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 $s$.