Proposition 21.9.27B4A7CC50CDC

Proof. Fix a vertex $v$ and use the chart $U_ v\cong \mathbb C^ n$ of theorem 21.8.3. For $\tau \in T$ we have $w_ i(\tau \cdot z)=\chi ^{e_ i}(\tau )w_ i(z)$, and $\tau \mapsto (\chi ^{e_ i}(\tau ))_ i$ maps $T$ onto $(\mathbb C^*)^ n$; so $T$ acts on $U_ v$ by independent scalings of the coordinates. The orbits in $U_ v$ are therefore the sets

and $O_{v,I}$ has dimension $n-|I|$. Let $Q_{v,I}=\{ m\in P:\ell _ i(m)=0\text{ for }i\in I\} $. It is a face, namely the set where $\sum _{i\in I}\langle \cdot ,u_ i\rangle $ is minimal on $P$. It contains $v$ and the points $v+se_ j$ for $j\notin I$ and small $s>0$, and it lies in $v+\operatorname {span}\{ e_ j:j\notin I\} $, because $m-v=\sum _ i\ell _ i(m)e_ i$; so $\dim Q_{v,I}=n-|I|$. Since $Z_ m/Z_ v=\prod _ iw_ i^{\ell _ i(m)}$, a homogeneous coordinate $Z_ m$, $m\in A$, is nonzero on $O_{v,I}$ if and only if $\ell _ i(m)=0$ for all $i\in I$, that is, if and only if $m\in Q_{v,I}$.

The charts $U_ v$ are $T$-invariant, because the action of $T$ multiplies $Z_ v$ by $\chi ^ v(\tau )\neq 0$. Hence every orbit $O$ in $X_ P$ meets, and therefore lies in, some chart $U_ v$, so it is one of the sets $O_{v,I}$. The set of $m\in A$ with $Z_ m\neq 0$ on $O$ does not depend on the chart, and it equals $Q_{v,I}\cap A$; since a face is the convex hull of its vertices, which are lattice points, $Q_{v,I}$ is the convex hull of this set. Thus each orbit $O$ determines a face $Q(O)$, with $\dim Q(O)=\dim O$. Distinct orbits determine distinct faces: if $Q(O)=Q(O')=Q$ and $v$ is a vertex of $Q$, then $Z_ v\neq 0$ on both orbits, so both lie in $U_ v$, where an orbit is determined by the set of indices $i$ with $v+e_ i\notin Q$. Every face $Q$ arises: choose a vertex $v\in Q$ and let $I$ be the set of indices of the facets at $v$ that contain $Q$; then $Q_{v,I}=Q$, because $Q$ is the intersection of $P$ with the facets containing it (proposition 21.9.1), and these facets all pass through $v$. We write $O_ Q$ for the orbit of the face $Q$; in particular $O_ P=\iota (T)$.

The closure $\overline{O_ Q}$ is $T$-invariant, hence a union of orbits, and it lies in the closed set $\{ Z_ m=0\text{ for }m\in A\setminus Q\} $. If $O_{Q'}\subset \overline{O_ Q}$, then $Q'\cap A\subset Q$, and so $Q'\subset Q$. Conversely, let $Q'\subset Q$ be faces, let $v$ be a vertex of $Q'$, and let $I\subset I'$ be the index sets of $Q$ and $Q'$ at $v$. A point $w\in O_{Q'}=O_{v,I'}$ is the limit as $\varepsilon \downarrow 0$ of the points obtained from $w$ by replacing the coordinates $w_ i=0$, $i\in I'\setminus I$, by $\varepsilon $, and these points lie in $O_{v,I}=O_ Q$. Hence $O_{Q'}\subset \overline{O_ Q}$.

Let $\rho $ be a facet and $D_\rho =\overline{O_\rho }$. If $v\in \rho $ and $\rho $ is the $i$th facet at $v$, the orbits in $U_ v$ indexed by faces contained in $\rho $ are the sets $O_{v,I}$ with $i\in I$, so $D_\rho \cap U_ v=\{ w_ i=0\} $; if $v\notin \rho $, then $D_\rho \cap U_ v=\varnothing $, because $Z_ v=0$ on $D_\rho $. Hence $D_\rho $ is a $T$-invariant closed submanifold of codimension one. Every proper face is contained in a facet, being the intersection of the facets containing it; hence $X_ P\setminus \iota (T)$, the union of the orbits of the proper faces, is the union of the divisors $D_\rho $.

Each $D_\rho $ is irreducible, that is, not the union of two analytic subsets different from $D_\rho $. First, $D_\rho $ is connected, being the closure of the orbit $O_\rho $: in a chart $U_ v$ in which $\rho $ is the $i$th facet at $v$, the coordinates $w_ j$, $j\neq i$, identify $O_\rho =O_{v,\{ i\} }$ with the connected set $(\mathbb C^*)^{n-1}$. Next, let $Z\subsetneq D_\rho $ be an analytic subset. The set of interior points of $Z$ in $D_\rho $ is open, and it is closed in $D_\rho $. Indeed, if $p$ lies in its closure, then $p\in Z$, and on a connected coordinate neighborhood of $p$ in $D_\rho $ the local holomorphic functions defining $Z$ vanish on a nonempty open set, hence identically by the identity theorem, so that $p$ is an interior point. As $D_\rho $ is connected and $Z\neq D_\rho $, the set $Z$ has empty interior in $D_\rho $. If $D_\rho =Z_1\cup Z_2$ with analytic subsets $Z_1,Z_2\neq D_\rho $, then the nonempty open set $D_\rho \setminus Z_1$ would be contained in $Z_2$, which has empty interior.

Let $m\in M$. Since $m=\sum _ j\langle m,u_ j\rangle e_ j$, the character $\chi ^ m$ is the Laurent monomial $\prod _ jw_ j^{\langle m,u_ j\rangle }$ in the coordinates of $U_ v$. Hence it extends to a meromorphic function on each $U_ v$, and these extensions agree on the dense set $\iota (T)$, hence on overlaps. Near each point of the dense orbit $O_\rho \cap U_ v$ of $D_\rho \cap U_ v=\{ w_ i=0\} $, it is $w_ i^{\langle m,u_ i\rangle }$ times a nowhere vanishing holomorphic function; so its order along $D_\rho $, the difference of the orders of numerator and denominator in the sense of definition 6.12.3, is $\langle m,u_ i\rangle =\langle m,u_\rho \rangle $. In particular, $\chi ^ m$ is holomorphic on $U_ v\cong \mathbb C^ n$ if and only if all the exponents $\langle m,u_ j\rangle $ are nonnegative, that is, if and only if $\langle m,x\rangle \geq 0$ for every $x$ in $\sigma _{\{ v\} }=\operatorname {cone}(u_1,\ldots ,u_ n)$. $\square$