Pome

Corollary 9.11.22E77092231FA9

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. $\square$

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