Theorem 21.8.3E49F0370E203

Proof. Every point of $X_ P$ has a nonzero vertex coordinate. Let $m\in A$. By lemma 21.8.2(1), $m$ lies in the convex hull of the vertices of $P$, so by lemma 21.2.3(3) it is a convex combination of affinely independent vertices. The coefficients of this combination are the unique solution of a linear system with integral coefficients, hence rational. Clearing denominators, we obtain

Since $\chi ^{dm}=\prod _ v(\chi ^ v)^{c_ v}$, the homogeneous equation $Z_ m^ d=\prod _ v Z_ v^{c_ v}$ of degree $d$ holds on $\iota (T)$, hence on its closure $X_ P$. If all vertex coordinates of a point of $X_ P$ vanished, these equations would force all its coordinates to vanish, which is impossible.

The vertex charts. Fix a vertex $v$. For $m\in A$, lemma 21.8.2(4) gives $m-v=\sum _ ib_ i(m)e_ i$ with $b_ i(m)=\ell _ i(m)=\langle m,u_ i\rangle +a_ i$, a nonnegative integer; we write $b(m)=(b_1(m),\ldots ,b_ n(m))$. In particular $b(v+e_ i)$ is the $i$th unit vector, and on $\iota (T)$

Let $\Phi _ v\colon \mathbb C^ n\to \{ Z_ v\neq 0\} $ be the map whose coordinates $Z_ m/Z_ v$ are the monomials $\prod _ iw_ i^{b_ i(m)}$. Its image $\Gamma _ v$ is the graph of a polynomial map $\gamma $ from the coordinates $w_ i=Z_{v+e_ i}/Z_ v$ of the affine chart $\{ Z_ v\neq 0\} $ to the remaining coordinates $y=(Z_ m/Z_ v)_{m\neq v,v+e_1,\ldots ,v+e_ n}$. Hence $\Gamma _ v$ is closed in the chart, and in the holomorphic coordinates $(w,y-\gamma (w))$ of the chart it is the coordinate plane $\{ y-\gamma (w)=0\} $; so $\Gamma _ v$ is a closed complex submanifold of the chart, and $w$ is a biholomorphism of $\Gamma _ v$ onto $\mathbb C^ n$, with inverse $\Phi _ v$. Since $e_1,\ldots ,e_ n$ is a basis of $M$, the map $t\mapsto (\chi ^{e_1}(t),\ldots ,\chi ^{e_ n}(t))$ is a bijection of $T$ onto $(\mathbb C^*)^ n$, and the last display shows that $\iota (T)=\Phi _ v((\mathbb C^*)^ n)$. As $(\mathbb C^*)^ n$ is dense in $\mathbb C^ n$ and $\Gamma _ v$ is closed in the chart, the closure of $\iota (T)$ in the chart is $\Gamma _ v$; that is, $U_ v=\Gamma _ v$. By the first step the sets $U_ v$ cover $X_ P$, so $X_ P$ is a complex submanifold of dimension $n$ with the asserted charts. The coordinates $w_ i$ recover $t$ from $\iota (t)$, so $\iota $ is injective with holomorphic inverse $w$ on $\iota (T)$, and $\iota (T)=\Phi _ v((\mathbb C^*)^ n)$ is open in $U_ v$. Thus $\iota $ is a biholomorphism of $T$ onto a dense open subset of $X_ P$.

Algebraicity and irreducibility. In addition to the equations of the first step, consider for each vertex $v$ and each $m\in A\setminus \{ v\} $ the homogeneous equation

where $\sum _ ib_ i(m)\geq 1$ because $m\neq v$. It holds on $\iota (T)$, hence on $X_ P$. Let $Y$ be the common zero set of all these equations in $\mathbb P^{|A|-1}$, so that $X_ P\subset Y$. By the equations of the first step, every point of $Y$ has a nonzero vertex coordinate $Z_ v$, and on $\{ Z_ v\neq 0\} $ the equations for $v$ say exactly that $Z_ m/Z_ v=\prod _ iw_ i^{b_ i(m)}$; thus $Y\cap \{ Z_ v\neq 0\} \subset \Gamma _ v=U_ v$. Hence $Y=X_ P$. To prove irreducibility, suppose that $X_ P=Y_1\cup Y_2$, where $Y_1,Y_2\subsetneq X_ P$ are common zero sets of homogeneous polynomials, and choose homogeneous polynomials $f_1,f_2$ with $f_ k=0$ on $Y_ k$ and $f_ k\not\equiv 0$ on $X_ P$. Then $g_ k(t)=f_ k((\chi ^ m(t))_{m\in A})$ is a Laurent polynomial in the coordinates of $T\cong (\mathbb C^*)^ n$. It is not identically zero: otherwise $f_ k$ would vanish on $\iota (T)$, hence on its closure $X_ P$, since the zero set of a homogeneous polynomial is closed. On the other hand $g_1g_2=0$ on $T$, because $f_1f_2=0$ on $X_ P$. This is impossible. Indeed, since $g_1$ is continuous and not identically zero, it does not vanish on some nonempty open set $W\subset (\mathbb C^*)^ n$, and $g_2$ vanishes on $W$. A suitable monomial multiple of $g_2$ is a polynomial that vanishes on $W$, so all its derivatives vanish at a point of $W$; being the sum of its Taylor expansion at that point, it is zero. Hence $g_2=0$, a contradiction. Thus $X_ P$ is irreducible.

The line bundle. The coordinate sections $s_ m=Z_ m|_{X_ P}$, $m\in A$, of $L$ have no common zero and define the inclusion $X_ P\subset \mathbb P^{|A|-1}$. By theorem 21.9.5 with $k=1$, whose proof uses only the vertex charts constructed above, they form a basis of $H^0(X_ P,L)$; in the notation of that theorem, $s_ m=\chi ^ ms_0$. Hence the full space of sections of $L$ defines an embedding, and $L$ is very ample in the sense of section 10.6. On $U_ v$ the section $s_ v$ has no zeros, and the Fubini–Study metric of example 10.6.6, example 10.6.7, restricted to $L$, gives it the squared norm

where $w^ b=\prod _ iw_ i^{b_ i}$. The curvature on $U_ v$ is $\mathrm{dd}^{\mathrm{c}}H_ v$, by definition 6.14.9. The complex Hessian of $H_ v$ is positive definite. Indeed, put $f_ m=w^{b(m)}$ and $S=\sum _ m|f_ m|^2$, and for $\xi \in \mathbb C^ n$ write $\xi f_ m=\sum _ j\xi _ j\partial f_ m/\partial w_ j$. Then

by the Cauchy–Schwarz inequality, with equality only if $(\xi f_ m)_ m=\lambda (f_ m)_ m$ for some $\lambda \in \mathbb C$. The constant monomial $f_ v=1$ forces $\lambda =0$, and then the monomials $f_{v+e_ i}=w_ i$ give $\xi _ i=0$ for every $i$. Thus the restricted Fubini–Study metric is a smooth quadratic metric on $L$ with positive definite curvature form. In dimension zero $X_ P$ is a point and there is nothing to prove. $\square$