3.3. Hyperplanes in the projective space [04SI]

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3.3. Hyperplanes in the projective space

This is a fundamental example for the main theorems. Let $H=\{z_{1}+\dots+z_{n+1}+1=0\}\subset{\mathbb{C}}{\mathbb{P}}^{n+1}$ be a hyperplane. Its toric part $H^{\circ}=H\cap(\mathbb{C}^{*})^{n+1}$ is an open pair-of-pants.

Let $\operatorname{Log}$ be the moment map for $(\mathbb{C}^{*})^{n+1}$ (see (3)).

Lemma 3.1.

$\Sigma_{n}\subset\operatorname{Log}(H^{\circ})$ .

Proof.

By [12] $\Sigma_{n}$ is a spine of the amoeba $\operatorname{Log}(H^{\circ})$ and, therefore, its subset. The lemma can alternatively be verified by writing explicit inequalities defining $\operatorname{Log}(H^{\circ})$ . ∎

The complement $\mathbb{R}^{n+1}\smallsetminus\Sigma_{n}$ consists of $n+2$ components. Each component is the region where one of the functions $0,x_{1},\dots,x_{n+1}$ is maximal. In the component corresponding to $x_{j}$ we consider the foliation into straight lines parallel to the gradient of $x_{j}$ (the $j$ th basis vector). In the component corresponding to $0$ we consider the foliation into straight lines parallel to $(1,\dots,1)$ . These foliations glue to a singular foliation $\mathcal{F}^{\prime}$ which has singularities at $\Sigma_{n}$ .

Refer to caption — Figure 6. The amoeba $\operatorname{Log}(H^{\circ})$ together with the foliation $\mathcal{F}^{\prime}$ and its deformation $\mathcal{F}$ .

It is easy to smooth out $\mathcal{F}^{\prime}$ (in a symmetric way with respect to the homogeneous coordinates permutations) at the open $n$ -cells of $\Sigma_{n}$ (see Figure 6). However, the singularities at the smaller-dimensional cells are essential. The leaves passing through an open $(n-k)$ -cell are homeomorphic to the cone over $k+2$ points.

We denote the resulting foliation with $\mathcal{F}$ . The foliation $\mathcal{F}$ is a singular fibration and defines the projection $\pi_{\mathcal{F}}:\mathbb{R}^{n+1}\to\Sigma_{n}$ .

The following statement is a key lemma in the proof of the main theorems of this paper.

Lemma 3.2.

The composition

\lambda_{H}=\pi_{\mathcal{F}}\circ\operatorname{Log}:H^{\circ}\to\Sigma_{n}

is a stratified $T^{n}$ -fibration in the sense of Definition 7. It satisfies to all conclusions of Theorem 3 except for the third one. The fibration $\lambda_{H}$ can be deformed so that the third condition will also hold.

The proof of this lemma occupies the rest of this subsection.

To figure out the fibers of $\lambda_{H}$ we need to understand the critical points of $\operatorname{Log}|_{H^{\circ}}$ . Following [6] and [11] for a hypersurface $V^{\circ}\subset(\mathbb{C}^{*})^{n+1}$ we define the logarithmic Gauss map

\gamma:V^{\circ}\to{\mathbb{C}}{\mathbb{P}}^{n}

by taking the composition of a branch of a holomorphic logarithm of each coordinate with the conventional Gauss map. This produces the following formula

\gamma(z_{1},\dots,z_{n+1})=[z_{1}\frac{\partial f}{\partial z_{1}}:\dots:z_{n+1}\frac{\partial f}{\partial z_{n+1}}],

where $f$ is the polynomial defining $V^{\circ}$ .

Note that the Newton polyhedron of $z_{j}\frac{\partial f}{\partial z_{j}}$ coincides with the Newton polyhedron $\Delta$ of $f$ . Therefore, by Kouchnirenko’s formula [9], $\deg\gamma=(n+1)!\operatorname{Vol}\Delta$ . In particular, if $V^{\circ}=H^{\circ}$ then $\deg\gamma=1$ .

Lemma 3.3 (cf. Lemma 3 of [11]).

The set of critical points of $\operatorname{Log}|_{V^{\circ}}$ coincides with $\gamma^{-1}(\mathbb{R}P^{n})$ .

Proof.

Let $z\in V^{\circ}$ and let $\mathcal{L}og$ be a branch of a holomorphic logarithm $(z_{1},\dots,z_{n+1})\mapsto(\log(z_{1}),\dots,\log(z_{n+1}))$ defined in a neighborhood of $z$ . The point $z$ is critical for $\operatorname{Log}|_{V^{\circ}}$ iff $V^{\circ}$ and the orbit of the real torus $T^{n}$ are not transversal at $z$ . But $\mathcal{L}og$ takes the tangent space to an orbit of $T^{n}$ to a translate of $i\mathbb{R}^{n+1}$ in $\mathbb{C}^{n+1}$ .

Therefore, $z$ is critical iff $\mathcal{L}og(T_{z}V^{\circ})$ contains at least $n$ purely imaginary vectors which is, in turn, equivalent to $\gamma(z)\in\mathbb{R}P^{n}$ . ∎

Corollary 3.4.

The set of critical points of $\operatorname{Log}|_{H^{\circ}}$ coincides with the real locus $\mathbb{R}H^{\circ}$ of $H^{\circ}$ (i.e. with the set of real solutions of $z_{1}+\dots+z_{n+1}+1=0$ ).

Proof.

Note that, since $H^{\circ}$ is defined over $\mathbb{R}$ , we have $\gamma(\mathbb{R}H^{\circ})\subset{\mathbb{R}}{\mathbb{P}}^{n}$ . Note that $\gamma$ extends to a map $H\to{\mathbb{C}}{\mathbb{P}}^{n}$ which is an isomorphism, since $\deg\gamma=1$ . ∎

Corollary 3.5.

The locus $\mathcal{D}\subset\operatorname{Log}(H^{\circ})$ of critical values of $\operatorname{Log}|_{H^{\circ}}$ is an immersed manifold transverse to the foliation $\mathcal{F}$ .

Proof.

The map $\operatorname{Log}|_{\mathbb{R}H^{\circ}}:\mathbb{R}H^{\circ}\to\mathcal{D}\subset\mathbb{R}^{n+1}$ is an immersion since the map $\operatorname{Log}|_{(\mathbb{R}^{*})^{n+1}}:(\mathbb{R}^{*})^{n+1}\to\mathbb{R}^{n+1}$ is an immersion (it is a trivial $2^{n+1}$ -covering of $\mathbb{R}^{n+1}$ ).

To see the transversality we recall the definition of the foliation $\mathcal{F}^{\prime}$ . For each component of $\mathbb{R}^{n+1}\smallsetminus\Sigma_{n}$ the foliation $\mathcal{F}^{\prime}$ is parallel to a vector $\stackrel{{\scriptstyle\to}}{{v}}$ normal to a facet of the Newton polyhedron of $H^{\circ}$ . Therefore, any hyperplane in the image $\gamma(\mathbb{R}H^{\circ})$ is transverse to $\stackrel{{\scriptstyle\to}}{{v}}$ . Furthermore, hyperplanes close to being parallel to $\stackrel{{\scriptstyle\to}}{{v}}$ are close to the hyperplane in ${\mathbb{C}}{\mathbb{P}}^{n+1}$ corresponding to this facet and therefore are far from the given component of $\mathbb{R}^{n+1}\smallsetminus\Sigma_{n}$ . Thus the result $\mathcal{F}$ of smoothing is also transverse to $\mathcal{D}$ and the angle between them in $\mathbb{R}^{n+1}$ is separated from 0. ∎

Note that $\pi_{\mathcal{F}}$ is a stratified $[-1,1]$ -fibration. Thus, the transversality of $\mathcal{D}$ and $\mathcal{F}$ implies that $\lambda_{H}$ is a stratified fibration for $\Sigma_{n}$ . We need to show that the restriction of $\pi_{F}$ to open $n$ -cells of $\Sigma$ is a torus fibration.

Consider a point $x=(-t,\dots,-t,0)$ for a large $t>0$ . Note that $\mathcal{D}$ is almost horizontal near $x$ . Thus the fiber of $\lambda_{H}$ over $x$ is diffeomorphic to the fiber $F$ of a composition of $\operatorname{Log}|_{H^{\circ}}$ and the linear projection onto the first $n$ coordinates. Note that the map $F\to T^{n}$ obtained by taking the arguments of the first $n$ coordinates is a diffeomorphism. Recall that $H^{\circ}$ is given by the equation $z_{1}+\dots+z_{n+1}+1=0$ . The absolute values of the coordinates $z_{1},\dots,z_{n}$ are fixed. For any value of their argument we take $z_{n+1}=1-z_{1}-\dots-z_{n}$ to get the unique point from $F$ corresponding to this choice of the arguments. Since $|z_{1}|,\dots,|z_{n}|$ are small $z_{n+1}\neq 0$ .

We verify the conclusions of Theorem 3 item-by-item. The first and the last conclusions are vacuous in this case, since $\Sigma_{n}$ (and, therefore, $\bar{\Sigma}_{n}$ as well) is contractible. The second one holds since $H^{\circ}$ is itself an open pair-of-pants.

To make the third conclusion true we have to modify $\lambda_{H}$ a little. The fiber $F$ is not Lagrangian, but it is close to a Lagrangian torus $\Lambda=\{|z_{j}|=\operatorname{const},\ j=1,\dots,n,\ z_{n+1}=-1\}$ . We can deform $H^{\circ}$ a little in a neighborhood of $\operatorname{Log}^{-1}(x)$ to make it intersect the fiber of $\pi_{\mathcal{F}}\circ\operatorname{Log}$ along $\Lambda$ . Therefore, $F$ is Lagrangian for a nearby symplectic structure. By Moser’s trick (see e.g. [2]) there exists a self-diffeomorphism $h$ of $H^{\circ}$ constant outside of a neighborhood of $\operatorname{Log}^{-1}(x)$ and taking one symplectic structure to another. We redefine $\lambda$ as $\lambda\circ h$ . This ensures a Lagrangian fiber over one of the $\begin{pmatrix}n+2\\ 2\end{pmatrix}$ open $n$ -cells of $\Sigma_{n}$ . We do the same for all other $n$ -cells.