The quotient calculation expresses integrability as a strict inequality on exponents. On an affine toric space the same inequality comes from the recession of a convex function, while the canonical bundle and boundary supply the translation of the exponent. This also applies to cones with more rays than their dimension.
In this subsection let $N$ be a lattice of rank $n\geq 1$, let $M=\operatorname {Hom}(N,\mathbb Z)$, and let $\sigma \subset N_{\mathbb R}$ be a full-dimensional strongly convex rational polyhedral cone. Its dual cone is $\sigma ^\vee =\{ m:\langle m,v\rangle \geq 0\text{ for all }v\in \sigma \} $. The normal affine toric space
\[ X_\sigma =\operatorname {Spec}\mathbb C[\sigma ^\vee \cap M] \]
is understood with its analytic structure. Its dense torus $T=\operatorname {Hom}(M,\mathbb C^*)$ has characters $\chi ^ m$, and its unique closed orbit is a vertex $o$. Choose nonzero generators $b_1,\ldots ,b_ s$ of the semigroup $\sigma ^\vee \cap M$; their characters embed $X_\sigma $ as a closed affine subset of $\mathbb C^ s$ and all vanish at $o$.
For each ray $\rho $ of $\sigma $, write $v_\rho $ for its primitive lattice generator and $D_\rho $ for the corresponding prime divisor. Fix an effective rational boundary $D=\sum _\rho a_\rho D_\rho $ such that $K_{X_\sigma }+D$ is $\mathbb Q$-Cartier. With $K_{X_\sigma }=-\sum _\rho D_\rho $, this means that there is $w_ D\in M_{\mathbb Q}$ with
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\begin{equation} \langle w_ D,v_\rho \rangle =1-a_\rho \quad \text{for every }\rho . \end{equation}
Take an integer $q>0$ with $qw_ D\in M$. For torus coordinates $t_1,\ldots ,t_ n$ determined by a basis of $M$, the meromorphic tensor
\[ B_ D=\chi ^{qw_ D} \left(\frac{\mathrm dt_1}{t_1}\wedge \cdots \wedge \frac{\mathrm dt_ n}{t_ n}\right)^{\! \otimes q} \]
generates $\omega _{X_\sigma }^{[q]}(qD)$. Indeed, its order along $D_\rho $ is $-qa_\rho $. The adapted volume $\mathrm dV_ D=|B_ D|^{2/q}$ is independent of this choice for local integrability, as in (9.11.11). We use the ideal $\mathcal I_ D^{\mathrm{ad}}(u)$ defined by the holomorphic-germ integral in (9.11.12) on this $X_\sigma $.
Let $u$ be a plurisubharmonic function invariant under the compact torus of coordinate rotations. The logarithmic map $\operatorname {Log}:T\to N_{\mathbb R}$ satisfies $\langle m,\operatorname {Log}t\rangle =\log |\chi ^ m(t)|$. There is a finite convex function $g$ on $N_{\mathbb R}$ with
\[ u(t)=2g(\operatorname {Log}t). \]
To see convexity, pull $u/2$ back by the exponential covering of $T$. It is independent of the imaginary coordinates, so its distributional complex Hessian is the real Hessian of $g$ up to a positive constant. A nontrivial plurisubharmonic function of this form gives a finite convex function. The factor two matches the weight $\mathrm e^{-u}$.
For a finite convex function $g$ write
\[ (\operatorname {rec}g)(v)= \lim _{t\to \infty }\frac{g(x+tv)-g(x)}t. \]
The convex difference quotients increase to this limit in $\mathbb R\cup \{ +\infty \} $; the limit does not depend on the base point $x$. The latter fact follows by comparing $g(x+tv)$ with convex combinations of $g(t v/(1-\delta ))$ and $g(x/\delta )$, for $0<\delta <1$, and then reversing the translation. Define the Newton set
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\begin{equation} P(u)=\left\{ \ell \in M_{\mathbb R}: \sup _{x\in -\sigma }\bigl (\langle \ell ,x\rangle -g(x)\bigr )<\infty \right\} . \end{equation}
Only its interior will enter the integrability test; the set itself need not be closed.
Lemma 9.11.19.5D5A847F99AC Let $C$ be a closed full-dimensional convex cone in a Euclidean space, let $g$ be finite and convex, and let $\ell $ be a linear form. Then
\[ \int _ C\exp \bigl (2\langle \ell ,x\rangle -2g(x)\bigr )\, \mathrm dx<\infty \quad \Longleftrightarrow \quad \langle \ell ,v\rangle <(\operatorname {rec}g)(v) \quad (v\in C\setminus \{ 0\} ). \]
This condition is also equivalent to $\ell $ being an interior point of $\{ a:\sup _ C(\langle a,x\rangle -g(x))<\infty \} $. The integral has the same finiteness on every translate of $C$.
Proof.
Set $F=g-\ell $. Suppose its recession is positive on $C\setminus \{ 0\} $. Then $F(x)\geq \gamma |x|-A$ on $C$ for some $\gamma >0$ and $A$. Otherwise there are $x_ j\in C$ with $|x_ j|\to \infty $ and $F(x_ j)/|x_ j|\leq 1/j$. After taking a subsequence, $x_ j/|x_ j|\to v\in C$ with $|v|=1$. For fixed $t>0$, convexity gives
\[ F\left(t\frac{x_ j}{|x_ j|}\right) \leq \left(1-\frac t{|x_ j|}\right)F(0) +\frac t{|x_ j|}F(x_ j). \]
Continuity gives $F(tv)\leq F(0)$ for every $t$, contradicting positive recession. A bound on the remaining compact set gives the asserted $A$. Exponential decay proves integrability. The same bound persists when $\ell $ is changed by a sufficiently small linear form, so $\ell $ lies in the stated interior.
Conversely, if $(\operatorname {rec}F)(v)\leq 0$ for some nonzero $v\in C$, then $F(tv)\leq F(0)$ for $t\geq 0$. Choose a compact ball $K$ in the interior of $C$. On the half-tube $\{ tv+x/2:t\geq 1,\ x\in K\} $, convexity gives
\[ F(tv+x/2)\leq \tfrac 12F(2tv)+\tfrac 12F(x). \]
Thus $\mathrm e^{-2F}$ has a positive lower bound on a set of infinite volume. If instead $\ell $ is in the stated interior, for each nonzero $v\in C$ one may perturb $\ell $ within that interior by a linear form positive on $v$. Dividing its defining upper bound along $tv$ by $t$ gives the strict recession inequality for $\ell $. Finally, translating $C$ replaces $g(x)$ by $g(x+b)$ and multiplies the exponential by a constant; recession is unchanged.
$\square$
Theorem 9.11.20.CD69C52538E4 For $m\in \sigma ^\vee \cap M$, the following are equivalent:
\[ \chi ^ m\in \mathcal I_ D^{\mathrm{ad}}(u)_ o, \qquad \chi ^ m\in \Gamma (X_\sigma ,\mathcal I_ D^{\mathrm{ad}}(u)), \qquad m+w_ D\in \operatorname {int}P(u). \]
Equivalently, $\langle m+w_ D,v\rangle <(\operatorname {rec}g)(v)$ for every nonzero $v\in -\sigma $.
Proof.
On the torus, polar coordinates turn the weighted monomial integral into a positive constant times
\[ \int _ Q\exp \bigl (2\langle m+w_ D,x\rangle -2g(x)\bigr )\, \mathrm dx, \]
where $Q$ is the logarithmic image of the domain. The omitted toric divisors have measure zero on the regular locus. A basis of neighborhoods of $o$ is given by $|\chi ^{b_ j}|<\epsilon $ for every $j$, with $0<\epsilon <1$. Their logarithmic domains satisfy
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\begin{equation} -\sigma -b\ \subset Q_\epsilon =\{ x:\langle b_ j,x\rangle <\log \epsilon \text{ for all }j\} \ \subset \ -\sigma , \end{equation}
for $b\in \operatorname {int}\sigma $ sufficiently far along an interior ray. Indeed, every $\langle b_ j,b\rangle $ is strictly positive, and the $b_ j$ generate the dual cone. The convex-cone lemma gives necessity and sufficiency at $o$, including divergence when a recession inequality is an equality.
For a compact subset of $X_\sigma $, the embedding coordinates have absolute value at most some $R>1$. Its logarithmic image is contained in $\{ x:\langle b_ j,x\rangle \leq \log R\} $, which is contained in $b_ R-\sigma $ for a sufficiently large $b_ R\in \operatorname {int}\sigma $. Under the same strict condition the integral over this translated cone is finite. This proves local integrability everywhere. The converse follows by testing at $o$, and the interior formulation follows from the lemma.
$\square$
Corollary 9.11.22.E77092231FA9 Let $\mathfrak a=(\chi ^{a_1},\ldots ,\chi ^{a_ r})$ be a nonzero monomial ideal on $X_\sigma $, with $a_ j\in \sigma ^\vee \cap M$, and let $c>0$ be real. For the measurable weight
\[ v=u-c\log \left(\sum _ j|\chi ^{a_ j}|^2\right), \]
a holomorphic character $\chi ^ m$ is locally integrable with weight $\mathrm e^{-v}\mathrm dV_ D$ at $o$, or everywhere on $X_\sigma $, exactly when
\[ m+w_ D+ca_ j\in \operatorname {int}P(u) \quad \text{for every }j. \]
Values on the common zero set do not affect these integrals.
Proof.
For a finite sum and $c>0$, its $c$-th power is comparable to the sum of the $c$-th powers of its terms. Thus the integrand is comparable to
\[ \sum _ j |\chi ^ m|^2|\chi ^{a_ j}|^{2c}\mathrm e^{-u}\, \mathrm dV_ D. \]
Each term gives the preceding logarithmic integral with exponent $m+w_ D+ca_ j$. The cone criterion applies to real exponents, so no rationality of $c$ is needed for this integrability assertion.
$\square$
Proposition 9.11.23.511E6797B7CD Let $X=\{ xw=yz\} \subset \mathbb C^4$, with vertex $o$, let $\mathfrak m_ o=(x,y,z,w)$, and take zero boundary. For $\lambda >0$ put
\[ u_\lambda =\lambda \log (|x|^2+|y|^2+|z|^2+|w|^2). \]
The adapted-volume ideal at the vertex is
\[ \mathcal I_0^{\mathrm{ad}}(u_\lambda )_ o =\mathfrak m_ o^{\, \max \{ 0,\lfloor \lambda \rfloor -1\} } \mathcal O_{X,o}. \]
In particular, the constant germ is integrable exactly when $\lambda <2$.
Proof.
On the torus use $x,y,z$ as characters, so $w=yz/x$. The character lattice is $\mathbb Z^3$, with semigroup generators
\[ (1,0,0),\quad (0,1,0),\quad (0,0,1),\quad (-1,1,1). \]
The dual cone has primitive rays $(0,1,0)$, $(0,0,1)$, $(1,1,0)$ and $(1,0,1)$. It is full-dimensional with four rays. The residue form $\Omega =\mathrm dx\wedge \mathrm dy\wedge \mathrm dz/x$ generates the canonical sheaf of this hypersurface; it equals $yz$ times the logarithmic torus form. Hence $w_0=(0,1,1)$, which pairs to one with each primitive ray.
Write $d(m)=m_1+m_2+m_3$. Each semigroup generator has degree one. The logarithmic function $g$ differs by a bounded term from $\lambda $ times the maximum of the four linear forms given by those generators. Their convex hull is the section of $\sigma ^\vee $ by $d=1$, so
\[ P(u_\lambda )=\{ \ell \in \sigma ^\vee :d(\ell )\geq \lambda \} . \]
Since $w_0\in \operatorname {int}\sigma ^\vee $ and $d(w_0)=2$, the character condition is $d(m)+2>\lambda $. The smallest allowed integer degree is $k=\max \{ 0,\lfloor \lambda \rfloor -1\} $.
This also gives the ideal for whole holomorphic germs. Lift a germ to a convergent power series in $x,y,z,w$. On a smaller invariant neighborhood, angular Parseval and Tonelli force every nonzero character coefficient to satisfy the same test. All homogeneous terms of degree less than $k$ therefore vanish in the coordinate ring. They are multiples of the homogeneous relation $xw-yz$, and subtracting those finitely many terms leaves a lift of order at least $k$. Thus the germ belongs to $\mathfrak m_ o^ k$. Conversely, every degree-$k$ monomial is integrable by the character test, and bounded holomorphic coefficient germs preserve integrability. This proves the identity and its strict endpoint.
$\square$
These ideals use the adapted canonical volume, whereas the resolution-ideal equivalence above uses smooth volumes on the models. Return now to the compact unibranch space and background fixed at the beginning of the appendix.
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