3.2. Amoebas of hypersurfaces [03DJ]

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3.2. Amoebas of hypersurfaces

Let $\mathrm{Log}_{s}:(\mathbb{C}\backslash\{0\})^{d}\rightarrow\mathbb{R}^{d}$ be the logarithmic map with the base $|s|$ :

\mathrm{Log}_{s}(x):=\frac{\log(|x|)}{\log|s|}=\left\{\frac{\log|x_{1}|}{\log|s|},\dots,\frac{\log|x_{d}|}{\log|s|}\right\}.

Definition.

([GKZ94, Ch. 6]) The amoeba associated to the family of affine hypersurfaces $H_{s}^{{\operatorname{af{}f}}}$ is the image of the log map:

\mathcal{A}^{\lambda}_{s}:=\mathrm{Log}_{s}(H_{s}^{\operatorname{af{}f}}).

The geometry of amoebas of affine hypersurfaces is a well developed subject that originated in the work of Gelfand, Kapranov and Zelevinsky [GKZ94]. We are going to review several useful facts about the amoebas, most of which are contained in (or can be easily deduced from) a nice survey paper by Mikhalkin [Mik01].

The limiting behavior of amoebas as $s\to\infty$ can be described in terms of the Legendre transform $L_{\lambda}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ of the vector $\lambda$ :

L_{\lambda}(n)=\max_{m\in\Delta\cap(\mathbb{Z}^{d})^{*}}\{\langle m,n\rangle+\lambda(m)\}.

Remark.

In the literature, the Legendre transform is sometimes defined with a “minus” rather than a “plus” sign. Those references work with convex (not concave) $\lambda$ .

$L_{\lambda}(n)$ is a piecewise linear convex function. Define the non-Archimedean amoeba $\mathcal{A}^{\lambda}_{\infty}\subset\mathbb{R}^{d}$ to be the corner locus of $L_{\lambda}(n)$ (the set of points where $L_{\lambda}(n)$ is not smooth). $\mathcal{A}^{\lambda}_{\infty}$ is a rational polyhedral complex of dimension $d-1$ (cf. [Mik02]), which gives a cell decomposition of $\mathbb{R}^{d}$ .

Lemma 3.1.

The decomposition of $\mathbb{R}^{d}$ by $\mathcal{A}^{\lambda}_{\infty}$ has the following description:

(1)

The cells are labeled by the simplices $\overline{\sigma}\in S\ast\{0\}$ .
(2)

(The closure of) a cell $Q^{\lambda}_{\overline{\sigma}}$ is the Minkowski sum of the polytope and the cone:

$Q^{\lambda}_{\overline{\sigma}}=F^{\vee}_{\sigma}+\operatorname{NC}_{\Delta}(\overline{\sigma}).$

Here $F^{\vee}_{\sigma}$ is the face of $\partial\Delta^{\vee}_{\lambda}$ dual to $\sigma=\overline{\sigma}\cap{\partial\Delta}\in S$ , (where we set $F^{\vee}_{\emptyset}=\Delta^{\vee}_{\lambda}$ for $\overline{\sigma}=\{0\}$ ), and $\operatorname{NC}_{\Delta}(F)$ is the normal cone to the face $F\prec\Delta$ , ( in particular $\operatorname{NC}_{\Delta}(F)=\{0\}$ for $F=\Delta$ ). Thus, $Q^{\lambda}_{\overline{\sigma}}$ is unbounded if and only if $\overline{\sigma}=\sigma\in S$ .
(3)

In particular, the $d$ -dimensional cells are labeled by the elements of $\operatorname{vert}(S)\cup\{0\}$ . That is, there is a bounded central cell $Q^{\lambda}_{\{0\}}=\Delta^{\vee}_{\lambda}$ and unbounded cells $Q^{\lambda}_{v}$ , one for each vertex $v\in\operatorname{vert}(S)$ (see Fig. 3.2).

Proof.

All statements follow easily from the definition of $L_{\lambda}$ . Namely, the cells correspond to the subsets $I\subset\Delta\cap(\mathbb{Z}^{d})^{*}$ : the corresponding linear functions $\langle m,n\rangle+\lambda(m),\ m\in I$ , saturate the maximum in $L_{\lambda}$ . Since $\lambda$ is a concave function this can happen only if $I$ is a set of vertices of some simplex $\overline{\sigma}\in S\ast\{0\}$ . This proves (1).

For (3) we notice that a $d$ -cell is a domain of linearity of $L_{\lambda}$ , labeled by the vertex $m\in\operatorname{vert}(S)\cup\{0\}$ whose corresponding linear function $\langle m,n\rangle+\lambda(m)$ is maximal. In particular, the central cell $Q^{\lambda}_{\{0\}}$ is the set of $n\in\mathbb{R}^{d}$ , such that the maximum is achieved by $\langle\{0\},n\rangle+\lambda(0)$ , i.e.

\langle m,n\rangle+\lambda(m)\leq\lambda(0),\ \text{ all }m\in\Delta\cap(\mathbb{Z}^{d})^{*},

which are exactly the defining inequalities for $\Delta^{\vee}_{\lambda}$ .

More generally, a point $n$ is in (the closure of) $Q^{\lambda}_{\overline{\sigma}}$ if and only if:

	$\displaystyle\langle m-v,n\rangle+\lambda(m)\leq\lambda(v)-\lambda(m),\ \text{ all }v\in\mathrm{vert}\overline{\sigma},\ m\in\Delta\cap(\mathbb{Z}^{d})^{*},$
	$\displaystyle\text{ and }\ \langle v_{1},n\rangle+\lambda(v_{1})=\langle v_{2},n\rangle+\lambda(v_{2}),\ v_{1},v_{2}\in\mathrm{vert}(\overline{\sigma}),$

which are exactly the defining inequalities for the polyhedron $F^{\vee}_{\sigma}+\operatorname{NC}_{\Delta}(\overline{\sigma})$ . ∎

The polyhedral complex $\mathcal{A}^{\lambda}_{\infty}$ is also called the spine of the amoeba $\mathcal{A}^{\lambda}_{s}$ because of the following fact (cf. [Mik02]):

Proposition 3.2.

As $s\to\infty$ the amoebas $\mathcal{A}^{\lambda}_{s}$ converge in the Hausdorff sense to the non-Archimedean amoeba $\mathcal{A}^{\lambda}_{\infty}$ .

Idea of the proof.

If we consider $s$ as a variable, we can think of the affine family $H_{s}^{\operatorname{af{}f}}$ as one hypersurface given by a single equation in $(\mathbb{C}\backslash\{0\})^{d+1}$ . Then the rescaled amoeba $\log|s|\cdot\mathcal{A}^{\lambda}_{s}$ sits inside the trace left by this extended $(d+1)$ -dimensional amoeba in the horizontal hyperplane $\log|s|=const$ . And the result follows from [GKZ94, Ch. 6, Prop. 1.9]. ∎

Given a $\mathbb{T}$ -invariant Kähler form on $X_{{\Delta_{\nu}}}$ in the class $[\nu]$ , we can consider the corresponding moment map $\mu\colon X_{{\Delta_{\nu}}}\to{{\Delta_{\nu}}}$ . In this case we can also define the compactified amoeba $\overline{\mathcal{A}^{\lambda}_{s}}:=\mu(H_{s})\subset{\Delta_{\nu}}$ .

For the proof of Theorem 3.7 we will need to introduce some domains in $\mathbb{R}^{d}$ , which are intimately connected with the amoebas and the function $L_{\lambda}$ . In some sense they are generalizations of the cells induced by $\mathcal{A}^{\lambda}_{\infty}$ .

Definition.

For $I$ and $J$ , two disjoint collections of integral points in $\Delta$ , and a real number $\epsilon\geq 0$ , we define the (possibly empty) polyhedron $Q^{\lambda}_{(I\mid J)}(\epsilon)$ in $\mathbb{R}^{d}$ by the conditions:

	$\displaystyle\langle m^{\prime\prime},n\rangle+\lambda(m^{\prime\prime})<\langle m,n\rangle+\lambda(m)-\epsilon,$
	$\displaystyle\langle m^{\prime},n\rangle+\lambda(m^{\prime})=\langle m,n\rangle+\lambda(m),$

for all $m,m^{\prime}\in I,m^{\prime\prime}\not\in I\cup J$ . We will abbreviate $Q^{\lambda}_{(I\mid J)}(\epsilon)$ by $Q^{\lambda}_{I}(\epsilon)$ when $J$ is empty.

Definition.

For $w\in{\partial\Delta^{\vee}}\cap\mathbb{Z}^{d}$ let $w^{\perp}$ be the set of integral points in $(\operatorname{carrier}_{\Delta^{\vee}}w)^{\vee}$ , that is

w^{\perp}=\{m\in\Delta\cap(\mathbb{Z}^{d})^{*}\ :\ \langle m,w\rangle=1\}.

Then, the truncated polytope $\Delta\backslash w^{\perp}$ is defined as the convex hull of integral points of $\Delta$ which are not in $w^{\perp}$ .

Notice that for small $\epsilon$ , $Q^{\lambda}_{(I\mid J)}(\epsilon)$ and $Q^{\lambda}_{(I\mid J)}(0)$ are combinatorially equivalent.

Lemma 3.3.

In certain special cases of later interest we can describe $Q^{\lambda}_{(I\mid J)}(0)$ as follows:

(1)

If $J=\emptyset$ , then $Q^{\lambda}_{I}(0)$ is non-empty if and only if $I$ is the set of vertices of some simplex $\overline{\sigma}\in S\ast\{0\}$ , in which case $Q^{\lambda}_{I}(0)=Q^{\lambda}_{\overline{\sigma}}$ .
(2)

$Q^{\lambda}_{(\{0\}\mid v)}(0)$ contains the relative interior of the facet $G_{v}\subset\Delta^{\vee}_{\lambda}$ .
(3)

$Q^{\lambda}_{(\{0\}\mid w^{\perp})}(0)$ is the Minkowski sum of the polytope $\Delta^{\vee}_{\lambda}$ and the normal cone to $\operatorname{carrier}_{\Delta\backslash w^{\perp}}\{0\}$ :

$Q^{\lambda}_{(\{0\}\mid w^{\perp})}(0)=\Delta^{\vee}_{\lambda}+\operatorname{NC}_{\Delta\backslash w^{\perp}}(\{0\}).$

In particular, it contains $\Delta^{\vee}_{\lambda}+\operatorname{cone}(w)$ .

Proof.

For (1) we notice that the set of defining inequalities of $Q^{\lambda}_{I}(0)$ is exactly the condition that the maximum in $L_{\lambda}$ is saturated by the linear functions $\langle m,n\rangle+\lambda(m),\ m\in I$ .

For (2), note that $Q^{\lambda}_{v\ast\{0\}}$ is defined by the same inequalities as $Q^{\lambda}_{(\{0\}\mid v)}(0)$ , plus an extra condition: $\langle v,n\rangle+\lambda(v)=\lambda(0)$ .

For (3) we can study the Legendre transform of the restriction of $\lambda$ to the truncated polytope $\Delta\backslash w^{\perp}$ (which is still a concave function). From this point of view, the Minkowski sum in (3) is in complete analogy with (2) of Lemma 3.1.

Note that in the process of truncation we removed all integral points of $\Delta$ with $\langle m,w\rangle=1$ . Hence, the remaining ones satisfy $\langle m,w\rangle\leq 0$ which implies that $\{0\}$ is on the boundary of $\Delta\backslash w^{\perp}$ , and $w$ is in (the boundary of) the normal cone $\operatorname{NC}_{\Delta\backslash w^{\perp}}(\{0\})$ . ∎