mathbb{R}^{k}$ . We define $Q^{n}\cap\operatorname{Log}^{-1}(\partial\Delta(R_{n}))$ to be equal to the union of these $Q^{k}$ for all faces of $\partial\Delta(R_{n})$ . By the induction hypothesis (and since $R_{n}$ was large enough) the choices over different faces agree.

Our next step is to extend $Q^{n}$ to the complement of $\operatorname{Log}^{-1}(\Delta(R_{n}))$ . For each face $\Delta^{\prime}$ of $\partial\Delta(R_{n})$ consider its outer normal cone $C_{\Delta^{\prime}}\subset\mathbb{R}^{n+1}$ (e.g. if $\Delta^{\prime}$ is a facet then $C_{\Delta^{\prime}}$ is a ray). We define

Q^{n}\cap\operatorname{Log}^{-1}(\Delta^{\prime}+C_{\Delta^{\prime}})=\bigcup\limits_{\stackrel{{\scriptstyle\to}}{{v}}\in C_{\Delta^{\prime}}}e^{\stackrel{{\scriptstyle\to}}{{v}}}Q^{n}\cap\operatorname{Log}^{-1}(\Delta^{\prime}).

In other words, we span the region above the normal cone of a $k$ -face $\Delta^{\prime}$ by the translates of the manifold $Q^{k}$ .

We set $Q^{n}\cap\operatorname{Log}^{-1}(\Delta(R_{n}-1))=H^{\circ}\cap\operatorname{Log}^{-1}(\Delta(R_{n}-1))$ . By now we have defined $Q^{n}$ everywhere, but $\operatorname{Log}^{-1}(\Delta(R_{n})\smallsetminus\Delta(R_{n}-1))$ .

Consider a facet $\Delta^{\prime}$ of $\partial\Delta(R_{n}-1)$ , e.g. the one sitting in the hyperplane $A=\{x_{n+1}=R_{n}-1\}$ . Since $R_{n}$ is large enough, $z_{n+1}^{R_{n}-1}$ is small enough and the intersection $H^{\circ}\cap\operatorname{Log}^{-1}(A)$ is close enough to the zero set of $z_{1}+\dots+z_{n}+1=0$ . By the induction hypothesis this zero set can be deformed to $Q^{n-1}$ . We define $Q^{n}\cap\{\operatorname{Log}|z_{n+1}|=t\}$ , $-R_{n}\leq t\leq-R_{n}+1$ using this deformation. We repeat the same procedure for all other facets of $\Delta(R_{n}-1)$ . ∎

Denote $\bar{Q}^{n}=Q^{n}\cap\operatorname{Log}^{-1}(\Delta(R_{n}+1))$ . This is “the kernel” of $Q^{n}$ and is diffeomorphic to a closed pair-of-pants $\bar{\mathcal{P}_{n}}$ (as a manifold with boundary and corners).

3.4. A localization Q n ⊂ ( ℂ ∗ ) n + 1 of the standard hyperplane [04ST]

3.4. A localization Qn⊂(ℂ∗)n+1Q^{n}\subset(\mathbb{C}^{*})^{n+1} of the standard hyperplane

Proposition 3.6.

Proof.

3.4. A localization $Q^{n}\subset(\mathbb{C}^{*})^{n+1}$ of the standard hyperplane