Let $N\simeq\mathbb{Z}^{n+1}$, and $M=\Hom(N,\mathbb{Z})$, and denote $N_{\mathbb{R}}=N\otimes\mathbb{R}$, $M_{\mathbb{R}}=M\otimes\mathbb{R}$. We regard $\mathbb{CP}^{n+1}$ as a toric Fano manifold $\mathbb{P}_{\Delta}$, with moment polytope $\Delta\subset M_{\mathbb{R}}$ corresponding to the anticanonical class $\mathcal{O}(n+2)$. More explicitly $\Delta$ is the $(n+1)$-simplex inside $M_{\mathbb{R}}\simeq\{\sum_{0}^{n+1}y_{i}=0\}\subset\mathbb{R}^{n+2}$ spanned by the vertices

$$(n+1,-1,\ldots-1),(-1,n+1,-1,\ldots-1),\ldots,(-1,\ldots,-1,n+1);$$

in particular $\Delta$ is a reflexive integral Delzant polytope, with dual polytope

$$\Delta^{\vee}=\{w\in N\otimes\mathbb{R}|\langle m,w\rangle\geq-1,\forall m\in\Delta\}\subset\mathbb{R}^{n+2}/\mathbb{R}(1,1,\ldots 1)$$

being the $(n+1)$-simplex spanned by the vertices $(1,0,\ldots,0),\ldots,(0,\ldots,0,1)$. The integral points $m\in\Delta_{\mathbb{Z}}=\Delta\cap M$ parametrize monomials $z^{m}$ in the anticanonical linear system $H^{0}(\mathbb{P}_{\Delta},\mathcal{O}(n+2))$. We study the family of hypersurfaces

$$X_{s}=\{F_{s}(z)=\sum_{m\in\Delta_{\mathbb{Z}}}a_{m}e^{s\lambda(m)}z^{m}=0\}\subset\mathbb{P}_{\Delta},\quad s\gg 1.$$

(8)

Here $a_{m}$ are a fixed collection of coefficients, with $a_{0}=1$ corresponding to the unique interior integral point $0\in\Delta_{\mathbb{Z}}$. For any vertex $m$ of $\Delta$, we require $a_{m}\neq 0$. The function $\lambda$ is defined for those $m\in\Delta_{\mathbb{Z}}$ for which $a_{m}\neq 0$; by assumption $\lambda(0)=0$, and $\lambda(m)<0$ otherwise. The natural piecewise linear extension of $\lambda$ to $M_{\mathbb{R}}$ is assumed to be concave, whose domains of linearity are by assumption simplices, producing a triangulation of $\Delta$. Using the adjunction formula, we can write down a holomorphic volume form $\Omega_{s}$ on $X_{s}$, such that along $X_{s}$

$$dF_{s}\wedge\Omega_{s}=d\log z^{1}\wedge\ldots d\log z^{n+1},$$

(9)

with $z^{1},z^{2},\ldots z^{n+1}$ the standard coordinates on $(\mathbb{C}^{*})^{n+1}\subset\mathbb{P}_{\Delta}$. We will always assume $s\gg 1$, and all the constants in the estimates are independent of $s$.

The key notion to describe the complex structure degeneration is a piecewise linear object called the tropicalisation of the hypersurfaces. Define the nonnegative piecewise linear function $L_{\lambda}$ on $N_{\mathbb{R}}$ by

$$L_{\lambda}(x)=\max_{m\in\Delta_{\mathbb{Z}},a_{m}\neq 0}\{\langle x,m\rangle+\lambda(m)\}.$$

The tropicalisation $\mathcal{A}_{\lambda}^{\infty}$ is defined as the nonsmooth locus of $L_{\lambda}$, or equivalently the locus inside $N_{\mathbb{R}}$ where the maximum $L_{\lambda}$ is achieved by at least two values of $m$. There is precisely one bounded component in the complement of $\mathcal{A}_{\lambda}^{\infty}$,

$$\Delta_{\lambda}^{\vee}=\{x|L_{\lambda}(x)=0\}\subset N_{\mathbb{R}},$$

whose boundary $\partial\Delta_{\lambda}^{\vee}\subset\mathcal{A}_{\lambda}^{\infty}$. The relation between the hypersurfaces and the tropicalisation is furnished by the rescaled log map,

$$\text{Log}_{s}:\mathbb{P}_{\Delta}\supset(\mathbb{C}^{*})^{n+1}\to\mathbb{R}^{n+1}\simeq N_{\mathbb{R}},\quad\text{Log}_{s}(z)=\frac{1}{s}(\log|z_{1}|,\ldots\log|z_{n+1}|).$$

The image $\mathcal{A}_{\lambda}^{s}=\text{Log}_{s}(X_{s}\cap(\mathbb{C}^{*})^{n+1})$ is called the amoeba. The following Prop. will be tacitly used frequently, as it allows us to think of regions on $X_{s}$ efficiently in terms of the regions on $\mathcal{A}_{\lambda}^{\infty}$, up to a tiny amount of fuzziness.

The tropicalisation $\mathcal{A}_{\lambda}^{\infty}$ is naturally stratified according to the subset of $m\in\Delta_{\mathbb{Z}}$ saturating the maximum $L_{\lambda}(x)$. This induces a kind of quantitative stratification structure on $\mathcal{A}_{\lambda}^{s}$ for $s\gg 1$.

Given a simplex $\sigma\subset\partial\Delta$ in the triangulation, we associate a subset $\mathcal{A}_{\lambda,\sigma}^{\infty}$:

$$\mathcal{A}_{\lambda,\sigma}^{\infty}=\{x\in\mathcal{A}_{\lambda}^{\infty}|L_{\lambda}(x)=\langle x,m\rangle+\lambda(m),\forall m\in\sigma\}.$$

Clearly if $\sigma\prec\sigma^{\prime}$, then $\mathcal{A}_{\lambda,\sigma}^{\infty}\supset\mathcal{A}_{\lambda,\sigma^{\prime}}^{\infty}$. The intuition is that larger $\sigma$ correspond to more nongeneric regions, and the complement of their neighbourhoods correspond to more generic regions.

Our next goal is to assign good holomorphic charts to $X_{s}$ related to the stratification structure. We first consider the toric region, which shall be covered by $(\mathbb{C}^{*})^{n}$-charts. Let $w\in N$ be the primitive integral outward normal vector to a facet $F(w)=\{m\in\Delta|\langle w,m\rangle=1\}$ of $\Delta$. The chart parametrised by $w$ is contained inside the region

$$U_{w}^{s,o}=\{z\in X_{s}|e^{s\lambda(m)}|z^{m}|\ll 1,\quad\forall m\in\Delta_{\mathbb{Z}}\setminus(F(w)\cup\{0\})\}.$$

(11)

Let $m_{0}\in F(w)\cap\Delta_{\mathbb{Z}}$, and choose an integral basis $m_{1},\ldots,m_{n}$ for $\{m\in M|\langle w,m\rangle=0\}$. Then the monomials $z^{m_{1}},\ldots,z^{m_{n}}$ provide the local $(\mathbb{C}^{*})^{n}$-coordinates on the chart, since by the implicit function theorem $X_{s}$ is locally a graph $\{z^{m_{0}}=f(z^{m_{1}},\ldots,z^{m_{n}})\}$. In fact by the defining equation (8) of the hypersurface

$$z^{-m_{0}}\approx-\sum_{m\in F(w)}a_{m}e^{s\lambda(m)}z^{m-m_{0}},$$

whence the holomorphic volume form is (cf. (9))

$$\Omega_{s}=\pm\frac{d\log z^{m_{0}}\wedge\ldots d\log z^{m_{n}}}{dF_{s}}\approx d\log z^{m_{1}}\wedge\ldots d\log z^{m_{n}}.$$

(12)

(Here $m_{i}$ are suitably oriented to take care of $\pm 1$.) We regard the above region as an open subset of $(\mathbb{C}^{*})^{n}$, and denote the chart $U^{s}_{w}$ as the largest $T^{n}$-invariant subset, delineated by a collection of affine linear inequalities on the variables $\log|z^{m_{i}}|$.

In the tropical limit $s=\infty$, the region $\text{Log}_{s}(U_{w}^{s,o})$ becomes

$$U_{w}^{\infty,o}=\{x\in\mathcal{A}_{\lambda}^{\infty}|\langle m,x\rangle+\lambda(m)<0,\forall m\in\Delta_{\mathbb{Z}}\setminus(F(w)\cup\{0\})\}$$

Inside this the limiting version of $\text{Log}_{s}(U_{w}^{s})$ is

$$U_{w}^{\infty}=\{(U_{w}^{\infty,o}\cap\partial\Delta_{\lambda}^{\vee})+\mathbb{R}_{\geq 0}w\}\cap\mathcal{A}_{\lambda}^{\infty}.$$

Later we shall also need the slightly shrinked regions for $0<\delta\ll\delta_{1}$: let

$$U^{s,o}_{w,\delta}=\{z\in X_{s}|e^{s\lambda(m)}|z^{m}|\ll e^{-s\delta},\quad\forall m\in\Delta_{\mathbb{Z}}\setminus(F(w)\cup\{0\})\},$$

whose largest $T^{n}$-invariant subset is $U^{s}_{w,\delta}$. The tropical limit of $\text{Log}_{s}(U_{w,\delta}^{s,o})$ is

$$U^{\infty,o}_{w,\delta}=\{x\in\mathcal{A}_{\lambda}^{\infty}|\langle m,x\rangle+\lambda(m)<-\delta,\forall m\in\Delta_{\mathbb{Z}}\setminus(F(w)\cup\{0\})\},$$

containing the limiting version of $\text{Log}_{s}(U_{w,\delta}^{s})$

$$U^{\infty}_{w,\delta}=\{(U_{w}^{\infty,o}\cap\partial\Delta_{\lambda}^{\vee})+\mathbb{R}_{\geq 0}w\}\cap\mathcal{A}_{\lambda}^{\infty}.$$

As the choice of $w$ varies, such regions $U_{w,\delta}^{\infty,o}$ cover a neighbourhood of $\partial\Delta_{\lambda}^{\vee}$ as a consequence of Lemma 3.4; so do $U_{w,\delta}^{\infty}$. This means the charts of toric type already cover part of the neighbourhood of the toric boundary.

We now consider the neighbourhood of the toric boundary near the stratum $\mathcal{A}_{\lambda,\sigma}^{\infty}$, but keeping away from higher strata and from $\partial\Delta_{\lambda}^{\vee}$. Here

$$|a_{m^{\prime}}e^{s\lambda(m^{\prime})}z^{m^{\prime}}|\ll|a_{m}e^{s\lambda(m)}z^{m}|,\quad\forall m^{\prime}\in\Delta_{\mathbb{Z}}\setminus\sigma,\forall m\in\sigma.$$

Since most terms in the defining equation (8) are negligible in our region, the hypersurface is locally approximately

$$\sum_{m\in\sigma\cap\Delta_{\mathbb{Z}}}a_{m}e^{s\lambda(m)}z^{m}\approx 0.$$

We focus on the subregion where $m_{0}^{\prime}\in\sigma$ achieves the maximal magnitude for $|a_{m}e^{s\lambda(m)}z^{m}|$, and $m_{1}^{\prime}\in\sigma$ achieves the second largest magnitude. These two magnitudes must be of comparable size by the hypersurface equation. Choose an integral basis $w_{1},\ldots w_{l}$ for the outward normal cone $NC_{\Delta}(\sigma)$, so $\langle m,w_{i}\rangle=1$ for $m\in\sigma\cap\Delta_{\mathbb{Z}}$. Denote the vertices of $\sigma$ as $m_{i}^{\prime}$ for $i=0,1,\ldots,\dim\sigma$, and choose $m_{1},\ldots m_{\dim\sigma-1}$ an integral basis of $\text{span}_{\mathbb{Q}}\{m_{2}^{\prime}-m_{0}^{\prime},\ldots,m_{\dim\sigma}^{\prime}-m_{0}^{\prime}\}\cap M$. Choose $m_{0}$ so that $m_{0},\ldots m_{\dim\sigma-1}$ is an integral basis of $\text{span}_{\mathbb{Q}}\{m_{1}^{\prime}-m_{0}^{\prime},\ldots,m_{\dim\sigma}^{\prime}-m_{0}^{\prime}\}\cap M$, and complete this into an integral basis $\{m_{0},\ldots,m_{n-l}\}$ for $\text{span}\{w_{1},\ldots w_{l}\}^{\perp}$, providing $(n+1-l)$ $\mathbb{C}^{*}$-variables $z^{m_{0}},\ldots z^{m_{n-l}}$. We then find $\mathfrak{m}_{j}$ for $j=1,2,\ldots l$, with $\langle\mathfrak{m}_{j},w_{i}\rangle=-\delta_{ij}$, and we can demand $\mathfrak{m}_{1}+\ldots\mathfrak{m}_{l}=-m_{0}^{\prime}$ because $\langle m_{0}^{\prime},w_{i}\rangle=1$. These provide the $\mathbb{C}$-variables $z^{\mathfrak{m}_{j}}$ for $1\leq j\leq l$, which can vanish on the toric boundary. On this local piece of $X_{s}$, the variables $z^{\mathfrak{m}_{j}}$ and $z^{m_{1}},\ldots z^{m_{n-l}}$ furnish a set of local coordinates as the $\mathbb{C}^{*}$-variable $z^{m_{0}}$ is expressible locally as a function of theirs.

The holomorphic volume form (9) is

$$\begin{split}\Omega_{s}=&\pm\frac{d\log z^{\mathfrak{m}_{1}}\wedge\ldots d\log z^{\mathfrak{m}_{l}}\wedge d\log z^{m_{0}}\wedge\ldots d\log z^{m_{n-l}}}{dF_{s}}\\ =&\pm\frac{dz^{\mathfrak{m}_{1}}\wedge\ldots dz^{\mathfrak{m}_{l}}}{z^{-m_{0}^{\prime}}}\bigwedge\frac{d\log z^{m_{0}}\wedge\ldots d\log z^{m_{n-l}}}{dF_{s}}\\ \approx&\pm dz^{\mathfrak{m}_{1}}\wedge\ldots dz^{\mathfrak{m}_{l}}\bigwedge d\log z^{m_{1}}\wedge\ldots d\log z^{m_{n-l}}\bigwedge\frac{d\log z^{m_{0}}}{a_{m_{1}^{\prime}}e^{s\lambda(m_{1}^{\prime})}dz^{m_{1}^{\prime}-m_{0}^{\prime}}}\\ =&\frac{dz^{\mathfrak{m}_{1}}\wedge\ldots dz^{\mathfrak{m}_{l}}}{a_{m_{1}^{\prime}}e^{s\lambda(m_{1}^{\prime})}z^{m_{1}^{\prime}-m_{0}^{\prime}}\mathfrak{d}}\bigwedge d\log z^{m_{1}}\wedge\ldots d\log z^{m_{n-l}},\end{split}$$

(13)

up to choosing appropriate ordering of the coordinates. Here $\mathfrak{d}$ is the divisibility of $m_{1}^{\prime}-m_{0}^{\prime}$ inside the group

$$\text{span}_{\mathbb{Q}}\{m_{1}^{\prime}-m_{0}^{\prime},\ldots,m_{\dim\sigma}^{\prime}-m_{0}^{\prime}\}\cap M/\text{span}_{\mathbb{Z}}\{m_{1},\ldots,m_{\dim\sigma-1}\}\simeq\mathbb{Z}.$$

Notice $a_{m_{1}^{\prime}}e^{s\lambda(m_{1}^{\prime})}z^{m_{1}^{\prime}-m_{0}^{\prime}}$ is uniformly equivalent to $a_{m_{0}^{\prime}}e^{s\lambda(m_{0}^{\prime})}$ in this region.

Local charts of the toric type and the boundary type cover the entire hypersurface $X_{s}$ for $s\gg 1$, and a substantial portion of any boundary type chart is in fact already covered by toric charts. Almost all the measure is contained in the toric type region.