ScalingStacks

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.

We now assign a a collection of charts to $\partial\Delta_{\lambda}^{\vee}$, whose transition functions are piecewise linear. (Some authors prefer the terminology ‘piecewise affine’.) These are closely related to the holomorphic charts on $X_{s}$ in section 3.1.

Let $w\in N$ be the primitive integral outward normal vector to a facet $F(w)$ of $\Delta$, and choose an integral basis $m_{1},\ldots m_{n}$ for $\{m\in M|\langle w,m\rangle=0\}$, suitably oriented to be compatible with (12). On the open subset of $\partial\Delta_{\lambda}^{\vee}$,

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

we regard $m_{1},\ldots m_{n}$ as the affine linear coordinates, also written as $x^{m_{1}},\ldots x^{m_{n}}$. Such charts cover $\partial\Delta_{\lambda}^{\vee}$. We denote $\widetilde{Sing}$ as the subset of points on $\partial\Delta_{\lambda}^{\vee}$ which do not lie on the interior of the top dimensional faces. It is easy to check that the transition functions on overlapping charts in $\partial\Delta_{\lambda}^{\vee}\setminus\widetilde{Sing}$ lie in $SL(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$, so the volume form $dx^{m_{1}}\wedge\ldots dx^{m_{n}}$ is defined independent of the choice of charts. We call the associated measure $d\mu_{\infty}$ the Lebesgue measure on $\partial\Delta_{\lambda}^{\vee}$, with respect to which $\widetilde{Sing}$ is a null set. The set $\widetilde{Sing}$ has real codimension 1, and the transition functions are in general only piecewise linear.

We now examine the normalised canonical measure on $X_{s}$

$$d\mu_{s}=\frac{1}{(4\pi s)^{n}}\sqrt{-1}^{n^{2}}\Omega_{s}\wedge\overline{\Omega}_{s}.$$

(14)

We specify a polarisation class $[\Delta]$ on the toric manifold $\mathbb{CP}^{n+1}=\mathbb{P}_{\Delta}$. A standard background Kähler metric is (a suitable multiple of) the Fubini-Study metric:

$$\begin{split}\omega_{FS}&=\frac{\sqrt{-1}(n+2)}{2}\partial\bar{\partial}\log(|Z_{0}|^{2}+\ldots|Z_{n+1}|^{2})\\ &=\frac{\sqrt{-1}(n+2)}{2}\partial\bar{\partial}\log(\sum_{m\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m,\text{Log}(z)\rangle}).\end{split}$$

Our normalisation guarantees that the potential has the asymptotic behaviour

$$\sup_{z}|\frac{(n+2)}{2}\log(\sum_{m\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m,\text{Log}(z)\rangle})-\max_{m\in\Delta}\langle m,\text{Log}(z)\rangle|<+\infty.$$

A general (singular) Kähler metric $\omega_{u}$ on $(\mathbb{P}_{\Delta},[\Delta])$ is given by a relative potential $u\in PSH(X,\omega_{FS})$. Alternatively, one thinks of $\omega_{u}$ as a collection of local absolute potentials:

$$\begin{cases}u_{0}=u+\frac{(n+2)}{2}\log(\sum_{m^{\prime}\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle}),\\ u_{m}=u+\frac{(n+2)}{2}\log(\sum_{m^{\prime}\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle})-\langle m,\text{Log}(z)\rangle,\end{cases}$$

(17)

where $u_{0}$ is a local potential in a compact region, and $u_{m}$ give the local potentials near the toric boundary.

We call a convex function $u$ on $N_{\mathbb{R}}=\mathbb{R}^{n+1}$ admissible if it satisfies the asymptotic growth condition

$$\sup_{x}|u(x)-\max_{m\in\Delta}\langle m,x\rangle|<+\infty,$$

(18)

which captures the information of the Kähler class.

A general (singular) Kähler metric $\omega_{\varphi}$ on $X_{s}$ in the polarisation class $s^{-1}[\Delta]$ is given by a potential $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$. The normalising factor $s^{-1}$ is aimed at extracting nontrivial limits as $s\to\infty$. We can completely analogous define the local potentials:

$$\begin{cases}\varphi_{0}=\varphi+\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex{(\Delta)}}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle}),\\ \varphi_{m}=\varphi+\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex{(\Delta)}}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle})-\langle m,\text{Log}_{s}(z)\rangle,\end{cases}$$

(19)

which are by definition psh on respective regions.

In particular, we can represent the Calabi-Yau metric $\omega_{CY,s}$ on $X_{s}$ by a potential $\varphi_{CY,s}$. The Calabi-Yau condition is

$$\omega_{CY,s}^{n}=a_{s}s^{-n}d\mu_{s},$$

(20)

where the normalising constant

$$a_{s}=\frac{\int_{X_{s}}[\Delta]^{n}}{\int_{X_{s}}d\mu_{s}}\to a_{\infty}=\frac{\int_{X_{s}}[\Delta]^{n}}{\text{Vol}(\partial\Delta_{\lambda}^{\vee})}$$

(21)

as $s\to+\infty$ (cf. Prop. 3.14).

We now discuss the issue of finding a tropical notion analogous to Kähler metrics. The concept of a Kähler metric is formulated in terms of a collection of local psh functions $\phi_{j}$ on overlapping complex charts, whose differences $\{\phi_{i}-\phi_{j}\}$ represent a given cocycle of local pluriharmonic function. Intuitively, the analogue should be a collection of local convex functions $u_{j}$ whose differences $\{u_{i}-u_{j}\}$ represent a given cocycle of local affine functions.

To the author’s awareness there is no definitive formulation of local convexity on polyhedral sets. In the case of interest, we need to define a class of ‘locally convex functions’ on $\partial\Delta_{\lambda}^{\vee}$. The problem is that on $\widetilde{Sing}\subset\partial\Delta_{\lambda}^{\vee}$, the transition functions between different charts are only piecewise linear, so convexity is not invariantly defined. This problem also prevents us from setting up a general global notion of real MA equation on $\partial\Delta_{\lambda}^{\vee}$, which is an essential ingredient in the SYZ conjecture in general. We will attempt to give a special definition in the Fermat case (cf. section 3.5).

However, the extension theorem 2.13 provides an alternative viewpoint: (1,1)-type Kähler currents can be defined extrinsically. By analogy, we propose that the correct notion should be equivalent to the following

The problem is to make this definition both intrinsic to $\partial\Delta_{\lambda}^{\vee}$, and local in nature. We do not fully succeed but shall make some partial progress.

We now introduce a local notion. The function $u$ below will be analogous to $\phi_{0}$ in (19). Recall the charts $\partial\Delta_{\lambda}^{\vee}\cap U_{w}^{\infty}$ associated to ourward normal vectors $w$ introduced in section 3.2, with local coordinates $x^{m_{1}},\ldots x^{m_{n}}$.