Theorem 21.9.569D3D97E7701

Proof. For vertices $v$ and $v'$, the coordinate sections satisfy $s_ v/s_{v'}=Z_ v/Z_{v'}=\chi ^{v-v'}$ on $\iota (T)$. Hence the section $s_0=\chi ^{-v}s_ v$ of $L$ over $\iota (T)$ does not depend on the vertex $v$. In the chart $U_ v$, its coefficient $\chi ^{-v}$ with respect to the frame $s_ v$ is a Laurent monomial in $w$, so $s_0$ extends to a meromorphic section on each $U_ v$; the extensions agree on overlaps, because they agree on the dense set $\iota (T)$. By proposition 21.9.2, if $v$ is a vertex of the facet $\rho $, then $\operatorname {ord}_{D_\rho }s_0=\operatorname {ord}_{D_\rho }\chi ^{-v}=-\langle v,u_\rho \rangle =a_\rho $, since $\ell _\rho (v)=0$. As $s_0$ has no zeros or poles on $\iota (T)$, its divisor is $D_ P$. Similarly, the meromorphic function $f_ v=\chi ^{-v}$ on $U_ v$ has divisor $\sum _{\rho \ni v}a_\rho (D_\rho \cap U_ v)$, which is the restriction of $D_ P$ to $U_ v$, because $D_\rho \cap U_ v=\varnothing $ for $v\notin \rho $. The line bundle $\mathcal O_{X_ P}(D_ P)$ of the divisor therefore has over $U_ v$ the local frame $1/f_ v$, and its transition functions, in the sense of definition 6.14.9, are $f_{v'}/f_ v=\chi ^{v-v'}$. These are the transition functions of $L$ in the frames $s_ v$, since $s_ v=\chi ^{v-v'}s_{v'}$. Hence the map sending $s_ v$ to $1/f_ v$ for every vertex $v$ is an isomorphism $L\cong \mathcal O_{X_ P}(D_ P)$.

Let $s$ be a holomorphic section of $L^ k$ and write $s=fs_0^ k$ on $\iota (T)$, with $f$ holomorphic on $T\cong (\mathbb C^*)^ n$. By lemma 21.9.4, applied in the coordinates of a basis of $M$, we have $f=\sum _{m\in M}c_ m\chi ^ m$, with normal convergence on compact subsets of $T$. The coefficients $c_ m$ do not depend on the basis: a change of basis replaces each character by the corresponding monomial in the new coordinates and preserves normal convergence on compact sets, so the uniqueness assertion of the lemma applies. Now fix a vertex $v$. The coefficient $g_ v$ of $s$ with respect to the frame $s_ v^ k$ is a holomorphic function on $U_ v\cong \mathbb C^ n$, and on $\iota (T)$

Since $u_1,\ldots ,u_ n$ is the basis of $N$ dual to the basis $e_1,\ldots ,e_ n$ of $M$, the map $m\mapsto (\langle m-kv,u_ i\rangle )_ i$ is a bijection of $M$ onto $\mathbb Z^ n$, so this is the Laurent expansion of $g_ v$ in the coordinates $w$. By the last assertion of lemma 21.9.4, $c_ m=0$ unless $\langle m-kv,u_ i\rangle \geq 0$ for every facet $i$ at $v$. Since $\langle v,u_\rho \rangle =-a_\rho $ for $v\in \rho $, this condition reads $\langle m,u_\rho \rangle +ka_\rho \geq 0$. Every facet contains a vertex, so for $k>0$ the conditions at all vertices say exactly that $m/k\in P$, that is, $m\in kP$. For $k=0$ they say that $\langle m,u_\rho \rangle \geq 0$ for every $\rho $; then $p+sm\in P$ for $p\in P$ and all $s\geq 0$, so $m=0$ by boundedness. Thus only the finitely many $m\in kP\cap M$ occur. Hence $s=\sum _{m\in kP\cap M}c_ m\chi ^ ms_0^ k$ on the dense set $\iota (T)$, and therefore on $X_ P$, once we know that the sections on the right are holomorphic. Indeed, for $m\in kP\cap M$ all exponents $\langle m-kv,u_ i\rangle $ are nonnegative, so $\chi ^ ms_0^ k$ extends holomorphically to every chart. Finally, the uniqueness of Laurent coefficients shows that the sections $\chi ^ ms_0^ k$ are linearly independent.

The action of $T$ on $\mathbb C^ A$ by $\tau \cdot (Z_ m)=(\chi ^ m(\tau )Z_ m)$ induces actions on the tautological line bundle (example 10.6.6), whose fiber over $[Z]$ is the line $\mathbb CZ$, on its dual $\mathcal O(1)$, on $L^ k$, and on $H^0(X_ P,L^ k)$ by $(\tau \cdot s)(x)=\tau \cdot s(\tau ^{-1}\cdot x)$. It multiplies the linear form $Z_ v$, and hence $s_ v$, by $\chi ^{-v}(\tau )$, and it maps the function $\chi ^ v$ to $\chi ^ v(\tau ^{-1}\, \cdot \, )=\chi ^{-v}(\tau )\chi ^ v$; therefore it fixes $s_0$ and multiplies $\chi ^ ms_0^ k$ by $\chi ^{-m}(\tau )$. So each summand is a line on which $T$ acts through a character. Finally, $h^0(X_ P,L^ k)=\# (kP\cap M)$, and the limit follows from theorem 21.7.3. In dimension zero there is one section, and the volume is one. $\square$