Proposition 12.15.1`37E6FFC01EDB`

Proposition 12.15.1.37E6FFC01EDB For smooth positive probability measures $\mu ,\nu $, choose constants $m,M>0$ such that $m\mu \leq \nu \leq M\mu $. Then for every pseudoeffective $\theta $,

\[ M^{-1/n}\lambda _\mu (\theta ) \leq \lambda _\nu (\theta ) \leq m^{-1/n}\lambda _\mu (\theta ). \]

In particular, whether $\lambda _\mu (\theta )$ vanishes does not depend on the smooth positive probability measure $\mu $.

Proof. Since $D_\nu (\gamma )=\int _ X(\nu /\mu )^{1/n}(\mu /\gamma ^ n)^{1/n}\gamma ^ n$, the inequalities $m\mu \leq \nu \leq M\mu $ give

\[ m^{1/n}D_\mu (\gamma )\leq D_\nu (\gamma )\leq M^{1/n}D_\mu (\gamma ) \]

for every admissible metric $\gamma $. The numerators of the quotients are nonnegative and do not involve the measure. Dividing and taking infima proves the assertion. $\square$

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@misc{pome-tag-37E6FFC01EDB,
  author = {{Pome contributors}},
  title = {Proposition 12.15.1},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/37E6FFC01EDB/}
}

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Proposition 12.15.137E6FFC01EDB

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Proposition 12.15.1`37E6FFC01EDB`