Corollary 19.4.55`C3CA9BD5D216`

Corollary 19.4.55.C3CA9BD5D216 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. If

\[ \Delta _{W_{\bullet }}(\pi ^*\theta ,\pi ^*\varphi )\subseteq \Delta _{W_{\bullet }}(\pi ^*\theta ,\pi ^*\psi ) \]

for all modifications $\pi \colon Y\rightarrow X$ and all admissible flags $W_{\bullet }$ on $Y$, then $\varphi \preceq _{\mathcal{I}} \psi $.

Proof. By theorem 19.4.51, the assumed inclusions imply that for every prime divisor $E$ on every modification $\pi $,

\[ \nu (\pi ^*\varphi ,E)\geq \nu (\pi ^*\psi ,E). \]

Hence $\varphi \preceq _{\mathcal{I}}\psi $ by proposition 16.8.9. $\square$

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BibTeX citation

@misc{pome-tag-C3CA9BD5D216,
  author = {{Pome contributors}},
  title = {Corollary 19.4.55},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/C3CA9BD5D216/}
}

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Corollary 19.4.55C3CA9BD5D216

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Corollary 19.4.55`C3CA9BD5D216`