Proof.
Put $\Delta =\Delta (\omega ,\varphi )$. By proposition 16.6.18, proposition 21.12.29, we may replace $\varphi $ by $P_{\omega }[\varphi ]$ without changing its multiplier ideal or Newton body. We may therefore assume that $F_{\varphi }-\operatorname{Supp}_{\Delta }$ is bounded on $N_{\mathbb {R}}$. Set
\[ H(\nu )=\langle m,\nu \rangle -\operatorname{Supp}_{\Delta }(\nu ),\quad \nu \in N_{\mathbb {R}}. \]
Since $\Delta $ is compact, $H$ is Lipschitz and positively homogeneous.
Observe also that $\mathrm{H}^0(X,L\otimes \mathcal{I}(\varphi ))$ is a $T_ c$-invariant subspace of $\mathrm{H}^0(X,L)$. Since the $T_ c$-weight spaces in $\mathrm{H}^0(X,L)$ are the lines $\mathbb {C}\chi ^ m$, it is the direct sum of suitable $\mathbb {C}\chi ^ m$'s.
If $m\notin P$, then $\chi ^ m$ is not a section of $L$. Suppose henceforth that $m\in M\cap P$. Membership in $\mathrm{H}^0(X,L\otimes \mathcal{I}(\varphi ))$ is equivalent to the finiteness of
39F3ED71C89D
21.12.33
\begin{equation} \int _{T(\mathbb {C})}|\chi ^ m|^2\exp (-\operatorname{Trop}^*F_0-\varphi )\, \omega ^ n. \end{equation}
Since $\operatorname{Trop}^*F_0+\varphi =\operatorname{Trop}^*F_{\varphi }$, the boundedness of $F_{\varphi }-\operatorname{Supp}_{\Delta }$ and proposition 21.12.24 show that this is equivalent to the finiteness of
\[ \int _{N_{\mathbb {R}}}\exp (H(\nu ))\, \operatorname{MA}_{\mathbb {R}}(F_0)(\nu ). \]
Changing variables by $\nu =\nabla G_0(m')$, we obtain the equivalent integral
428CD3ABF6EF
21.12.34
\begin{equation} \int _ P\exp \left(\langle m,\nabla G_0(m')\rangle -\operatorname{Supp}_{\Delta }(\nabla G_0(m'))\right)\, \mathrm{d}m'. \end{equation}
If $m\in \Delta $, its integrand is at most $1$, proving 1.
To prove 2, extend $\sigma $ to a maximal cone $\tau \in \Sigma $. Its primitive ray generators $u_1,\ldots ,u_ n$ form a basis of $N$, and we may arrange that $u_ i=u_{\rho _ i}$ for $1\leq i\leq p$. If $e_1,\ldots ,e_ n$ is the dual basis of $M$, then $z_ i=\chi ^{e_ i}$ are coordinates on the affine chart $U_{\tau }\cong \mathbb {C}^ n$.
Let $v\in P\cap M$ be the vertex corresponding to $\tau $. The character $\chi ^ v$ is a local frame of $L$ on $U_{\tau }$. Writing
\[ \nu =\operatorname{Trop}(z)=\sum _{i=1}^ n\log |z_ i|^2\, u_ i, \]
the weight of the singular metric in this frame is $F_{\varphi }(\nu )-\langle v,\nu \rangle $. The local representative of $\chi ^ m$ is $\chi ^{m-v}$, so its squared norm is
\[ \exp \left(\langle m,\nu \rangle -F_{\varphi }(\nu )\right). \]
On the unit polydisc, $\omega ^ n$ is comparable to Euclidean volume. Set $t_ i=-\log |z_ i|^2$. Integrating out the angular variables and using the boundedness of $F_{\varphi }-\operatorname{Supp}_{\Delta }$, the weighted integral over this polydisc is finite if and only if the following integral is finite:
\[ \int _{[0,\infty )^ n}\exp \left(H\left(-\sum _{i=1}^ n t_ i u_ i\right)-\sum _{i=1}^ n t_ i\right)\, \mathrm{d}t. \]
Restrict this integral to the tube parametrized by $s>1$ and $0<y_ i<1$ for $2\leq i\leq n$, with
\[ t_1=c_1s,\qquad t_ i=c_ is+y_ i\quad (2\leq i\leq p),\qquad t_ i=y_ i\quad (p<i\leq n). \]
The Jacobian is $c_1>0$. Since $H$ is Lipschitz and positively homogeneous and $\sum _{i=1}^ p c_ i=1$, the exponent on this tube is
\[ H\left(-\sum _{i=1}^ n t_ i u_ i\right)-\sum _{i=1}^ n t_ i =s\bigl (H(-u)-1\bigr )+\mathcal{O}(1), \]
with a bounded error uniform in $s$ and the $y_ i$'s. The local integral is therefore bounded from below by a positive multiple of
\[ \int _1^{\infty }\exp \left(s\bigl (H(-u)-1\bigr )\right)\, \mathrm{d}s, \]
which diverges under (21.12.32). This proves 2.
$\square$
Pome
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